Gecko [ fernandoquevedodop.com ]

Name	Size	Permission
2to3.html	70.96 KB	-rw-r--r--
__future__.html	20.75 KB	-rw-r--r--
__main__.html	44.87 KB	-rw-r--r--
_thread.html	30.39 KB	-rw-r--r--
abc.html	46.87 KB	-rw-r--r--
aifc.html	35.56 KB	-rw-r--r--
allos.html	34.33 KB	-rw-r--r--
archiving.html	17 KB	-rw-r--r--
argparse.html	300.89 KB	-rw-r--r--
array.html	37.28 KB	-rw-r--r--
ast.html	293.08 KB	-rw-r--r--
asynchat.html	37.41 KB	-rw-r--r--
asyncio-api-index.html	27.68 KB	-rw-r--r--
asyncio-dev.html	36.1 KB	-rw-r--r--
asyncio-eventloop.html	204.43 KB	-rw-r--r--
asyncio-exceptions.html	17.11 KB	-rw-r--r--
asyncio-future.html	38.23 KB	-rw-r--r--
asyncio-llapi-index.html	61.97 KB	-rw-r--r--
asyncio-platforms.html	21.77 KB	-rw-r--r--
asyncio-policy.html	39.56 KB	-rw-r--r--
asyncio-protocol.html	118.33 KB	-rw-r--r--
asyncio-queue.html	31.86 KB	-rw-r--r--
asyncio-stream.html	69.98 KB	-rw-r--r--
asyncio-subprocess.html	49.25 KB	-rw-r--r--
asyncio-sync.html	42.79 KB	-rw-r--r--
asyncio-task.html	109.14 KB	-rw-r--r--
asyncio.html	17.31 KB	-rw-r--r--
asyncore.html	49.36 KB	-rw-r--r--
atexit.html	22.87 KB	-rw-r--r--
audioop.html	46 KB	-rw-r--r--
audit_events.html	72.85 KB	-rw-r--r--
base64.html	48.09 KB	-rw-r--r--
bdb.html	66 KB	-rw-r--r--
binary.html	15.97 KB	-rw-r--r--
binascii.html	36.3 KB	-rw-r--r--
binhex.html	15.97 KB	-rw-r--r--
bisect.html	43.38 KB	-rw-r--r--
builtins.html	14.87 KB	-rw-r--r--
bz2.html	50.68 KB	-rw-r--r--
calendar.html	69.35 KB	-rw-r--r--
cgi.html	59.95 KB	-rw-r--r--
cgitb.html	19.96 KB	-rw-r--r--
chunk.html	22.21 KB	-rw-r--r--
cmath.html	43.89 KB	-rw-r--r--
cmd.html	50.42 KB	-rw-r--r--
code.html	35.38 KB	-rw-r--r--
codecs.html	157.96 KB	-rw-r--r--
codeop.html	18.26 KB	-rw-r--r--
collections.abc.html	79.17 KB	-rw-r--r--
collections.html	189.43 KB	-rw-r--r--
colorsys.html	17.31 KB	-rw-r--r--
compileall.html	50.68 KB	-rw-r--r--
concurrency.html	24.15 KB	-rw-r--r--
concurrent.futures.html	79.03 KB	-rw-r--r--
concurrent.html	11.56 KB	-rw-r--r--
configparser.html	159.4 KB	-rw-r--r--
constants.html	22.41 KB	-rw-r--r--
contextlib.html	117.2 KB	-rw-r--r--
contextvars.html	38.38 KB	-rw-r--r--
copy.html	17.92 KB	-rw-r--r--
copyreg.html	18.63 KB	-rw-r--r--
crypt.html	28.29 KB	-rw-r--r--
crypto.html	13.93 KB	-rw-r--r--
csv.html	82.89 KB	-rw-r--r--
ctypes.html	295.49 KB	-rw-r--r--
curses.ascii.html	32.04 KB	-rw-r--r--
curses.html	221.24 KB	-rw-r--r--
curses.panel.html	22.16 KB	-rw-r--r--
custominterp.html	11.85 KB	-rw-r--r--
dataclasses.html	113.16 KB	-rw-r--r--
datatypes.html	31.82 KB	-rw-r--r--
datetime.html	374.92 KB	-rw-r--r--
dbm.html	51.16 KB	-rw-r--r--
debug.html	17 KB	-rw-r--r--
decimal.html	270.12 KB	-rw-r--r--
development.html	32.34 KB	-rw-r--r--
devmode.html	31.72 KB	-rw-r--r--
dialog.html	43.03 KB	-rw-r--r--
difflib.html	111.99 KB	-rw-r--r--
dis.html	136.07 KB	-rw-r--r--
distribution.html	13.67 KB	-rw-r--r--
distutils.html	14.73 KB	-rw-r--r--
doctest.html	199.32 KB	-rw-r--r--
email.charset.html	31.92 KB	-rw-r--r--
email.compat32-message.html	97.74 KB	-rw-r--r--
email.contentmanager.html	37.02 KB	-rw-r--r--
email.encoders.html	18.47 KB	-rw-r--r--
email.errors.html	24.45 KB	-rw-r--r--
email.examples.html	60.49 KB	-rw-r--r--
email.generator.html	48.79 KB	-rw-r--r--
email.header.html	37.61 KB	-rw-r--r--
email.headerregistry.html	66.26 KB	-rw-r--r--
email.html	30.75 KB	-rw-r--r--
email.iterators.html	18.23 KB	-rw-r--r--
email.message.html	99.75 KB	-rw-r--r--
email.mime.html	44.89 KB	-rw-r--r--
email.parser.html	57.39 KB	-rw-r--r--
email.policy.html	84.24 KB	-rw-r--r--
email.utils.html	42.57 KB	-rw-r--r--
ensurepip.html	23.9 KB	-rw-r--r--
enum.html	151 KB	-rw-r--r--
errno.html	60.17 KB	-rw-r--r--
exceptions.html	108.44 KB	-rw-r--r--
faulthandler.html	31.5 KB	-rw-r--r--
fcntl.html	36.67 KB	-rw-r--r--
filecmp.html	32.08 KB	-rw-r--r--
fileformats.html	13.52 KB	-rw-r--r--
fileinput.html	40.37 KB	-rw-r--r--
filesys.html	16.78 KB	-rw-r--r--
fnmatch.html	21.02 KB	-rw-r--r--
fractions.html	35.29 KB	-rw-r--r--
frameworks.html	16.15 KB	-rw-r--r--
ftplib.html	71.97 KB	-rw-r--r--
functional.html	12.19 KB	-rw-r--r--
functions.html	260.33 KB	-rw-r--r--
functools.html	92.96 KB	-rw-r--r--
gc.html	40.97 KB	-rw-r--r--
getopt.html	29.4 KB	-rw-r--r--
getpass.html	16.01 KB	-rw-r--r--
gettext.html	98.02 KB	-rw-r--r--
glob.html	27.2 KB	-rw-r--r--
graphlib.html	34.82 KB	-rw-r--r--
grp.html	16.03 KB	-rw-r--r--
gzip.html	45.65 KB	-rw-r--r--
hashlib.html	93.19 KB	-rw-r--r--
heapq.html	42.43 KB	-rw-r--r--
hmac.html	24.74 KB	-rw-r--r--
html.entities.html	14.91 KB	-rw-r--r--
html.html	14.81 KB	-rw-r--r--
html.parser.html	47.7 KB	-rw-r--r--
http.client.html	85.63 KB	-rw-r--r--
http.cookiejar.html	113.94 KB	-rw-r--r--
http.cookies.html	47.64 KB	-rw-r--r--
http.html	45.52 KB	-rw-r--r--
http.server.html	73.05 KB	-rw-r--r--
i18n.html	13.44 KB	-rw-r--r--
idle.html	77.18 KB	-rw-r--r--
imaplib.html	87.08 KB	-rw-r--r--
imghdr.html	17.94 KB	-rw-r--r--
imp.html	54.32 KB	-rw-r--r--
importlib.html	239.96 KB	-rw-r--r--
importlib.metadata.html	43.14 KB	-rw-r--r--
index.html	75.06 KB	-rw-r--r--
inspect.html	168.74 KB	-rw-r--r--
internet.html	29.89 KB	-rw-r--r--
intro.html	13.81 KB	-rw-r--r--
io.html	154.15 KB	-rw-r--r--
ipaddress.html	137.75 KB	-rw-r--r--
ipc.html	12.6 KB	-rw-r--r--
itertools.html	153.27 KB	-rw-r--r--
json.html	98.9 KB	-rw-r--r--
keyword.html	14.88 KB	-rw-r--r--
language.html	16.71 KB	-rw-r--r--
linecache.html	18.09 KB	-rw-r--r--
locale.html	77.35 KB	-rw-r--r--
logging.config.html	100.3 KB	-rw-r--r--
logging.handlers.html	146.28 KB	-rw-r--r--
logging.html	169.67 KB	-rw-r--r--
lzma.html	66.89 KB	-rw-r--r--
mailbox.html	176.34 KB	-rw-r--r--
mailcap.html	20.86 KB	-rw-r--r--
markup.html	22.31 KB	-rw-r--r--
marshal.html	24.43 KB	-rw-r--r--
math.html	85.82 KB	-rw-r--r--
mimetypes.html	40.27 KB	-rw-r--r--
mm.html	11.16 KB	-rw-r--r--
mmap.html	55.53 KB	-rw-r--r--
modulefinder.html	23.67 KB	-rw-r--r--
modules.html	15.88 KB	-rw-r--r--
msilib.html	78.38 KB	-rw-r--r--
msvcrt.html	29.22 KB	-rw-r--r--
multiprocessing.html	405.24 KB	-rw-r--r--
multiprocessing.shared_memory....	60 KB	-rw-r--r--
netdata.html	20.24 KB	-rw-r--r--
netrc.html	20.39 KB	-rw-r--r--
nis.html	16.76 KB	-rw-r--r--
nntplib.html	84.93 KB	-rw-r--r--
numbers.html	45.67 KB	-rw-r--r--
numeric.html	19.29 KB	-rw-r--r--
operator.html	107.32 KB	-rw-r--r--
optparse.html	256.96 KB	-rw-r--r--
os.html	594.39 KB	-rw-r--r--
os.path.html	65.54 KB	-rw-r--r--
ossaudiodev.html	56.35 KB	-rw-r--r--
pathlib.html	170.01 KB	-rw-r--r--
pdb.html	68.67 KB	-rw-r--r--
persistence.html	19.13 KB	-rw-r--r--
pickle.html	154.45 KB	-rw-r--r--
pickletools.html	22.92 KB	-rw-r--r--
pipes.html	21.7 KB	-rw-r--r--
pkgutil.html	42.76 KB	-rw-r--r--
platform.html	45.08 KB	-rw-r--r--
plistlib.html	30.06 KB	-rw-r--r--
poplib.html	41.42 KB	-rw-r--r--
posix.html	20.96 KB	-rw-r--r--
pprint.html	55.35 KB	-rw-r--r--
profile.html	84.63 KB	-rw-r--r--
pty.html	23.7 KB	-rw-r--r--
pwd.html	16.84 KB	-rw-r--r--
py_compile.html	29 KB	-rw-r--r--
pyclbr.html	24.06 KB	-rw-r--r--
pydoc.html	19.59 KB	-rw-r--r--
pyexpat.html	104.38 KB	-rw-r--r--
python.html	22.46 KB	-rw-r--r--
queue.html	43.39 KB	-rw-r--r--
quopri.html	18.64 KB	-rw-r--r--
random.html	86.31 KB	-rw-r--r--
re.html	214.24 KB	-rw-r--r--
readline.html	50.19 KB	-rw-r--r--
reprlib.html	30.85 KB	-rw-r--r--
resource.html	48.26 KB	-rw-r--r--
rlcompleter.html	18.06 KB	-rw-r--r--
runpy.html	30.99 KB	-rw-r--r--
sched.html	27.04 KB	-rw-r--r--
secrets.html	30.2 KB	-rw-r--r--
security_warnings.html	16.21 KB	-rw-r--r--
select.html	74.58 KB	-rw-r--r--
selectors.html	41.91 KB	-rw-r--r--
shelve.html	36.54 KB	-rw-r--r--
shlex.html	61.03 KB	-rw-r--r--
shutil.html	100.13 KB	-rw-r--r--
signal.html	95.39 KB	-rw-r--r--
site.html	38.16 KB	-rw-r--r--
smtpd.html	44.09 KB	-rw-r--r--
smtplib.html	81.4 KB	-rw-r--r--
sndhdr.html	16.4 KB	-rw-r--r--
socket.html	256.95 KB	-rw-r--r--
socketserver.html	91.95 KB	-rw-r--r--
spwd.html	16.71 KB	-rw-r--r--
sqlite3.html	225.51 KB	-rw-r--r--
ssl.html	328.83 KB	-rw-r--r--
stat.html	57.79 KB	-rw-r--r--
statistics.html	123.22 KB	-rw-r--r--
stdtypes.html	615.43 KB	-rw-r--r--
string.html	108.03 KB	-rw-r--r--
stringprep.html	24.94 KB	-rw-r--r--
struct.html	70.58 KB	-rw-r--r--
subprocess.html	199.03 KB	-rw-r--r--
sunau.html	39.84 KB	-rw-r--r--
superseded.html	26.21 KB	-rw-r--r--
symtable.html	33.26 KB	-rw-r--r--
sys.html	198.83 KB	-rw-r--r--
sysconfig.html	39.81 KB	-rw-r--r--
syslog.html	27.36 KB	-rw-r--r--
tabnanny.html	16.18 KB	-rw-r--r--
tarfile.html	161.92 KB	-rw-r--r--
telnetlib.html	38.21 KB	-rw-r--r--
tempfile.html	58.39 KB	-rw-r--r--
termios.html	22.86 KB	-rw-r--r--
test.html	222.85 KB	-rw-r--r--
text.html	16.8 KB	-rw-r--r--
textwrap.html	48.83 KB	-rw-r--r--
threading.html	128.33 KB	-rw-r--r--
time.html	107.46 KB	-rw-r--r--
timeit.html	52.63 KB	-rw-r--r--
tk.html	28.08 KB	-rw-r--r--
tkinter.colorchooser.html	14.49 KB	-rw-r--r--
tkinter.dnd.html	17.09 KB	-rw-r--r--
tkinter.font.html	22.21 KB	-rw-r--r--
tkinter.html	106.85 KB	-rw-r--r--
tkinter.messagebox.html	20.09 KB	-rw-r--r--
tkinter.scrolledtext.html	14.52 KB	-rw-r--r--
tkinter.tix.html	59.69 KB	-rw-r--r--
tkinter.ttk.html	138.37 KB	-rw-r--r--
token.html	45.73 KB	-rw-r--r--
tokenize.html	39.86 KB	-rw-r--r--
trace.html	38.85 KB	-rw-r--r--
traceback.html	75.61 KB	-rw-r--r--
tracemalloc.html	122.33 KB	-rw-r--r--
tty.html	14.46 KB	-rw-r--r--
turtle.html	271.71 KB	-rw-r--r--
types.html	64.85 KB	-rw-r--r--
typing.html	315.69 KB	-rw-r--r--
unicodedata.html	27.3 KB	-rw-r--r--
unittest.html	310.01 KB	-rw-r--r--
unittest.mock-examples.html	177.75 KB	-rw-r--r--
unittest.mock.html	363.51 KB	-rw-r--r--
unix.html	13.46 KB	-rw-r--r--
urllib.error.html	18.56 KB	-rw-r--r--
urllib.html	12.92 KB	-rw-r--r--
urllib.parse.html	104.65 KB	-rw-r--r--
urllib.request.html	208.87 KB	-rw-r--r--
urllib.robotparser.html	22.63 KB	-rw-r--r--
uu.html	18.2 KB	-rw-r--r--
uuid.html	42.88 KB	-rw-r--r--
venv.html	90.43 KB	-rw-r--r--
warnings.html	69.13 KB	-rw-r--r--
wave.html	36.12 KB	-rw-r--r--
weakref.html	73.36 KB	-rw-r--r--
webbrowser.html	33.98 KB	-rw-r--r--
windows.html	12.32 KB	-rw-r--r--
winreg.html	89.48 KB	-rw-r--r--
winsound.html	26.3 KB	-rw-r--r--
wsgiref.html	111.52 KB	-rw-r--r--
xdrlib.html	42.07 KB	-rw-r--r--
xml.dom.html	114.66 KB	-rw-r--r--
xml.dom.minidom.html	49.6 KB	-rw-r--r--
xml.dom.pulldom.html	30.06 KB	-rw-r--r--
xml.etree.elementtree.html	186.49 KB	-rw-r--r--
xml.html	22.34 KB	-rw-r--r--
xml.sax.handler.html	55.78 KB	-rw-r--r--
xml.sax.html	31.74 KB	-rw-r--r--
xml.sax.reader.html	56.23 KB	-rw-r--r--
xml.sax.utils.html	22.28 KB	-rw-r--r--
xmlrpc.client.html	79.73 KB	-rw-r--r--
xmlrpc.html	11.9 KB	-rw-r--r--
xmlrpc.server.html	63.56 KB	-rw-r--r--
zipapp.html	52.45 KB	-rw-r--r--
zipfile.html	116.59 KB	-rw-r--r--
zipimport.html	33.76 KB	-rw-r--r--
zlib.html	42.29 KB	-rw-r--r--
zoneinfo.html	57.25 KB	-rw-r--r--

Code Editor : statistics.html

<!DOCTYPE html>

<title>statistics — Mathematical statistics functions &#8212; Python 3.10.12 documentation</title><meta name="viewport" content="width=device-width, initial-scale=1.0">
 
 <link rel="stylesheet" type="text/css" href="../_static/pygments.css" />
 <link rel="stylesheet" type="text/css" href="../_static/pydoctheme.css?2022.1" />
 
 <script data-url_root="../" id="documentation_options" src="../_static/documentation_options.js"></script>
 <script src="../_static/jquery.js"></script>
 <script src="../_static/underscore.js"></script>
 <script src="../_static/doctools.js"></script>
 
 <script src="../_static/sidebar.js"></script>
 
 <link rel="search" type="application/opensearchdescription+xml"
 title="Search within Python 3.10.12 documentation"
 href="../_static/opensearch.xml"/>
 <link rel="author" title="About these documents" href="../about.html" />
 <link rel="index" title="Index" href="../genindex.html" />
 <link rel="search" title="Search" href="../search.html" />
 <link rel="copyright" title="Copyright" href="../copyright.html" />
 <link rel="next" title="Functional Programming Modules" href="functional.html" />
 <link rel="prev" title="random — Generate pseudo-random numbers" href="random.html" />
 <link rel="canonical" href="file:///usr/share/doc/python3.10/html/library/statistics.html" />

</head>
<body>
<div class="mobile-nav">
 <input type="checkbox" id="menuToggler" class="toggler__input" aria-controls="navigation"
 aria-pressed="false" aria-expanded="false" role="button" aria-label="Menu" />
 <label for="menuToggler" class="toggler__label">
 
 </label>
 <nav class="nav-content" role="navigation">
 <a href="https://www.python.org/" class="nav-logo">
 <img src="../_static/py.svg" alt="Logo"/>
 </a>
 <div class="version_switcher_placeholder"></div>
 <form role="search" class="search" action="../search.html" method="get">
 <svg xmlns="http://www.w3.org/2000/svg" width="20" height="20" viewBox="0 0 24 24" class="search-icon">
 <path fill-rule="nonzero"
 d="M15.5 14h-.79l-.28-.27a6.5 6.5 0 001.48-5.34c-.47-2.78-2.79-5-5.59-5.34a6.505 6.505 0 00-7.27 7.27c.34 2.8 2.56 5.12 5.34 5.59a6.5 6.5 0 005.34-1.48l.27.28v.79l4.25 4.25c.41.41 1.08.41 1.49 0 .41-.41.41-1.08 0-1.49L15.5 14zm-6 0C7.01 14 5 11.99 5 9.5S7.01 5 9.5 5 14 7.01 14 9.5 11.99 14 9.5 14z" fill="#444"></path>
 </svg>
 <input type="text" name="q" aria-label="Quick search"/>
 <input type="submit" value="Go"/>
 </form>
 </nav>
 <div class="menu-wrapper">
 <nav class="menu" role="navigation" aria-label="main navigation">
 <div class="language_switcher_placeholder"></div>
 <h3><a href="../contents.html">Table of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate">statistics</code> — Mathematical statistics functions</a><ul>
<li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>
<li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>
<li><a class="reference internal" href="#statistics-for-relations-between-two-inputs">Statistics for relations between two inputs</a></li>
<li><a class="reference internal" href="#function-details">Function details</a></li>
<li><a class="reference internal" href="#exceptions">Exceptions</a></li>
<li><a class="reference internal" href="#normaldist-objects"><code class="xref py py-class docutils literal notranslate">NormalDist</code> objects</a><ul>
<li><a class="reference internal" href="#normaldist-examples-and-recipes"><code class="xref py py-class docutils literal notranslate">NormalDist</code> Examples and Recipes</a></li>
</ul>
</li>
</ul>
</li>
</ul>

<div class="related" role="navigation" aria-label="related navigation">
 <h3>Navigation</h3>
 <ul>
 <li class="right" style="margin-right: 10px">
 <a href="../genindex.html" title="General Index"
 accesskey="I">index</a></li>
 <li class="right" >
 <a href="../py-modindex.html" title="Python Module Index"
 >modules</a> |</li>
 <li class="right" >
 <a href="functional.html" title="Functional Programming Modules"
 accesskey="N">next</a> |</li>
 <li class="right" >
 <a href="random.html" title="random — Generate pseudo-random numbers"
 accesskey="P">previous</a> |</li>

<li><img src="../_static/py.svg" alt="python logo" style="vertical-align: middle; margin-top: -1px"/></li>
 <li><a href="https://www.python.org/">Python</a> &#187;</li>
 <li class="switchers">
 <div class="language_switcher_placeholder"></div>
 <div class="version_switcher_placeholder"></div>
 </li>
 <li>
 
 </li>
 <li id="cpython-language-and-version">
 <a href="../index.html">3.10.12 Documentation</a> &#187;
 </li>

<li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> &#187;</li>
 <li class="nav-item nav-item-2"><a href="numeric.html" accesskey="U">Numeric and Mathematical Modules</a> &#187;</li>
 <li class="nav-item nav-item-this"><a href=""><code class="xref py py-mod docutils literal notranslate">statistics</code> — Mathematical statistics functions</a></li>
 <li class="right">

<div class="document">
 <div class="documentwrapper">
 <div class="bodywrapper">
 <div class="body" role="main">
 
 <section id="module-statistics">
<h1><a class="reference internal" href="#module-statistics" title="statistics: Mathematical statistics functions"><code class="xref py py-mod docutils literal notranslate">statistics</code></a> — Mathematical statistics functions<a class="headerlink" href="#module-statistics" title="Permalink to this headline">¶</a></h1>
<div class="versionadded">
New in version 3.4.
</div>
Source code: <a class="reference external" href="https://github.com/python/cpython/tree/3.10/Lib/statistics.py">Lib/statistics.py</a>
<hr class="docutils" />
This module provides functions for calculating mathematical statistics of
numeric (<a class="reference internal" href="numbers.html#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate">Real</code></a>-valued) data.
The module is not intended to be a competitor to third-party libraries such
as <a class="reference external" href="https://numpy.org">NumPy</a>, <a class="reference external" href="https://www.scipy.org/">SciPy</a>, or
proprietary full-featured statistics packages aimed at professional
statisticians such as Minitab, SAS and Matlab. It is aimed at the level of
graphing and scientific calculators.
Unless explicitly noted, these functions support <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate">int</code></a>,
<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a>, <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate">Decimal</code></a> and <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate">Fraction</code></a>.
Behaviour with other types (whether in the numeric tower or not) is
currently unsupported. Collections with a mix of types are also undefined
and implementation-dependent. If your input data consists of mixed types,
you may be able to use <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map()</code></a> to ensure a consistent result, for
example: <code class="docutils literal notranslate">map(float, input_data)</code>.
Some datasets use <code class="docutils literal notranslate">NaN</code> (not a number) values to represent missing data.
Since NaNs have unusual comparison semantics, they cause surprising or
undefined behaviors in the statistics functions that sort data or that count
occurrences. The functions affected are <code class="docutils literal notranslate">median()</code>, <code class="docutils literal notranslate">median_low()</code>,
<code class="docutils literal notranslate">median_high()</code>, <code class="docutils literal notranslate">median_grouped()</code>, <code class="docutils literal notranslate">mode()</code>, <code class="docutils literal notranslate">multimode()</code>, and
<code class="docutils literal notranslate">quantiles()</code>. The <code class="docutils literal notranslate">NaN</code> values should be stripped before calling these
functions:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from statistics import median
&gt;&gt;&gt; from math import isnan
&gt;&gt;&gt; from itertools import filterfalse

&gt;&gt;&gt; data = [20.7, float(&#39;NaN&#39;),19.2, 18.3, float(&#39;NaN&#39;), 14.4]
&gt;&gt;&gt; sorted(data) # This has surprising behavior
[20.7, nan, 14.4, 18.3, 19.2, nan]
&gt;&gt;&gt; median(data) # This result is unexpected
16.35

&gt;&gt;&gt; sum(map(isnan, data)) # Number of missing values
2
&gt;&gt;&gt; clean = list(filterfalse(isnan, data)) # Strip NaN values
&gt;&gt;&gt; clean
[20.7, 19.2, 18.3, 14.4]
&gt;&gt;&gt; sorted(clean) # Sorting now works as expected
[14.4, 18.3, 19.2, 20.7]
&gt;&gt;&gt; median(clean) # This result is now well defined
18.75
</pre></div>
</div>
<section id="averages-and-measures-of-central-location">
<h2>Averages and measures of central location<a class="headerlink" href="#averages-and-measures-of-central-location" title="Permalink to this headline">¶</a></h2>
These functions calculate an average or typical value from a population
or sample.
<table class="docutils align-default">
<colgroup>
<col style="width: 27%" />
<col style="width: 73%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate">mean()</code></a></td>
<td>Arithmetic mean (“average”) of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate">fmean()</code></a></td>
<td>Fast, floating point arithmetic mean.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.geometric_mean" title="statistics.geometric_mean"><code class="xref py py-func docutils literal notranslate">geometric_mean()</code></a></td>
<td>Geometric mean of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.harmonic_mean" title="statistics.harmonic_mean"><code class="xref py py-func docutils literal notranslate">harmonic_mean()</code></a></td>
<td>Harmonic mean of data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate">median()</code></a></td>
<td>Median (middle value) of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate">median_low()</code></a></td>
<td>Low median of data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate">median_high()</code></a></td>
<td>High median of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate">median_grouped()</code></a></td>
<td>Median, or 50th percentile, of grouped data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate">mode()</code></a></td>
<td>Single mode (most common value) of discrete or nominal data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.multimode" title="statistics.multimode"><code class="xref py py-func docutils literal notranslate">multimode()</code></a></td>
<td>List of modes (most common values) of discrete or nominal data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.quantiles" title="statistics.quantiles"><code class="xref py py-func docutils literal notranslate">quantiles()</code></a></td>
<td>Divide data into intervals with equal probability.</td>
</tr>
</tbody>
</table>
</section>
<section id="measures-of-spread">
<h2>Measures of spread<a class="headerlink" href="#measures-of-spread" title="Permalink to this headline">¶</a></h2>
These functions calculate a measure of how much the population or sample
tends to deviate from the typical or average values.
<table class="docutils align-default">
<colgroup>
<col style="width: 34%" />
<col style="width: 66%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.pstdev" title="statistics.pstdev"><code class="xref py py-func docutils literal notranslate">pstdev()</code></a></td>
<td>Population standard deviation of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a></td>
<td>Population variance of data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate">stdev()</code></a></td>
<td>Sample standard deviation of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate">variance()</code></a></td>
<td>Sample variance of data.</td>
</tr>
</tbody>
</table>
</section>
<section id="statistics-for-relations-between-two-inputs">
<h2>Statistics for relations between two inputs<a class="headerlink" href="#statistics-for-relations-between-two-inputs" title="Permalink to this headline">¶</a></h2>
These functions calculate statistics regarding relations between two inputs.
<table class="docutils align-default">
<colgroup>
<col style="width: 32%" />
<col style="width: 68%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.covariance" title="statistics.covariance"><code class="xref py py-func docutils literal notranslate">covariance()</code></a></td>
<td>Sample covariance for two variables.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.correlation" title="statistics.correlation"><code class="xref py py-func docutils literal notranslate">correlation()</code></a></td>
<td>Pearson’s correlation coefficient for two variables.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.linear_regression" title="statistics.linear_regression"><code class="xref py py-func docutils literal notranslate">linear_regression()</code></a></td>
<td>Slope and intercept for simple linear regression.</td>
</tr>
</tbody>
</table>
</section>
<section id="function-details">
<h2>Function details<a class="headerlink" href="#function-details" title="Permalink to this headline">¶</a></h2>
Note: The functions do not require the data given to them to be sorted.
However, for reading convenience, most of the examples show sorted sequences.
<dl class="py function">
<dt class="sig sig-object py" id="statistics.mean">
statistics.mean(data)<a class="headerlink" href="#statistics.mean" title="Permalink to this definition">¶</a></dt>
<dd>Return the sample arithmetic mean of data which can be a sequence or iterable.
The arithmetic mean is the sum of the data divided by the number of data
points. It is commonly called “the average”, although it is only one of many
different mathematical averages. It is a measure of the central location of
the data.
If data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> will be raised.
Some examples of use:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mean([1, 2, 3, 4, 4])
2.8
&gt;&gt;&gt; mean([-1.0, 2.5, 3.25, 5.75])
2.625

&gt;&gt;&gt; from decimal import Decimal as D
&gt;&gt;&gt; mean([D(&quot;0.5&quot;), D(&quot;0.75&quot;), D(&quot;0.625&quot;), D(&quot;0.375&quot;)])
Decimal(&#39;0.5625&#39;)
</pre></div>
</div>
<div class="admonition note">
Note
The mean is strongly affected by <a class="reference external" href="https://en.wikipedia.org/wiki/Outlier">outliers</a> and is not necessarily a
typical example of the data points. For a more robust, although less
efficient, measure of <a class="reference external" href="https://en.wikipedia.org/wiki/Central_tendency">central tendency</a>, see <a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate">median()</code></a>.
The sample mean gives an unbiased estimate of the true population mean,
so that when taken on average over all the possible samples,
<code class="docutils literal notranslate">mean(sample)</code> converges on the true mean of the entire population. If
data represents the entire population rather than a sample, then
<code class="docutils literal notranslate">mean(data)</code> is equivalent to calculating the true population mean μ.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.fmean">
statistics.fmean(data)<a class="headerlink" href="#statistics.fmean" title="Permalink to this definition">¶</a></dt>
<dd>Convert data to floats and compute the arithmetic mean.
This runs faster than the <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate">mean()</code></a> function and it always returns a
<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a>. The data may be a sequence or iterable. If the input
dataset is empty, raises a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a>.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; fmean([3.5, 4.0, 5.25])
4.25
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.geometric_mean">
statistics.geometric_mean(data)<a class="headerlink" href="#statistics.geometric_mean" title="Permalink to this definition">¶</a></dt>
<dd>Convert data to floats and compute the geometric mean.
The geometric mean indicates the central tendency or typical value of the
data using the product of the values (as opposed to the arithmetic mean
which uses their sum).
Raises a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if the input dataset is empty,
if it contains a zero, or if it contains a negative value.
The data may be a sequence or iterable.
No special efforts are made to achieve exact results.
(However, this may change in the future.)
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; round(geometric_mean([54, 24, 36]), 1)
36.0
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.harmonic_mean">
statistics.harmonic_mean(data, weights=None)<a class="headerlink" href="#statistics.harmonic_mean" title="Permalink to this definition">¶</a></dt>
<dd>Return the harmonic mean of data, a sequence or iterable of
real-valued numbers. If weights is omitted or None, then
equal weighting is assumed.
The harmonic mean is the reciprocal of the arithmetic <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate">mean()</code></a> of the
reciprocals of the data. For example, the harmonic mean of three values a,
b and c will be equivalent to <code class="docutils literal notranslate">3/(1/a + 1/b + 1/c)</code>. If one of the
values is zero, the result will be zero.
The harmonic mean is a type of average, a measure of the central
location of the data. It is often appropriate when averaging
ratios or rates, for example speeds.
Suppose a car travels 10 km at 40 km/hr, then another 10 km at 60 km/hr.
What is the average speed?
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; harmonic_mean([40, 60])
48.0
</pre></div>
</div>
Suppose a car travels 40 km/hr for 5 km, and when traffic clears,
speeds-up to 60 km/hr for the remaining 30 km of the journey. What
is the average speed?
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; harmonic_mean([40, 60], weights=[5, 30])
56.0
</pre></div>
</div>
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised if data is empty, any element
is less than zero, or if the weighted sum isn’t positive.
The current algorithm has an early-out when it encounters a zero
in the input. This means that the subsequent inputs are not tested
for validity. (This behavior may change in the future.)
<div class="versionadded">
New in version 3.6.
</div>
<div class="versionchanged">
Changed in version 3.10: Added support for weights.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median">
statistics.median(data)<a class="headerlink" href="#statistics.median" title="Permalink to this definition">¶</a></dt>
<dd>Return the median (middle value) of numeric data, using the common “mean of
middle two” method. If data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
data can be a sequence or iterable.
The median is a robust measure of central location and is less affected by
the presence of outliers. When the number of data points is odd, the
middle data point is returned:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median([1, 3, 5])
3
</pre></div>
</div>
When the number of data points is even, the median is interpolated by taking
the average of the two middle values:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median([1, 3, 5, 7])
4.0
</pre></div>
</div>
This is suited for when your data is discrete, and you don’t mind that the
median may not be an actual data point.
If the data is ordinal (supports order operations) but not numeric (doesn’t
support addition), consider using <a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate">median_low()</code></a> or <a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate">median_high()</code></a>
instead.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median_low">
statistics.median_low(data)<a class="headerlink" href="#statistics.median_low" title="Permalink to this definition">¶</a></dt>
<dd>Return the low median of numeric data. If data is empty,
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised. data can be a sequence or iterable.
The low median is always a member of the data set. When the number of data
points is odd, the middle value is returned. When it is even, the smaller of
the two middle values is returned.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median_low([1, 3, 5])
3
&gt;&gt;&gt; median_low([1, 3, 5, 7])
3
</pre></div>
</div>
Use the low median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median_high">
statistics.median_high(data)<a class="headerlink" href="#statistics.median_high" title="Permalink to this definition">¶</a></dt>
<dd>Return the high median of data. If data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a>
is raised. data can be a sequence or iterable.
The high median is always a member of the data set. When the number of data
points is odd, the middle value is returned. When it is even, the larger of
the two middle values is returned.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median_high([1, 3, 5])
3
&gt;&gt;&gt; median_high([1, 3, 5, 7])
5
</pre></div>
</div>
Use the high median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median_grouped">
statistics.median_grouped(data, interval=1)<a class="headerlink" href="#statistics.median_grouped" title="Permalink to this definition">¶</a></dt>
<dd>Return the median of grouped continuous data, calculated as the 50th
percentile, using interpolation. If data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a>
is raised. data can be a sequence or iterable.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median_grouped([52, 52, 53, 54])
52.5
</pre></div>
</div>
In the following example, the data are rounded, so that each value represents
the midpoint of data classes, e.g. 1 is the midpoint of the class 0.5–1.5, 2
is the midpoint of 1.5–2.5, 3 is the midpoint of 2.5–3.5, etc. With the data
given, the middle value falls somewhere in the class 3.5–4.5, and
interpolation is used to estimate it:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
3.7
</pre></div>
</div>
Optional argument interval represents the class interval, and defaults
to 1. Changing the class interval naturally will change the interpolation:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median_grouped([1, 3, 3, 5, 7], interval=1)
3.25
&gt;&gt;&gt; median_grouped([1, 3, 3, 5, 7], interval=2)
3.5
</pre></div>
</div>
This function does not check whether the data points are at least
interval apart.
<div class="impl-detail compound">
CPython implementation detail: Under some circumstances, <a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate">median_grouped()</code></a> may coerce data points to
floats. This behaviour is likely to change in the future.
</div>
<div class="admonition seealso">
See also
<ul class="simple">
<li>“Statistics for the Behavioral Sciences”, Frederick J Gravetter and
Larry B Wallnau (8th Edition).</li>
<li>The <a class="reference external" href="https://help.gnome.org/users/gnumeric/stable/gnumeric.html#gnumeric-function-SSMEDIAN">SSMEDIAN</a>
function in the Gnome Gnumeric spreadsheet, including <a class="reference external" href="https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html">this discussion</a>.</li>
</ul>
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.mode">
statistics.mode(data)<a class="headerlink" href="#statistics.mode" title="Permalink to this definition">¶</a></dt>
<dd>Return the single most common data point from discrete or nominal data.
The mode (when it exists) is the most typical value and serves as a
measure of central location.
If there are multiple modes with the same frequency, returns the first one
encountered in the data. If the smallest or largest of those is
desired instead, use <code class="docutils literal notranslate">min(multimode(data))</code> or <code class="docutils literal notranslate">max(multimode(data))</code>.
If the input data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
<code class="docutils literal notranslate">mode</code> assumes discrete data and returns a single value. This is the
standard treatment of the mode as commonly taught in schools:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mode([1, 1, 2, 3, 3, 3, 3, 4])
3
</pre></div>
</div>
The mode is unique in that it is the only statistic in this package that
also applies to nominal (non-numeric) data:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mode([&quot;red&quot;, &quot;blue&quot;, &quot;blue&quot;, &quot;red&quot;, &quot;green&quot;, &quot;red&quot;, &quot;red&quot;])
&#39;red&#39;
</pre></div>
</div>
<div class="versionchanged">
Changed in version 3.8: Now handles multimodal datasets by returning the first mode encountered.
Formerly, it raised <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> when more than one mode was
found.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.multimode">
statistics.multimode(data)<a class="headerlink" href="#statistics.multimode" title="Permalink to this definition">¶</a></dt>
<dd>Return a list of the most frequently occurring values in the order they
were first encountered in the data. Will return more than one result if
there are multiple modes or an empty list if the data is empty:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; multimode(&#39;aabbbbccddddeeffffgg&#39;)
[&#39;b&#39;, &#39;d&#39;, &#39;f&#39;]
&gt;&gt;&gt; multimode(&#39;&#39;)
[]
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.pstdev">
statistics.pstdev(data, mu=None)<a class="headerlink" href="#statistics.pstdev" title="Permalink to this definition">¶</a></dt>
<dd>Return the population standard deviation (the square root of the population
variance). See <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a> for arguments and other details.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
0.986893273527251
</pre></div>
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.pvariance">
statistics.pvariance(data, mu=None)<a class="headerlink" href="#statistics.pvariance" title="Permalink to this definition">¶</a></dt>
<dd>Return the population variance of data, a non-empty sequence or iterable
of real-valued numbers. Variance, or second moment about the mean, is a
measure of the variability (spread or dispersion) of data. A large
variance indicates that the data is spread out; a small variance indicates
it is clustered closely around the mean.
If the optional second argument mu is given, it is typically the mean of
the data. It can also be used to compute the second moment around a
point that is not the mean. If it is missing or <code class="docutils literal notranslate">None</code> (the default),
the arithmetic mean is automatically calculated.
Use this function to calculate the variance from the entire population. To
estimate the variance from a sample, the <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate">variance()</code></a> function is usually
a better choice.
Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if data is empty.
Examples:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
&gt;&gt;&gt; pvariance(data)
1.25
</pre></div>
</div>
If you have already calculated the mean of your data, you can pass it as the
optional second argument mu to avoid recalculation:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mu = mean(data)
&gt;&gt;&gt; pvariance(data, mu)
1.25
</pre></div>
</div>
Decimals and Fractions are supported:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from decimal import Decimal as D
&gt;&gt;&gt; pvariance([D(&quot;27.5&quot;), D(&quot;30.25&quot;), D(&quot;30.25&quot;), D(&quot;34.5&quot;), D(&quot;41.75&quot;)])
Decimal(&#39;24.815&#39;)

&gt;&gt;&gt; from fractions import Fraction as F
&gt;&gt;&gt; pvariance([F(1, 4), F(5, 4), F(1, 2)])
Fraction(13, 72)
</pre></div>
</div>
<div class="admonition note">
Note
When called with the entire population, this gives the population variance
σ². When called on a sample instead, this is the biased sample variance
s², also known as variance with N degrees of freedom.
If you somehow know the true population mean μ, you may use this
function to calculate the variance of a sample, giving the known
population mean as the second argument. Provided the data points are a
random sample of the population, the result will be an unbiased estimate
of the population variance.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.stdev">
statistics.stdev(data, xbar=None)<a class="headerlink" href="#statistics.stdev" title="Permalink to this definition">¶</a></dt>
<dd>Return the sample standard deviation (the square root of the sample
variance). See <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate">variance()</code></a> for arguments and other details.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
1.0810874155219827
</pre></div>
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.variance">
statistics.variance(data, xbar=None)<a class="headerlink" href="#statistics.variance" title="Permalink to this definition">¶</a></dt>
<dd>Return the sample variance of data, an iterable of at least two real-valued
numbers. Variance, or second moment about the mean, is a measure of the
variability (spread or dispersion) of data. A large variance indicates that
the data is spread out; a small variance indicates it is clustered closely
around the mean.
If the optional second argument xbar is given, it should be the mean of
data. If it is missing or <code class="docutils literal notranslate">None</code> (the default), the mean is
automatically calculated.
Use this function when your data is a sample from a population. To calculate
the variance from the entire population, see <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a>.
Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if data has fewer than two values.
Examples:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
&gt;&gt;&gt; variance(data)
1.3720238095238095
</pre></div>
</div>
If you have already calculated the mean of your data, you can pass it as the
optional second argument xbar to avoid recalculation:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; m = mean(data)
&gt;&gt;&gt; variance(data, m)
1.3720238095238095
</pre></div>
</div>
This function does not attempt to verify that you have passed the actual mean
as xbar. Using arbitrary values for xbar can lead to invalid or
impossible results.
Decimal and Fraction values are supported:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from decimal import Decimal as D
&gt;&gt;&gt; variance([D(&quot;27.5&quot;), D(&quot;30.25&quot;), D(&quot;30.25&quot;), D(&quot;34.5&quot;), D(&quot;41.75&quot;)])
Decimal(&#39;31.01875&#39;)

&gt;&gt;&gt; from fractions import Fraction as F
&gt;&gt;&gt; variance([F(1, 6), F(1, 2), F(5, 3)])
Fraction(67, 108)
</pre></div>
</div>
<div class="admonition note">
Note
This is the sample variance s² with Bessel’s correction, also known as
variance with N-1 degrees of freedom. Provided that the data points are
representative (e.g. independent and identically distributed), the result
should be an unbiased estimate of the true population variance.
If you somehow know the actual population mean μ you should pass it to the
<a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a> function as the mu parameter to get the variance of a
sample.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.quantiles">
statistics.quantiles(data, *, n=4, method='exclusive')<a class="headerlink" href="#statistics.quantiles" title="Permalink to this definition">¶</a></dt>
<dd>Divide data into n continuous intervals with equal probability.
Returns a list of <code class="docutils literal notranslate">n - 1</code> cut points separating the intervals.
Set n to 4 for quartiles (the default). Set n to 10 for deciles. Set
n to 100 for percentiles which gives the 99 cuts points that separate
data into 100 equal sized groups. Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if n
is not least 1.
The data can be any iterable containing sample data. For meaningful
results, the number of data points in data should be larger than n.
Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if there are not at least two data points.
The cut points are linearly interpolated from the
two nearest data points. For example, if a cut point falls one-third
of the distance between two sample values, <code class="docutils literal notranslate">100</code> and <code class="docutils literal notranslate">112</code>, the
cut-point will evaluate to <code class="docutils literal notranslate">104</code>.
The method for computing quantiles can be varied depending on
whether the data includes or excludes the lowest and
highest possible values from the population.
The default method is “exclusive” and is used for data sampled from
a population that can have more extreme values than found in the
samples. The portion of the population falling below the i-th of
m sorted data points is computed as <code class="docutils literal notranslate">i / (m + 1)</code>. Given nine
sample values, the method sorts them and assigns the following
percentiles: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.
Setting the method to “inclusive” is used for describing population
data or for samples that are known to include the most extreme values
from the population. The minimum value in data is treated as the 0th
percentile and the maximum value is treated as the 100th percentile.
The portion of the population falling below the i-th of m sorted
data points is computed as <code class="docutils literal notranslate">(i - 1) / (m - 1)</code>. Given 11 sample
values, the method sorts them and assigns the following percentiles:
0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre># Decile cut points for empirically sampled data
&gt;&gt;&gt; data = [105, 129, 87, 86, 111, 111, 89, 81, 108, 92, 110,
... 100, 75, 105, 103, 109, 76, 119, 99, 91, 103, 129,
... 106, 101, 84, 111, 74, 87, 86, 103, 103, 106, 86,
... 111, 75, 87, 102, 121, 111, 88, 89, 101, 106, 95,
... 103, 107, 101, 81, 109, 104]
&gt;&gt;&gt; [round(q, 1) for q in quantiles(data, n=10)]
[81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.covariance">
statistics.covariance(x, y, /)<a class="headerlink" href="#statistics.covariance" title="Permalink to this definition">¶</a></dt>
<dd>Return the sample covariance of two inputs x and y. Covariance
is a measure of the joint variability of two inputs.
Both inputs must be of the same length (no less than two), otherwise
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
Examples:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
&gt;&gt;&gt; y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
&gt;&gt;&gt; covariance(x, y)
0.75
&gt;&gt;&gt; z = [9, 8, 7, 6, 5, 4, 3, 2, 1]
&gt;&gt;&gt; covariance(x, z)
-7.5
&gt;&gt;&gt; covariance(z, x)
-7.5
</pre></div>
</div>
<div class="versionadded">
New in version 3.10.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.correlation">
statistics.correlation(x, y, /)<a class="headerlink" href="#statistics.correlation" title="Permalink to this definition">¶</a></dt>
<dd>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Pearson_correlation_coefficient">Pearson’s correlation coefficient</a>
for two inputs. Pearson’s correlation coefficient r takes values
between -1 and +1. It measures the strength and direction of the linear
relationship, where +1 means very strong, positive linear relationship,
-1 very strong, negative linear relationship, and 0 no linear relationship.
Both inputs must be of the same length (no less than two), and need
not to be constant, otherwise <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
Examples:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
&gt;&gt;&gt; y = [9, 8, 7, 6, 5, 4, 3, 2, 1]
&gt;&gt;&gt; correlation(x, x)
1.0
&gt;&gt;&gt; correlation(x, y)
-1.0
</pre></div>
</div>
<div class="versionadded">
New in version 3.10.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.linear_regression">
statistics.linear_regression(x, y, /)<a class="headerlink" href="#statistics.linear_regression" title="Permalink to this definition">¶</a></dt>
<dd>Return the slope and intercept of <a class="reference external" href="https://en.wikipedia.org/wiki/Simple_linear_regression">simple linear regression</a>
parameters estimated using ordinary least squares. Simple linear
regression describes the relationship between an independent variable x and
a dependent variable y in terms of this linear function:
<blockquote>
<div>y = slope * x + intercept + noise
</div></blockquote>
where <code class="docutils literal notranslate">slope</code> and <code class="docutils literal notranslate">intercept</code> are the regression parameters that are
estimated, and <code class="docutils literal notranslate">noise</code> represents the
variability of the data that was not explained by the linear regression
(it is equal to the difference between predicted and actual values
of the dependent variable).
Both inputs must be of the same length (no less than two), and
the independent variable x cannot be constant;
otherwise a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
For example, we can use the <a class="reference external" href="https://en.wikipedia.org/wiki/Monty_Python#Films">release dates of the Monty
Python films</a>
to predict the cumulative number of Monty Python films
that would have been produced by 2019
assuming that they had kept the pace.
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; year = [1971, 1975, 1979, 1982, 1983]
&gt;&gt;&gt; films_total = [1, 2, 3, 4, 5]
&gt;&gt;&gt; slope, intercept = linear_regression(year, films_total)
&gt;&gt;&gt; round(slope * 2019 + intercept)
16
</pre></div>
</div>
<div class="versionadded">
New in version 3.10.
</div>
</dd></dl>

</section>
<section id="exceptions">
<h2>Exceptions<a class="headerlink" href="#exceptions" title="Permalink to this headline">¶</a></h2>
A single exception is defined:
<dl class="py exception">
<dt class="sig sig-object py" id="statistics.StatisticsError">
exception statistics.StatisticsError<a class="headerlink" href="#statistics.StatisticsError" title="Permalink to this definition">¶</a></dt>
<dd>Subclass of <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate">ValueError</code></a> for statistics-related exceptions.
</dd></dl>

</section>
<section id="normaldist-objects">
<h2><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> objects<a class="headerlink" href="#normaldist-objects" title="Permalink to this headline">¶</a></h2>
<a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> is a tool for creating and manipulating normal
distributions of a <a class="reference external" href="http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm">random variable</a>. It is a
class that treats the mean and standard deviation of data
measurements as a single entity.
Normal distributions arise from the <a class="reference external" href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a> and have a wide range
of applications in statistics.
<dl class="py class">
<dt class="sig sig-object py" id="statistics.NormalDist">
class statistics.NormalDist(mu=0.0, sigma=1.0)<a class="headerlink" href="#statistics.NormalDist" title="Permalink to this definition">¶</a></dt>
<dd>Returns a new NormalDist object where mu represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic
mean</a> and sigma
represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a>.
If sigma is negative, raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a>.
<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.mean">
mean<a class="headerlink" href="#statistics.NormalDist.mean" title="Permalink to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.median">
median<a class="headerlink" href="#statistics.NormalDist.median" title="Permalink to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Median">median</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.mode">
mode<a class="headerlink" href="#statistics.NormalDist.mode" title="Permalink to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Mode_(statistics)">mode</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.stdev">
stdev<a class="headerlink" href="#statistics.NormalDist.stdev" title="Permalink to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.variance">
variance<a class="headerlink" href="#statistics.NormalDist.variance" title="Permalink to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Variance">variance</a> of a normal
distribution. Equal to the square of the standard deviation.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.from_samples">
classmethod from_samples(data)<a class="headerlink" href="#statistics.NormalDist.from_samples" title="Permalink to this definition">¶</a></dt>
<dd>Makes a normal distribution instance with mu and sigma parameters
estimated from the data using <a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate">fmean()</code></a> and <a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate">stdev()</code></a>.
The data can be any <a class="reference internal" href="../glossary.html#term-iterable">iterable</a> and should consist of values
that can be converted to type <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a>. If data does not
contain at least two elements, raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> because it
takes at least one point to estimate a central value and at least two
points to estimate dispersion.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.samples">
samples(n, *, seed=None)<a class="headerlink" href="#statistics.NormalDist.samples" title="Permalink to this definition">¶</a></dt>
<dd>Generates n random samples for a given mean and standard deviation.
Returns a <a class="reference internal" href="stdtypes.html#list" title="list"><code class="xref py py-class docutils literal notranslate">list</code></a> of <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a> values.
If seed is given, creates a new instance of the underlying random
number generator. This is useful for creating reproducible results,
even in a multi-threading context.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.pdf">
pdf(x)<a class="headerlink" href="#statistics.NormalDist.pdf" title="Permalink to this definition">¶</a></dt>
<dd>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Probability_density_function">probability density function (pdf)</a>, compute
the relative likelihood that a random variable X will be near the
given value x. Mathematically, it is the limit of the ratio <code class="docutils literal notranslate">P(x &lt;=
X &lt; x+dx) / dx</code> as dx approaches zero.
The relative likelihood is computed as the probability of a sample
occurring in a narrow range divided by the width of the range (hence
the word “density”). Since the likelihood is relative to other points,
its value can be greater than <code class="docutils literal notranslate">1.0</code>.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.cdf">
cdf(x)<a class="headerlink" href="#statistics.NormalDist.cdf" title="Permalink to this definition">¶</a></dt>
<dd>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">cumulative distribution function (cdf)</a>,
compute the probability that a random variable X will be less than or
equal to x. Mathematically, it is written <code class="docutils literal notranslate">P(X &lt;= x)</code>.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.inv_cdf">
inv_cdf(p)<a class="headerlink" href="#statistics.NormalDist.inv_cdf" title="Permalink to this definition">¶</a></dt>
<dd>Compute the inverse cumulative distribution function, also known as the
<a class="reference external" href="https://en.wikipedia.org/wiki/Quantile_function">quantile function</a>
or the <a class="reference external" href="https://web.archive.org/web/20190203145224/https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/">percent-point</a>
function. Mathematically, it is written <code class="docutils literal notranslate">x : P(X &lt;= x) = p</code>.
Finds the value x of the random variable X such that the
probability of the variable being less than or equal to that value
equals the given probability p.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.overlap">
overlap(other)<a class="headerlink" href="#statistics.NormalDist.overlap" title="Permalink to this definition">¶</a></dt>
<dd>Measures the agreement between two normal probability distributions.
Returns a value between 0.0 and 1.0 giving <a class="reference external" href="https://www.rasch.org/rmt/rmt101r.htm">the overlapping area for
the two probability density functions</a>.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.quantiles">
quantiles(n=4)<a class="headerlink" href="#statistics.NormalDist.quantiles" title="Permalink to this definition">¶</a></dt>
<dd>Divide the normal distribution into n continuous intervals with
equal probability. Returns a list of (n - 1) cut points separating
the intervals.
Set n to 4 for quartiles (the default). Set n to 10 for deciles.
Set n to 100 for percentiles which gives the 99 cuts points that
separate the normal distribution into 100 equal sized groups.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.zscore">
zscore(x)<a class="headerlink" href="#statistics.NormalDist.zscore" title="Permalink to this definition">¶</a></dt>
<dd>Compute the
<a class="reference external" href="https://www.statisticshowto.com/probability-and-statistics/z-score/">Standard Score</a>
describing x in terms of the number of standard deviations
above or below the mean of the normal distribution:
<code class="docutils literal notranslate">(x - mean) / stdev</code>.
<div class="versionadded">
New in version 3.9.
</div>
</dd></dl>

Instances of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> support addition, subtraction,
multiplication and division by a constant. These operations
are used for translation and scaling. For example:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; temperature_february = NormalDist(5, 2.5) # Celsius
&gt;&gt;&gt; temperature_february * (9/5) + 32 # Fahrenheit
NormalDist(mu=41.0, sigma=4.5)
</pre></div>
</div>
Dividing a constant by an instance of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> is not supported
because the result wouldn’t be normally distributed.
Since normal distributions arise from additive effects of independent
variables, it is possible to <a class="reference external" href="https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables">add and subtract two independent normally
distributed random variables</a>
represented as instances of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a>. For example:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; birth_weights = NormalDist.from_samples([2.5, 3.1, 2.1, 2.4, 2.7, 3.5])
&gt;&gt;&gt; drug_effects = NormalDist(0.4, 0.15)
&gt;&gt;&gt; combined = birth_weights + drug_effects
&gt;&gt;&gt; round(combined.mean, 1)
3.1
&gt;&gt;&gt; round(combined.stdev, 1)
0.5
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<section id="normaldist-examples-and-recipes">
<h3><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> Examples and Recipes<a class="headerlink" href="#normaldist-examples-and-recipes" title="Permalink to this headline">¶</a></h3>
<a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> readily solves classic probability problems.
For example, given <a class="reference external" href="https://nces.ed.gov/programs/digest/d17/tables/dt17_226.40.asp">historical data for SAT exams</a> showing
that scores are normally distributed with a mean of 1060 and a standard
deviation of 195, determine the percentage of students with test scores
between 1100 and 1200, after rounding to the nearest whole number:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sat = NormalDist(1060, 195)
&gt;&gt;&gt; fraction = sat.cdf(1200 + 0.5) - sat.cdf(1100 - 0.5)
&gt;&gt;&gt; round(fraction * 100.0, 1)
18.4
</pre></div>
</div>
Find the <a class="reference external" href="https://en.wikipedia.org/wiki/Quartile">quartiles</a> and <a class="reference external" href="https://en.wikipedia.org/wiki/Decile">deciles</a> for the SAT scores:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; list(map(round, sat.quantiles()))
[928, 1060, 1192]
&gt;&gt;&gt; list(map(round, sat.quantiles(n=10)))
[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]
</pre></div>
</div>
To estimate the distribution for a model than isn’t easy to solve
analytically, <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> can generate input samples for a <a class="reference external" href="https://en.wikipedia.org/wiki/Monte_Carlo_method">Monte
Carlo simulation</a>:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; def model(x, y, z):
... return (3*x + 7*x*y - 5*y) / (11 * z)
...
&gt;&gt;&gt; n = 100_000
&gt;&gt;&gt; X = NormalDist(10, 2.5).samples(n, seed=3652260728)
&gt;&gt;&gt; Y = NormalDist(15, 1.75).samples(n, seed=4582495471)
&gt;&gt;&gt; Z = NormalDist(50, 1.25).samples(n, seed=6582483453)
&gt;&gt;&gt; quantiles(map(model, X, Y, Z)) 
[1.4591308524824727, 1.8035946855390597, 2.175091447274739]
</pre></div>
</div>
Normal distributions can be used to approximate <a class="reference external" href="https://mathworld.wolfram.com/BinomialDistribution.html">Binomial
distributions</a>
when the sample size is large and when the probability of a successful
trial is near 50%.
For example, an open source conference has 750 attendees and two rooms with a
500 person capacity. There is a talk about Python and another about Ruby.
In previous conferences, 65% of the attendees preferred to listen to Python
talks. Assuming the population preferences haven’t changed, what is the
probability that the Python room will stay within its capacity limits?
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; n = 750 # Sample size
&gt;&gt;&gt; p = 0.65 # Preference for Python
&gt;&gt;&gt; q = 1.0 - p # Preference for Ruby
&gt;&gt;&gt; k = 500 # Room capacity

&gt;&gt;&gt; # Approximation using the cumulative normal distribution
&gt;&gt;&gt; from math import sqrt
&gt;&gt;&gt; round(NormalDist(mu=n*p, sigma=sqrt(n*p*q)).cdf(k + 0.5), 4)
0.8402

&gt;&gt;&gt; # Solution using the cumulative binomial distribution
&gt;&gt;&gt; from math import comb, fsum
&gt;&gt;&gt; round(fsum(comb(n, r) * p**r * q**(n-r) for r in range(k+1)), 4)
0.8402

&gt;&gt;&gt; # Approximation using a simulation
&gt;&gt;&gt; from random import seed, choices
&gt;&gt;&gt; seed(8675309)
&gt;&gt;&gt; def trial():
... return choices((&#39;Python&#39;, &#39;Ruby&#39;), (p, q), k=n).count(&#39;Python&#39;)
&gt;&gt;&gt; mean(trial() &lt;= k for i in range(10_000))
0.8398
</pre></div>
</div>
Normal distributions commonly arise in machine learning problems.
Wikipedia has a <a class="reference external" href="https://en.wikipedia.org/wiki/Naive_Bayes_classifier#Person_classification">nice example of a Naive Bayesian Classifier</a>.
The challenge is to predict a person’s gender from measurements of normally
distributed features including height, weight, and foot size.
We’re given a training dataset with measurements for eight people. The
measurements are assumed to be normally distributed, so we summarize the data
with <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a>:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; height_male = NormalDist.from_samples([6, 5.92, 5.58, 5.92])
&gt;&gt;&gt; height_female = NormalDist.from_samples([5, 5.5, 5.42, 5.75])
&gt;&gt;&gt; weight_male = NormalDist.from_samples([180, 190, 170, 165])
&gt;&gt;&gt; weight_female = NormalDist.from_samples([100, 150, 130, 150])
&gt;&gt;&gt; foot_size_male = NormalDist.from_samples([12, 11, 12, 10])
&gt;&gt;&gt; foot_size_female = NormalDist.from_samples([6, 8, 7, 9])
</pre></div>
</div>
Next, we encounter a new person whose feature measurements are known but whose
gender is unknown:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; ht = 6.0 # height
&gt;&gt;&gt; wt = 130 # weight
&gt;&gt;&gt; fs = 8 # foot size
</pre></div>
</div>
Starting with a 50% <a class="reference external" href="https://en.wikipedia.org/wiki/Prior_probability">prior probability</a> of being male or female,
we compute the posterior as the prior times the product of likelihoods for the
feature measurements given the gender:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; prior_male = 0.5
&gt;&gt;&gt; prior_female = 0.5
&gt;&gt;&gt; posterior_male = (prior_male * height_male.pdf(ht) *
... weight_male.pdf(wt) * foot_size_male.pdf(fs))

&gt;&gt;&gt; posterior_female = (prior_female * height_female.pdf(ht) *
... weight_female.pdf(wt) * foot_size_female.pdf(fs))
</pre></div>
</div>
The final prediction goes to the largest posterior. This is known as the
<a class="reference external" href="https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation">maximum a posteriori</a> or MAP:
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; &#39;male&#39; if posterior_male &gt; posterior_female else &#39;female&#39;
&#39;female&#39;
</pre></div>
</div>
</section>
</section>
</section>

<div class="clearer"></div>
 </div>
 </div>
 </div>
 <div class="sphinxsidebar" role="navigation" aria-label="main navigation">
 <div class="sphinxsidebarwrapper">
 <h3><a href="../contents.html">Table of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate">statistics</code> — Mathematical statistics functions</a><ul>
<li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>
<li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>
<li><a class="reference internal" href="#statistics-for-relations-between-two-inputs">Statistics for relations between two inputs</a></li>
<li><a class="reference internal" href="#function-details">Function details</a></li>
<li><a class="reference internal" href="#exceptions">Exceptions</a></li>
<li><a class="reference internal" href="#normaldist-objects"><code class="xref py py-class docutils literal notranslate">NormalDist</code> objects</a><ul>
<li><a class="reference internal" href="#normaldist-examples-and-recipes"><code class="xref py py-class docutils literal notranslate">NormalDist</code> Examples and Recipes</a></li>
</ul>
</li>
</ul>
</li>
</ul>

<h4>Previous topic</h4>
 <a href="random.html"
 title="previous chapter"><code class="xref py py-mod docutils literal notranslate">random</code> — Generate pseudo-random numbers</a>
 <h4>Next topic</h4>
 <a href="functional.html"
 title="next chapter">Functional Programming Modules</a>
 <div role="note" aria-label="source link">
 <h3>This Page</h3>
 <ul class="this-page-menu">
 <li><a href="../bugs.html">Report a Bug</a></li>
 <li>
 <a href="https://github.com/python/cpython/blob/3.10/Doc/library/statistics.rst"
 rel="nofollow">Show Source
 </a>
 </li>
 </ul>
 </div>
 </div>
 </div>
 <div class="clearer"></div>
 </div> 
 <div class="related" role="navigation" aria-label="related navigation">
 <h3>Navigation</h3>
 <ul>
 <li class="right" style="margin-right: 10px">
 <a href="../genindex.html" title="General Index"
 >index</a></li>
 <li class="right" >
 <a href="../py-modindex.html" title="Python Module Index"
 >modules</a> |</li>
 <li class="right" >
 <a href="functional.html" title="Functional Programming Modules"
 >next</a> |</li>
 <li class="right" >
 <a href="random.html" title="random — Generate pseudo-random numbers"
 >previous</a> |</li>

<li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> &#187;</li>
 <li class="nav-item nav-item-2"><a href="numeric.html" >Numeric and Mathematical Modules</a> &#187;</li>
 <li class="nav-item nav-item-this"><a href=""><code class="xref py py-mod docutils literal notranslate">statistics</code> — Mathematical statistics functions</a></li>
 <li class="right">

<div class="inline-search" role="search">
 <form class="inline-search" action="../search.html" method="get">
 <input placeholder="Quick search" aria-label="Quick search" type="text" name="q" />
 <input type="submit" value="Go" />
 <input type="hidden" name="check_keywords" value="yes" />
 <input type="hidden" name="area" value="default" />
 </form>
 </div>
 |
 </li>
 
 </ul>
 </div> 
 <div class="footer">
 &copy; <a href="../copyright.html">Copyright</a> 2001-2026, Python Software Foundation.
 
 This page is licensed under the Python Software Foundation License Version 2.
 
 Examples, recipes, and other code in the documentation are additionally licensed under the Zero Clause BSD License.
 
 See <a href="/license.html">History and License</a> for more information.

The Python Software Foundation is a non-profit corporation.
<a href="https://www.python.org/psf/donations/">Please donate.</a>

Last updated on January 26, 2026.
 <a href="/bugs.html">Found a bug</a>?

Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 4.3.2.
 </div>

</body>
</html>

${this.title}

Code Editor : statistics.html