How to calculate probability in a normal distribution given mean standard deviation

Question

How to calculate probability in normal distribution given mean  std in Python  I can always explicitly code my own function according to the definition like the OP in this question did  Calculating Probability of a Random Variable in a Distribution in Python Just wondering if there is a library function call will allow you to do this  In my imagine it would like this  nd   NormalDistribution mu 100  std 12  p   nd prob 98   There is a similar question in Perl  How can I compute the probability at a point given a normal distribution in Perl   But I didn t see one in Python  Numpy has a random normal function  but it s like sampling  not exactly what I want

User · Answer

Scipy stats is a great module  Just to offer another approach  you can calculate it directly using  import math def normpdf x  mean  sd       var   float sd   2     denom    2 math pi var    5     num   math exp - float x -float mean    2  2 var       return num denom   This uses the formula found here  http   en wikipedia org wiki Normal distribution Probability density function  to test    gt  gt  gt  normpdf 7 5 5    0 07365402806066466  gt  gt  gt  norm 5 5  pdf 7  0 073654028060664664

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Here is more info  First you are dealing with a frozen distribution  frozen in this case means its parameters are set to specific values   To create a frozen distribution   import scipy stats scipy stats norm loc 100  scale 12   where loc is the mean and scale is the std dev  if you wish to pull out a random number from your distribution scipy stats norm rvs loc 100  scale 12    To find the probability that the variable has a value LESS than or equal  let s say 113  you d use CDF cumulative Density Function scipy stats norm cdf 113 100 12  Output  0 86066975255037792  or 86 07  probability   To find the probability that the variable has a value GREATER than or  equal to let s say 125  you d use SF Survival Function  scipy stats norm sf 125 100 12  Output  0 018610425189886332  or 1 86    To find the variate for which the probability is given  let s say the   value which needed to provide a 98  probability  you d use the   PPF Percent Point Function scipy stats norm ppf  98 100 12  Output  124 64498692758187

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Starting Python 3 8  the standard library provides the NormalDist object as part of the statistics module   It can be used to get the probability density function  pdf - likelihood that a random sample X will be near the given value x  for a given mean  mu  and standard deviation  sigma    from statistics import NormalDist  NormalDist mu 100  sigma 12  pdf 98    0 032786643008494994   Also note that the NormalDist object also provides the cumulative distribution function  cdf - probability that a random sample X will be less than or equal to x    NormalDist mu 100  sigma 12  cdf 98    0 43381616738909634

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I wrote this program to do the math for you  Just enter in the summary statistics  No need to provide an array   One-Sample Z-Test for a Population Proportion   To do this for mean rather than proportion  change the formula for z accordingly  EDIT  Here is the content from the link   import scipy stats as stats import math  def one sample ztest pop proportion tail  p  pbar  n  alpha        Calculate test stat      sigma   math sqrt  p  1-p    n       z   round  pbar - p    sigma  2       if tail     lower           pval   round stats norm p  sigma  cdf pbar  4          print  Results for a lower tailed z-test           elif tail     upper           pval   round 1 - stats norm p  sigma  cdf pbar  4          print  Results for an upper tailed z-test           elif tail     two           pval   round stats norm p  sigma  cdf pbar  2 4          print  Results for a two tailed z-test            Print test results     print  Test statistic       format z          print  P-value       format pval       print  Confidence       format alpha         Compare p-value to confidence level     if pval  lt   alpha          print      lt        Reject the null hypothesis   format pval  alpha       else          print      gt      Do not reject the null hypothesis   format pval  alpha      one sample ztest pop proportion  upper    20   25  400   05    one sample ztest pop proportion  two    64   52  100   05

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The formula cited from wikipedia mentioned in the answers cannot be used to calculate normal probabilites   You would have to write a numerical integration approximation function using that formula in order to calculate the probability   That formula computes the value for the probability density function   Since the normal distribution is continuous  you have to compute an integral to get probabilities   The wikipedia site mentions the CDF  which does not have a closed form for the normal distribution

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You can just use the error function that s built in to the math library  as stated on their website

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There s one in scipy stats    gt  gt  gt  import scipy stats  gt  gt  gt  scipy stats norm 0  1   lt scipy stats distributions rv frozen object at 0x928352c gt   gt  gt  gt  scipy stats norm 0  1  pdf 0  0 3989422804014327  gt  gt  gt  scipy stats norm 0  1  cdf 0  0 5  gt  gt  gt  scipy stats norm 100  12   lt scipy stats distributions rv frozen object at 0x928352c gt   gt  gt  gt  scipy stats norm 100  12  pdf 98  0 032786643008494994  gt  gt  gt  scipy stats norm 100  12  cdf 98  0 43381616738909634  gt  gt  gt  scipy stats norm 100  12  cdf 100  0 5    One thing to beware of -- just a tip -- is that the parameter passing is a little broad   Because of the way the code is set up  if you accidentally write scipy stats norm mean 100  std 12  instead of scipy stats norm 100  12  or scipy stats norm loc 100  scale 12   then it ll accept it  but silently discard those extra keyword arguments and give you the default  0 1

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Note that probability is different than probability density pdf    which some of the previous answers refer to  Probability is the chance that the variable has a specific value  whereas the probability density is the chance that the variable will be near a specific value  meaning probability over a range  So to obtain the probability you need to compute the integral of the probability density function over a given interval  As an approximation  you can simply multiply the probability density by the interval you re interested in and that will give you the actual probability  import numpy as np from scipy stats import norm  data start   -10 data end   10 data points   21 data   np linspace data start  data end  data points   point of interest   5 mu   np mean data  sigma   np std data                                     interval    data end - data start     data points - 1  probability   norm pdf point of interest  loc mu  scale sigma    interval  The code above will give you the probability that the variable will have an exact value of 5 in a normal distribution between -10 and 10 with 21 data points  meaning interval is 1   You can play around with a fixed interval value  depending on the results you want to achieve

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In case you would like to find the area between 2 values of x mean   1   standard deviation   2  the probability of x between  0 5 2   import scipy stats scipy stats norm 1  2  cdf 2  - scipy stats norm 1 2  cdf 0 5

[python] How to calculate probability in a normal distribution given mean & standard deviation?

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