# Source code for simplestatistics.statistics.normal

```
"""
Implements normal() function with norm_single_calculation() helper function.
"""
# I need sane division that returns a float not int
from __future__ import division
import math
def norm_single_calculation(x, mean, standard_deviation):
"""
Helper function that calculates normal density function for a particular
x with a given mean and stadard deviation.
"""
# break the normal distribution probability density function
# into two components
# component 1
# 1 / (sqrt(2 * sd^2 * pi))
# component 2
# e ^ (- (x - mean) ^ 2 / 2 sd^2)
# x = decimalize(x)
# mean = decimalize(mean)
# standard_deviation = decimalize(standard_deviation)
# before we do a lot of math, let's check if sd is 0
if standard_deviation == 0:
return(0)
# component 1
comp1 = 1 / (math.sqrt(2 * pow(standard_deviation, 2) * math.pi))
# component 2
comp2 = pow(math.e, (-1 * (pow(x - mean, 2)) / (2 * pow(standard_deviation, 2))))
# return the product of the two components
return(comp1 * comp2)
[docs]def normal(x, mean, standard_deviation):
"""
The `Normal Distribution`_ is, quoting from the Wikipedia page:
A very common continuous probability distribution. Normal distributions
are important in statistics and are often used in the natural and social
sciences to represent real-valued random variables whose distributions are not known.
The normal distribution is useful because of the central limit theorem.
In its most general form, under some conditions (which include finite variance),
it states that averages of random variables independently drawn from independent
distributions converge in distribution to the normal, that is, become normally
distributed when the number of random variables is sufficiently large. Physical
quantities that are expected to be the sum of many independent processes (such
as measurement errors) often have distributions that are nearly normal. Moreover,
many results and methods (such as propagation of uncertainty and least squares
parameter fitting) can be derived analytically in explicit form when the
relevant variables are normally distributed.
.. _`Normal Distribution`: https://en.wikipedia.org/wiki/Normal_distribution
Probability density function equation:
.. math::
f(x \\mid \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2 \\sigma^2 \\pi}}
e^{-\\frac{(x - \\mu)^2}{2\\sigma^2}}
Args:
x: Int or float or list of ints or floats representing values for which
the normal distribution density is desired.
mean: Mean of the sample or population.
standard_deviation: Standard deviation of the sample or population.
Returns:
Float or (or list of floats if provided x was a list)
representing the normal distribution density function for x
or each value in x.
Examples:
>>> normal(4, 8, 2)
0.02699548325659403
>>> normal([1, 4], 8, 2)
[0.00043634134752288024, 0.02699548325659403]
>>> normal(4, 8, 0)
0
"""
if type(x) in [int, float]:
return(norm_single_calculation(x, mean, standard_deviation))
elif type(x) in [list, tuple]:
norm_distribution_list = [norm_single_calculation(value, mean, standard_deviation)
for value in x]
return(norm_distribution_list)
```