9.2 Normal Probability Distribution

The normal probability distribution with mean $\mu$ and standard deviation $\sigma$ is given by:

$f(x)=\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ It is bell shaped and is symmetric about the mean $\mu$ .

Function graph showing y = 100*exp(-((x-m)^2)/(2*s^2))/(s*sqrt(2*pi)) and y = 100*exp(-((x-73)^2)/(2*5^2))/(5*sqrt(2*pi)). Adjustable parameters: Mean mu (m) = 78 points, Standard deviation sigma (s) = 8 points. Viewing window: x from 58.38 to 87.62, y from -1.44 to 16.64. — The book's picture of the normal curve is one curve. This one has both of its parameters on sliders, against a dashed reference fixed at the mu = 73, sigma = 5 exam distribution the rest of this section uses. The vertical axis is the density in PERCENT PER EXAM POINT (100 times f(x)), so the reference curve peaks at 7.98 - 100/(5 x sqrt(2 pi)) - and the area under it is still 100. Drag mu and the curve slides without changing shape; drag sigma and the peak height moves in exact inverse proportion, because the area has to stay 1: at sigma = 4 the peak is 9.97, at sigma = 10 it is 3.99. That trade-off is the thing a single printed bell curve cannot show, and it is what makes a z-score comparable across two different normal distributions.

The normal curve: a bell-shaped density symmetric about the mean, with tick marks at μ − 3σ through μ + 3σ and a horizontal arrow marking one standard deviation σ from the mean to μ + σ.

The syntax for the normal cumulative probability $P(X<x)$ with $\mu$ =mean and $\sigma$ =std is:

from scipy import stats
stats.norm.cdf(x,mean,std)

The syntax for inverse normal or p $^{th}$ percentile (with area to the left of p) and $\mu$ =mean and $\sigma$ =std is:

from scipy import stats
stats.norm.ppf(p,mean,std)

Note: The standard normal distribution has a mean $\mu$ of zero and standard deviation $\sigma$ of one.

Example: In a standard normal distribution what is the probability that the z value is less than 2?

Watch demo video

from scipy import stats
stats.norm.cdf(2,0,1)    #for standard normal mean is zero and std is one

Show expected output

0.9772498680518208

Interpretation: $P(z<2)=0.9772498680518208$

Example: In a standard normal distribution what is the probability that the z value is between -2 and 2.

$P(-2<z<2)=P(z<2)-P(z<-2)$

=stats.norm.cdf(2,0,1)-stats.norm.cdf(-2,0,1)

from scipy import stats
stats.norm.cdf(2,0,1)-stats.norm.cdf(-2,0,1)

Show expected output

0.9544997361036416

Interpretation: $P(-2<z<2)=0.9544997361036416$

Example: The midterm exam scores in a math class is normally distributed with a mean of 73 and a standard deviation of 5.

Find the probability that a randomly selected student scored more than 65 on the exam.

$P(X>65)=1 - P(X<65)$ =1-stats.norm.cdf(65,73,5)

Watch demo video

from scipy import stats
1-stats.norm.cdf(65,73,5)

Show expected output

0.945200708300442

Interpretation: $P(X>65)=0.945200708300442$

Example: The midterm exam scores in a math class is normally distributed with a mean of 73 and a standard deviation of 5.

Find the probability that a randomly selected student scored less than 85.

$P(X<85)$ =stats.norm.cdf(85,73,5)

from scipy import stats
stats.norm.cdf(85,73,5)

Show expected output

0.9918024640754038

Interpretation: $P(X<85)=0.9918024640754038$

Example: The midterm exam scores in a math class is normally distributed with a mean of 73 and a standard deviation of 5.

Find the 90th percentile (that is, find the score k that has 90% of the scores below k and 10% of the scores above k).

$90^{th} \text{percentile}$ = stats.norm.ppf(.9,73,5)

Watch demo video

from scipy import stats
stats.norm.ppf(.90,73,5)

Show expected output

79.407757827723

Interpretation: Ninety percent of scores are less than or equal to 79.407757827723

Example: The midterm exam scores in a math class is normally distributed with a mean of 73 and a standard deviation of 5.

Find the probability that a randomly selected student has a score between 70 and 90.

$P(70 < X < 90) = P(X<90) - P(X<70)$

stats.norm.cdf(90,73,5) - stats.norm.cdf(70,73,5)

Show expected output

0.7254099529842495

Interpretation: $P(70<X<90)=0.7254099529842495$

SciPy.Stats Summary of Probability Distributions

from scipy import stats

procedure	scipy.stats
Binomial pdf $P(X=x)$	`stats.binom.pmf(k,n,p)`
Binomial cdf $P(X\le x)$	`stats.binom.cdf(k,n,p)`
Normal cdf $P(X\le x)$	`stats.norm.cdf(x,mean,std)`
Inverse Normal or p-th Percentile	`stats.norm.ppf(p,mean,std)`
t distribution cdf $P(X \le x)$	`stats.t.cdf(t,df,0,1)`
Inverse t or p-th Percentile	`stats.t.ppf(p,df,0,1)`
chi-square cdf $P(\chi^2 \le ch2)$	`stats.chi2.cdf(ch2,k-1)`
Inverse chi-square or p-th Percentile	`stats.chi2.ppf(p,k-1)`

Adapted from Python for Introductory Statistics, by Simon Aman (Truman College, City Colleges of Chicago), licensed under CC BY 4.0. Changes were made: reformatted as an accessible XYZ web edition with live in-browser code cells. License: CC-BY-4.0.