The Normal Probability Distribution

Chapter 4. Probability Distributions: Continuous Probability Distributions

The Normal Probability Distribution

Normal Probability Distribution

If #X# is a value sampled from a population having a #N(\mu,\sigma)# distribution, then #X# is a random variable having a #N(\mu, \sigma)# probability distribution.

Then for any number #k#:

#\mathbb{P}(X \leq k)# is the area to the left of #k#
#\mathbb{P}(X \gt k)# is the area to the right of #k#

Important Properties

For any continuous probability distribution, the following properties always hold:

#\mathbb{P}(X=k)=0#
#\mathbb{P}(X\leq k)=\mathbb{P}(X\lt k)#
#\mathbb{P}(X\geq k)=\mathbb{P}(X\gt k)#
#\mathbb{P}(X\gt k)= 1 - \mathbb{P}(X \leq k)#
#\mathbb{P}(X\lt k)= 1 - \mathbb{P}(X \geq k)#

Additionally, for any #2# numbers #j# and #k# (with #j<k#), the following properties hold:

#\mathbb{P}(j\lt X\lt k)=\mathbb{P}(j\leq X \lt k)=\mathbb{P}(j \lt X\leq k)=\mathbb{P}(j\leq X\leq k)#
#\mathbb{P}(k\gt X \gt j)=\mathbb{P}(k\geq X \gt j)=\mathbb{P}(k\gt X\geq j)=\mathbb{P}(k \geq X\geq j)#
#\mathbb{P}(j \leq X \leq k) = \mathbb{P}(X \leq k) - \mathbb{P}(X \leq j)#

Computation of Normal Probabilities with Statistical Software

Let #X# be a normal random variable with parameters #\mu# and #\sigma#.

To calculate cumulative probabilities for a normal distribution in Excel, make use of the following function:

NORM.DIST(x, mean, standard_dev, cumulative)

x: The value at which you wish to evaluate the distribution function.

mean: The mean of the distribution.

standard_dev: The standard deviation of the distribution.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution function, #\mathbb{P}(X \leq x)#

FALSE - uses the probability density function, #\mathbb{P}(X = x)#

To calculate cumulative probabilities for a normal distribution in R, make use of the following function:

pnorm(q, mean, sd, lower.tail)

q: The value at which you wish to evaluate the distribution function.

mean: The mean of the distribution.

sd: The standard deviation of the distribution.

lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

IQ scores have a #N(100,15)# distribution.

Choose someone at random from the population and let #X# be their IQ score.

What is #\mathbb{P}(X \lt 89)#? Round your answer to #3# decimal places.

#\mathbb{P}(X \lt 89)=0.232#

There are a number of different ways we can calculate #\mathbb{P}(X \lt 89)#. Click on one of the panels to toggle a specific solution.

Excel Calculation

Since the normal distribution is a continuous distribution, it is true that:
\[\mathbb{P}(X \lt 89)=\mathbb{P}(X \leq 89)\]
To calculate #\mathbb{P}(X \leq 89)# in Excel, make use of the following function:

NORM.DIST(x, mean, standard_dev, cumulative)

x: The value at which you wish to evaluate the distribution function.

mean: The mean of the distribution.

standard_dev: The standard deviation of the distribution.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution function, #\mathbb{P}(X \leq x)#

FALSE - uses the probability density function

Thus, to calculate #\mathbb{P}(X \leq 89)#, run the following command:
\[= \text{NORM.DIST}(89, 100, 15, \text{TRUE})\]
This gives:
\[\mathbb{P}(X \lt 89) = \mathbb{P}(X \leq 89) =0.232\]

R Calculation

Since the normal distribution is a continuous distribution, it is true that:
\[\mathbb{P}(X \lt 89)=\mathbb{P}(X \leq 89)\]
To calculate #\mathbb{P}(X \leq 89)# in R, make use of the following function:

pnorm(q, mean, sd, lower.tail)

q: The value at which you wish to evaluate the distribution function.

mean: The mean of the distribution.

sd: The standard deviation of the distribution.

lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

Thus, to calculate #\mathbb{P}(X \leq 89)#, run the following command:
\[\text{pnorm}(q = 89, mean = 100, sd = 15, lower.tail = \text{TRUE})\]
This gives:
\[\mathbb{P}(X \lt 89) = \mathbb{P}(X \leq 89) =0.232\]

New example

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pth Quantile

The #\boldsymbol{p^{th}}# quantile of a probability distribution is the number #q# such that #\mathbb{P}(X\leq q)=p#.

Finding the #p^{th}# quantile for a given #p# in #(0,1)# is the inverse of finding a probability.

Finding the pth Quantile with Excel

Let #X# be a normal random variable with parameters #\mu# and #\sigma#.

To calculate the number #q# such that #\mathbb{P}(X \leq q)=p# in Excel, make use of the following function:

NORM.INV(probability, mean, standard_dev)

probability: A probability corresponding to the normal distribution.

mean: The mean of the distribution.

standard_dev: The standard deviation of the distribution.

To calculate the number #q# such that #\mathbb{P}(X \leq q)=p# in R, make use of the following function:

qnorm(p, mean, sd, lower.tail)

p: A probability corresponding to the normal distribution.

mean: The mean of the distribution.

sd: The standard deviation of the distribution.

lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

Suppose #X# has a #N(150, 18)# distribution.

Find the number #q# such that #\mathbb{P}(X \leq q) = 0.29#. Round your answer to #2# decimal places.

#q=140.04#

There are a number of different ways we can calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.29#. Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.29# in Excel, make use of the following function:

NORM.INV(probability, mean, standard_dev)

probability: A probability corresponding to the normal distribution.

mean: The mean of the distribution.

standard_dev: The standard deviation of the distribution.

Thus, to calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.29#, run the following command:
\[=\text{NORM.INV}(0.29, 150, 18)\]
This gives:
\[q = 140.04\]

R Calculation

To calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.29# in R, make use of the following function:

qnorm(p, mean, sd, lower.tail)

p: A probability corresponding to the normal distribution.

mean: The mean of the distribution.

sd: The standard deviation of the distribution.

lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

Thus, to calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.29#, run the following command:
\[\text{qnorm}(p = 0.29, mean = 150, sd = 18, lower.tail = \text{TRUE})\]
This gives:
\[q = 140.04\]

New example