The Normal Probability Distribution

4. Probability Distributions: Common 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 property hold:

$\mathbb{P}(j \leq X \leq k) = \mathbb{P}(X \leq k) - \mathbb{P}(X \leq j)$

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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

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 $\mathbb{P}(X \lt 81)$ ? Round your answer to $3$ decimal places.

$\mathbb{P}(X \lt 81)=0.103$

There are a number of different ways we can calculate $\mathbb{P}(X \lt 81)$ . 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:
$\mathbb{P}(X \lt 81)=\mathbb{P}(X \leq 81)$
To calculate $\mathbb{P}(X \leq 81)$ 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 $\mathbb{P}(X \leq 81)$ , run the following command:
$= \text{NORM.DIST}(81, 100, 15, \text{TRUE})$
This gives:
$\mathbb{P}(X \lt 81) = \mathbb{P}(X \leq 81) =0.103$

R Calculation

Since the normal distribution is a continuous distribution, it is true that:
$\mathbb{P}(X \lt 81)=\mathbb{P}(X \leq 81)$
To calculate $\mathbb{P}(X \leq 81)$ 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 $\mathbb{P}(X \leq 81)$ , run the following command:
$\text{pnorm}(q = 81, mean = 100, sd = 15, lower.tail = \text{TRUE})$
This gives:
$\mathbb{P}(X \lt 81) = \mathbb{P}(X \leq 81) =0.103$

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 Statistical Software

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 $N(150, 13)$ distribution.

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

$q=142.81$

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:
$=\text{NORM.INV}(0.29, 150, 13)$
This gives:
$q = 142.81$

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:
$\text{qnorm}(p = 0.29, mean = 150, sd = 13, lower.tail = \text{TRUE})$
This gives:
$q = 142.81$

New example