StATS: What are normal probabilities?

The normal distribution appears frequently in the study of Statistics. The normal distribution is a continuous distribution. Probabilities are represented by area.

Here is a graph of the normal distribution, which you should recognize as the classic bell curve. This graph was drawn using an Excel spreadsheet.

Examples of the normal distribution.

Diastolic blood pressure is normally distributed with a mean of 80 mm Hg and a standard deviation of 12. We can write this statement in shorthand as

or even more briefly as X is N(80,12).

At the bottom of this page is a graph of the bell curve associated with this particular distribution. Notice that it is centered around the value of the mean (80) and the ends of the curve are roughly three standard deviations away from the mean (44 and 116).

Blood cotinine levels in adults who smoke a pack a day is normally distributed with mean 260 and standard deviation 50. X is N(260,50)

At the bottom of this page is a graph of the bell curve associated with this particular distribution. Notice again that it is centered around the value of the mean (260) and the ends of the curve are roughly three standard deviations away from the mean (110 and 410).

Normal probabilities

To compute probabilities from a normal distribution, we need to subtract the mean and divide by the standard deviation. This calculation is known as "standardization". The standardized value is a measure of how many standard deviations that value is away from the mean.

We then compare the standardized value to a table of standard normal probabilities. Any statistics book should have this table, or you can refer to this Excel spreadsheet.

Diastolic BP is N(80,12). Compute P[X>100].

There is a 4.75% chance that an individual diastolic BP would be greater than 100.

Cotinine level is N(260,50). Compute P[X < 210].

There is a 15.87% probability that an individual cotinine level in a smoker will be less than 210.

Normal percentiles

The pth percentile is a value so that the probability of being smaller than that value is p. For example, the 25th percentile is a value so that the probability of being smaller is 0.25. If

then the pth percentile of the normal distribution is computed as

where Zp is the percentile of a standard normal distribution.

Diastolic blood pressure is N(80,12). Compute the 75th percentile.

Cotinine level is N(260,50). Compute the 10th percentile

Group exercises

1. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

a. Calculate the probability of finding a genius (IQ > 130).
b. Calculate the probability of average intelligence (between 90 and 110).
c. Calculate the 60th percentile of IQ.
d. Calculate the 99th percentile of IQ.

2. Serum albumin in healthy 20 year olds is normally distributed with a mean of 4.4 and a standard deviation of 0.2.

a. What is the probability that a healthy 20 year old patient will have a serum albumin of 3.8 or less?
b. A "reference range" is defined as the 2.5th to the 97.5th percentiles. Calculate a reference range for serum albumin in healthy 20 year olds.

3. The height of a 5 year old girl is normally distributed with mean of 106 cm and a standard deviation of 5 cm.

a. What is the probability that a randomly selected 5 year old girl has a height greater than 110 cm.
b. What is the probability for a height greater than 120 cm?
c. Calculate the 10th percentile for height.
d. Calculate the 3rd percentile for height.

This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Definitions, Category: Probability concepts.