Standard Normal Distribution
KEY

Even if two variables can both be described by a normal distribution, it can also be difficult to compare these variables if their mean and or standard deviations are different, for example heights in centimeters and weights in kilograms. As the empirical rule suggests, all normal distributions share many properties. In fact, all normal distributions are the same if we measure in units of size from the mean μ as the center. To solve this conflict, we can create a new variable, z, which defines a standardized value that can apply to all normal distributions.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. This can be expressed as Z. Our new variable, z, gives a measure of how far the variable is from the mean (x−μ)then "normalizes" it by dividing by the standard deviation (σ). This new variable gives us a way of comparing different variables. The z-value tells us how many standard deviations or "how many sigmas" the variable is from its respective mean. In essence, we are just "rescaling" the original distribution to fit the properties of a standard normal distribution.

Example 1. The percentages scored in an exam are normally distributed with a mean of 70% and a standard deviation of 10%.

  1. Victoria scored 90% on the exam. Calculate her z-score and explain what it means.

A z-score of 2 means that Victoria’s grade was 2 standard deviations above the mean.

  1. Ethan scored 55% on the exam. Calculate his z-score and explain what it means.

A z-score of -1.5 means that Ethan’s grade was 1.5 standard deviations below the mean.

Example 2.The table shows Emma’s midyear exam results. The exam results for each subject are normally distributed with the mean and standard deviation shown in the table.

Subject / Emma’s grade / /
English / 48 / 40 / 4.4
Mandarin / 81 / 60 / 9
Geography / 84 / 55 / 18
Biology / 68 / 50 / 20
Algebra / 84 / 50 / 15
  1. Find the z-score for each of Emma’s subjects.

Subject / z-score
English /
Mandarin /
Geography /
Biology /
Algebra /
  1. Relatively speaking, in what subject did Emma get the “best” grade? The “worst”?

Emma’s “best” score was in Algebra, and her “worst” score was in biology.

Using the Normal Distribution

The normal distribution is useful when finding an unknown mean or standard deviation for a normal distribution. You may be given cumulative probabilities and be asked to find the mean (if is known) or the standard deviation (if is known).

Example 3. Suppose X is normally distributed with a mean of 40, and P(X 45) = 0.9. Find the standard deviation.

We will need to convert our data to a standard normal curve to figure out the standard deviation.

Original distribution

is unknown
/ Standard distribution



z = invNorm(0.9, 0, 1) = 1.28155 = 1.28

Since z = 1.28, we know that 45 is 1.28 standard deviations away from the mean of 40. To find the standard deviation, we will use the formula

Example 4. Sacks of potatoes with a mean weight of 5 kg are packed by an automatic loader. In a test, it was found that 15% of the bags were over 5.2 kg. Use this information to find the standard deviation of the process.

Original distribution

is unknown
/ Standard distribution



invNorm(0.85, 0, 1) = 1.036433 = 1.04

Since z = 1.04, we know that 5.2 is 1.04 standard deviations away from the mean of 5. To find the standard deviation, we will use the formula