What is the Standard Normal Distribution?

Learning Outcomes

  • Calculate and interpret z-scores

Recall: Distance

Distance is how far one number on a number line is from another number on a number line. Distance is positive. In statistics we find the distance from a data point to the mean when we are finding z-scores. Z-scores are described below. If you have a negative z-score, the negative is directional, meaning below the mean.

The standard normal distribution is a normal distribution of standardized values called z-scores. A z-score is measured in units of the standard deviation. For example, if the mean of a normal distribution is five and the standard deviation is two, the value 11 is three standard deviations above (or to the right of) the mean. The calculation is as follows:

x = μ + (z)(σ) = 5 + (3)(2) = 11

The z-score is three.

The mean for the standard normal distribution is zero, and the standard deviation is one. The transformation [latex]\displaystyle{z}=\frac{{x - \mu}}{{\sigma}}[/latex] produces the distribution Z ~ N(0, 1). The value x comes from a normal distribution with mean μ and standard deviation σ.

The following two videos give a description of what it means to have a data set that is “normally” distributed.

Z-Scores

If X is a normally distributed random variable and X ~ N(μ, σ), then the z-score is:

[latex]\displaystyle{z}=\frac{{x - \mu}}{{\sigma}}[/latex]

The z-score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z-scores, and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of zero.

Example

Suppose X ~ N(5, 6). This says that x is a normally distributed random variable with mean μ = 5 and standard deviation σ = 6. Suppose x = 17. Then:

[latex]\displaystyle{z}=\frac{{x - \mu}}{{\sigma}}[/latex]= [latex]\displaystyle{z}=\frac{{17-5}}{{6}}={2}[/latex]

This means that x = 17 is two standard deviations (2σ) above or to the right of the mean μ = 5. The standard deviation is σ = 6.

Notice that: 5 + (2)(6) = 17 (The pattern is μ + zσ = x)

Now suppose x = 1. Then: [latex]\displaystyle{z}=\frac{{x - \mu}}{{\sigma}}[/latex] = [latex]\displaystyle {z}=\frac{{1-5}}{{6}} = -{0.67}[/latex]

(rounded to two decimal places)

This means that x = 1 is 0.67 standard deviations (–0.67σ) below or to the left of the mean μ = 5. Notice that: 5 + (–0.67)(6) is approximately equal to one (This has the pattern μ + (–0.67)σ = 1)

Summarizing, when z is positive, x is above or to the right of μ and when zis negative, x is to the left of or below μ. Or, when z is positive, x is greater than μ, and when z is negative x is less than μ.

try it

What is the z-score of x, when x = 1 andX ~ N(12,3)?

 

Example

Some doctors believe that a person can lose five pounds, on the average, in a month by reducing his or her fat intake and by exercising consistently. Suppose weight loss has a normal distribution. Let X = the amount of weight lost(in pounds) by a person in a month. Use a standard deviation of two pounds. X ~ N(5, 2). Fill in the blanks.

  1. Suppose a person lost ten pounds in a month. The z-score when x = 10 pounds is z = 2.5 (verify). This z-score tells you that x = 10 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).
  2. Suppose a person gained three pounds (a negative weight loss). Then z = __________. This z-score tells you that x = –3 is ________ standard deviations to the __________ (right or left) of the mean.
  3. Suppose the random variables X and Y have the following normal distributions: X ~ N(5, 6) and Y ~ N(2, 1). If x = 17, then z = 2. (This was previously shown.) If y = 4, what is z?

The z-score for y = 4 is z = 2. This means that four is z = 2 standard deviations to the right of the mean. Therefore, x = 17 and y = 4 are both two (of their own) standard deviations to the right of their respective means.

The z-score allows us to compare data that are scaled differently. To understand the concept, suppose X ~ N(5, 6) represents weight gains for one group of people who are trying to gain weight in a six week period and Y ~ N(2, 1) measures the same weight gain for a second group of people. A negative weight gain would be a weight loss. Since x = 17 and y= 4 are each two standard deviations to the right of their means, they represent the same, standardized weight gain relative to their means.

Try It

Fill in the blanks.

Jerome averages 16 points a game with a standard deviation of four points. X ~N(16,4). Suppose Jerome scores ten points in a game. The z–score when x = 10 is –1.5. This score tells you that x = 10 is _____ standard deviations to the ______(right or left) of the mean______(What is the mean?).