z Scores

Definition

The standard score (or z score) is the number of standard deviations that a value x is above or below the mean.

For a sample with a mean of and standard deviation of s, the z score for x is given by

For populations z scores are computed using

Example 1

Suppose a sample of adults had a mean height of 69 inches with a standard deviation of 3 inches. Find standard scores (or z scores) for individuals in this sample with heights of 72 inches, 64.5 inches, and 79.5 inches.

Solution .

For the first person the z score is z = (72-69)/3 = 3/3 = 1. This z score says that the individual is one standard deviation above the mean.

For the second person the z score is (64.5-69)/3 = (-4.50)/3 = -1.5. This person's height is 1.5 standard deviations below the average.

The third person has a z score of (79.5-69)/3 = 10.5/3 = 3.5 which means that this person is 3.5 standard deviations above the mean.

Recall that the empirical rule of the previous section says that if the data has a bell-shaped distribution then about 95% of all scores fall within 2 standard deviations of the mean and about 99.7% of the scores fall within 3 standard deviations of the mean. We can now rephrase this in terms of z scores.

Emprical Rule (using z scores)

For data that has a bell-shaped distribution,

about 68% of the data has z scores with

about 95% of the data has z scores with

about 99.7% of the data has z scores with

Would you expect to find many people like the third person in example one in the general population?

As the next example shows, z scores make it possible to make comparisons between different populations.

Example 2

Battery A lasted for 37.8 months and is from a population that has an average life of 36 months with a standard deviation of 1.3 months. Battery B lasted for 38.1 months and is from a population with a mean of 36 months and a standard deviation of 1.9 months. Compute the z score for each battery.

Solution .

For A, z = (37.8 - 36)/1.3 = 1.8/1.3 = 1.38

For B, z = (38.1 - 36)/1.9 = 2.1/1.9 = 1.11

Because it has a larger z score, battery A is relatively better than battery B.

Example 3

Convert the data 4.6, 7, 8.3, 9.2, 9.8, 10.3, 11.5, 13.4, 15.9 to z scores.

Solution .

We must first find the sample mean and standard deviation.

The following table shows the details of the computations involved in obtaining the mean of 10 and standard deviation of 3.36. Note that .4E-1 stands for 0.04

       x    ((x - mean)^2)  

--------------------------------

       4.6    29.2              

       7.     9.00              

       8.3    2.89              

       9.2    .64               

       9.8    .4E-1             

       10.3   .9E-1             

       11.5   2.25              

       13.4   11.6              

       15.9   34.8              

SUM    90.0   90.5              

COUNT  9.                       

       10.0   11.3              

       MEAN   VARIANCE          

              3.36              

              STANDARD DEVIATION

Using the sample mean of 10 and sample standard deviation of 3.36, we convert 4.6 to its z score by finding (4.6 - 10)/3.36 = (-5.4)/(3.36) = -1.61. After converting the other data values to standard or z scores we obtain the following table.

sorted data    z-scores  

---------------------------

 4.6           -1.61       

 7             -.893       

 8.3           -.506       

 9.2           -.238       

 9.8           -0.595E-1   

 10.3           0.893E-1   

 11.5           .446       

 13.4           1.01       

 15.9           1.76