Often we want to compare values from different data sets. However, unless the datasets have the same location and spread, it is not possible to compare the results directly.

Exploration

Let's say Sam got $64$64 on his biology exam and $78$78 on his chemistry exam. On first glance, it would seem that he did better on his chemistry exam. However, the chemistry exam could have been much harder than the biology exam.

If we know the mean and standard deviation of the test results for both tests then we can calculate standard scores which might allow us to make a more meaningful comparison.

A standard score, also known as a $z$z-score, measures the difference between a measured value and the mean as a multiple of the standard deviation. In other words, it is the number of standard deviations from the mean to the measured value. We can use standard scores to compare measured values from different data sets in terms of performance within their respective data sets.

The formula to calculate the standard score $z$z is

$z=\frac{x-\mu}{\sigma}$z=x−μσ

where $x$x is the data value, $\mu$μ is the population mean for the data set, and $\sigma$σ is the standard deviation.

Going back to Sam's exam results, lets say, we find out the mean and standard deviation for all results in the chemistry and biology exams:

	Mean	Standard deviation
Chemistry	$58$58	$10$10
Biology	$52$52	$4$4

Then Sam's standard score for the chemistry exam is $z=\frac{78-58}{10}=2$z=78−5810=2. That is, Sam's result is $2$2 standard deviations above the mean.

Sam's standard score for biology is $z=\frac{64-52}{4}=3$z=64−524=3. So, in the biology test, Sam's result is $3$3 standard deviations above the mean.

Compared to the other students who sat the exam, Sam achieved a better result in the biology exam, even though the raw mark was lower.

Standard scores

A standard score (or $z$z-score) measures the difference between a value and the mean as a multiple of standard deviations.

$z=\frac{x-\mu}{\sigma}$z=x−μσ

A positive $z$z-score indicates the raw value was above the mean.
A $z$z-score of $0$0 indicates the raw value was equal to the mean.
A negative $z$z-score indicates the raw value was below the mean.

While standard scores are useful to compare results, we still need to take care comparing results. In the previous example, the comparison would not be valid if the groups of students doing both tests had greatly different abilities.

Practice questions

Question 1

A general ability test has a mean score of $100$100 and a standard deviation of $15$15.

If $Paul$ received a score of $102$102 in the test, what was his $z$z-score correct to two decimal places?
If $Georgia$ had a $z$z-score of $3.13$3.13, what was her score in the test, correct to the nearest integer?

Question 2

$Kathleen$ scored $83.4$83.4 in her $Biology$ exam, in which the mean score and standard deviation were $81$81 and $2$2 respectively. She also scored $60$60 in her $Geography$ exam, in which the mean score was $46$46 and the standard deviation was $4$4.

Find $Kathleen$ ’s $z$z-score in $Biology$ . Give your answer to one decimal place if needed.
Find $Kathleen$ ’s $z$z-score in $Geography$ . Give your answer to one decimal place if needed.
Which exam did $Kathleen$ do better in?
$Biology$
A
$Geography$
B

Question 3

If $Sean$ 's $z$z-score in a test is $-1.1$−1.1, the mean mark is $76%$76% and standard deviation is $3%$3%, what is his test score, $X$X?

Outcomes

2.1.6

use the number of deviations from the mean (standard scores) to describe deviations from the mean in normally distributed data sets

7.045 Calculating standard scores

Calculating standard scores