# 2.2.8 - z-scores

Often we want to describe an observation in relation to the distribution of all observations. We can do this using a z-score. By converting observations to z-scores, we can compare observations from different distributions.

z-score

Distance between an individual score and the mean in standard deviation units; also known as a standardized score.

z-score
$$z=\dfrac{x - \overline{x}}{s}$$

$$x$$ = original data value
$$\overline{x}$$ = mean of the original distribution
$$s$$ = standard deviation of the original distribution

This equation could also be rewritten in terms of population values: $$z=\frac{x-\mu}{\sigma}$$

Later in the course, we will learn more about the z-distribution, which is a special case of the normal distribution.

z-distribution

A bell-shaped distribution with a mean of 0 and standard deviation of 1, also known as the standard normal distribution.

## Example: Milk Section

A study of 66,831 dairy cows found that the mean milk yield was 12.5 kg per milking with a standard deviation of 4.3 kg per milking (data from Berry, et al., 2013).

A cow produces 18.1 kg per milking. What is this cow’s z-score?

$$z=\frac{x-\overline{x}}{s} =\frac{18.1-12.5}{4.3}=1.302$$

This cow’s z-score is 1.302; her milk production was 1.302 standard deviations above the mean.

A cow produces 12.5 kg per milking. What is this cow’s z-score?

$$z=\frac{x-\overline{x}}{s} =\frac{12.5-12.5}{4.3}=0$$

This cow’s z-score is 0; her milk production was the same as the mean.

A cow produces 8 kg per milking. What is this cow’s z-score?

$$z=\frac{x-\overline{x}}{s} =\frac{8-12.5}{4.3}=-1.047$$

This cow’s z-score is -1.047; her milk production was 1.047 standard deviations below the mean.

Berry, D. P., Coyne, J., Boughlan, B., Burke, M., McCarthy, J., Enright, B., Cromie, A. R., McParland, S. (2013). Genetics of milking characteristics in dairy cows. Animal, 7(11), 1750-1758.

## Example: Comparing Test Scores Section

SAT-Math scores are normally distributed with a mean of 500 and standard deviation of 100. ACT-Math scores are normally distributed with a mean of 18 and standard deviation of 6. A student has taken both tests. They scored 600 on the SAT-Math and 22 on the ACT-Math. Which score is more impressive?

We can't directly compare the student's SAT and ACT scores because they are on different scales. We can convert these test scores into z-scores so we can directly compare them.

$$z_{SAT}=\frac{600-500}{100}=1$$

This student scored 1 standard deviation above the mean on the SAT-Math.

$$z_{ACT}=\frac{22-18}{6}=0.667$$

This student scored 0.667 standard deviations above the mean on the ACT-Math.

The student's SAT-Math score is more impressive than their ACT-Math score because the z-score is higher. They scored better than a larger proportion of other test takers on the SAT-Math.

## Practice: Computing z-scores Section

Type in the answer you think is correct - then click the 'Check' button to see how you did.

Click the right arrow to proceed to the next question.  When you have completed all of the questions you will see how many you got right and the correct answers.

For each question, compute the z-score.

Type in the answer you think is correct - then click the 'Check' button to see how you did.

Click the right arrow to proceed to the next question.  When you have completed all of the questions you will see how many you got right and the correct answers.

For each question, compute the z-score.