2.2.8 - z-scores

Often we want to describe one observation in relation to the distribution of all observations. We can do this using a z-score.

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: \(z=\frac{x-\mu}{\sigma}\)

z-distribution

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

Distribution Plot

Example: Milk Section

Dairy Cow

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.

Example: BMI of Boys Section

A recent study examined the relationship between sedentary behavior and academic performance in youth. In a sample of 582 boys, the average weight was 49.8 kg with a standard deviation of 15.7 kg (data from Esteban-Cornejo, et al., 2015).

A boy in this sample weighs 73.35 kg. What is this boy's z-score?

\(z=\frac{x-\overline{x}}{s} =\frac{73.35-49.8}{15.7}=1.5\)

This boy's z-score is 1.5; he weighs 1.5 standard deviations above the mean.

 

A boy in this sample weighs 38.5 kg. What is this boy's z-score?

\(z=\frac{x-\overline{x}}{s} =\frac{38.5-49.8}{15.7}=-0.720\)

This boy's z score is -0.720; he weighs 0.720 standard deviations less than the mean.

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.

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.

Esteban-Cornejo, I., Martinez-Gomez, D., Sallis, J. F., Cabanas-Sanchez, V., Fernandez-Santos, J., Costro-Pinero, J., & Veiga, O. L. (2015). Objectively measured and self-reported leisure-time sedentary behavior and academic performance in youth: The UP&DOWN Study. Preventive Medicine, 77, 106-111.