# Measuring “Position” – Percentiles, Quartiles, and Z-Scores

If Bob gets a 50% on his final history exam, that’s pretty bad right?

Well, that depends on what the average exam score was for his entire class. If the average score was a 90%, then Bob’s 50% is pretty bad. But if the average score was a 30%, then Bob’s 50% is actually pretty good.

In statistics, the position of a value relative to other values in the same dataset can be important.

The most common ways to measure the position of a value are by using percentiles, quartiles, and z-scores.

## Percentiles

Percentiles are values that divide a rank-ordered (values are arranged smallest to largest) dataset into 100 equal parts.

Percentiles are useful because they tell us what percentage of values in a dataset lie below a certain value.

For example, if your exam score is at the 80th percentile of your class, this means your score is better than 80% of your classmates.  Or if your score is at the 20th percentile, it means your score is only better than 20% of your classmates.

The 50th percentile corresponds to the median value in a dataset.

### Quartiles

Quartiles are values that divide a rank-ordered dataset into four equal parts. The values that divide each part are the first, second, and third quartiles, and are denoted as Q1, Q2, and Q3.

In the following dataset, Q1 = 3.5, Q2 = 6.5 , and Q3 = 9.5

Note: Q1 corresponds to the 25th percentile, Q2 corresponds to the 50th percentile, and Q3 corresponds to the 75th percentile.

## Z-Scores

Z-scores tell us how many standard deviations away a value is from the mean. We use the following formula to calculate a z-score:

z = (X – μ) / σ

where X is the value we are analyzing, μ is the mean, and σ is the standard deviation.

Consider the following dataset:

The mean score is 83.2. The standard deviation among the scores is 10.65

Suppose we want to find the z-score of Tyler’s final exam score. Here’s how we would calculate it:

z = (92 – 83.2) / 10.65 = 0.826

This indicates that Tyler’s exam score is 0.826 standard deviations greater than the mean exam score.

A z-score of 0 indicates a value that is equal to the mean.

A z-score below 0 indicates a value that is less than the mean.

A z-score above 0 indicates a value that is greater than the mean.