# Measures of Position: Percentiles, Quartiles, z-Scores

Statisticians often talk about the **position** of a value,
relative to other values in a
set of observations.
The most common measures of position are percentiles, quartiles,
and standard scores (aka, z-scores).

## Percentiles

Assume that the
elements
in a data set are rank ordered from the
smallest to the largest. The values that divide a rank-ordered
set of elements into 100 equal parts are called
**percentiles**.

An element having a percentile rank of P_{i} would
have a greater value than i percent of all the elements
in the set. Thus, the observation at the 50th percentile would be
denoted P_{50}, and it would be greater than
50 percent of the observations in
the set. An observation at the 50th percentile would correspond to the
median value
in the set.

## Quartiles

**Quartiles** divide a rank-ordered data set into
four equal parts. The values that divide each part are called
the first, second, and third quartiles; and they are denoted by
Q_{1}, Q_{2}, and Q_{3}, respectively.

Note the relationship between quartiles and percentiles.
Q_{1} corresponds to P_{25},
Q_{2} corresponds to P_{50},
Q_{3} corresponds to P_{75}. Q_{2}
is the median value in the set.

## Standard Scores (z-Scores)

A **standard score** (aka, a **z-score**)
indicates how many
standard deviations
an element is from the mean. A standard score can be
calculated from the following formula.

z = (X - μ) / σ

where z is the z-score, X is the value of the element, μ is the mean of the population, and σ is the standard deviation.

Here is how to interpret z-scores.

- A z-score less than 0 represents an element less than the mean.
- A z-score greater than 0 represents an element greater than the mean.
- A z-score equal to 0 represents an element equal to the mean.
- A z-score equal to 1 represents an element that is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean; etc.
- A z-score equal to -1 represents an element that is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean; etc.

## Test Your Understanding

**Problem 1**

A national achievement test is administered annually to 3rd graders. The test has a mean score of 100 and a standard deviation of 15. If Jane's z-score is 1.20, what was her score on the test?

(A) 82

(B) 88

(C) 100

(D) 112

(E) 118

**Solution**

The correct answer is (E). From the z-score equation, we know

z = (X - μ) / σ

where z is the z-score, X is the value of the element, μ is the mean of the population, and σ is the standard deviation.

Solving for Jane's test score (X), we get

X = ( z * σ) + 100 = ( 1.20 * 15) + 100 = 18 + 100 = 118