# Measures of Position

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 quartiles, percentiles, and standard 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.
- If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; and about 99% have a z-score between -3 and 3.

## Test Your Understanding of This Lesson

**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

(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