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 Pi 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 P50, 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 Q1, Q2, and Q3, respectively.

Note the relationship between quartiles and percentiles. Q1 corresponds to P25, Q2 corresponds to P50, Q3 corresponds to P75. Q2 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 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

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