Measures of Position: Percentiles, Quartiles, Standard Scores

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

Percentiles

Assume that the elements in a dataset 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 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 dataset 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. The chart below shows a set of eight numbers divided into quartiles. Q1 is equal to 2.5; Q2, to 4.5; and Q3; to 6.5.

Note the relationship between quartiles and percentiles. Q₁ corresponds to P₂₅, Q₂ corresponds to P₅₀, and Q₃ corresponds to P₇₅. Q₂ is the median value in the set.

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Standard Scores (z-Scores)

A standard score (aka, a z-score) indicates how many standard deviations an data point is from the mean of a dataset. A standard score for a population can be calculated from the following formula.

z = (X - μ) / σ

where z is the z-score, X is the value of an individual data point from the population, μ is the mean of the population, and σ is the standard deviation of the population.

A standard score for a sample can be calculated from the following formula.

z = (x - x ) / s

where z is the z-score, x is the value of an individual data point from the sample, x is the sample mean, and s is the standard deviation of the sample.

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) + μ = ( 1.20 * 15) + 100 = 18 + 100 = 118

Problem 2

What is the 40th percentile of 10, 20, 25, 15, 30.

(A) 15
(B) 16
(C) 17
(D) 18
(E) 19

Solution

The correct answer is (C). To find the 40th percentile, we take the following steps:

1. Sort the data in ascending order.
   • Raw data: 10, 20, 25, 15, 30
   • Sorted data: 10, 15, 20, 25, 30
2. Calculate rank of desired percentile.

   Rank = (P/100) * (n + 1) = (40/100) * 6 = 2.4

   where P is the desired percentile (in this case, 40) and n is the number of data points (5).

Since the rank is 2.4 (not a whole number), interpolate between the second value (15) and the third value (20) to find the 40th percentile.

40th percentile = 15 + 0.4 * (20 - 15) = 17