Measures of Position
Statisticians often talk about the position of a value,
relative to other values in a
set of data.
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. The
chart below shows a set of four numbers divided into quartiles.
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.
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