Transcript Anatomy: Measures of Position

ANATOMY OF MEASURES OF POSITION
Five data values, including the Minimum value, First Quartile (Q1), Median (Q2), Third Quartile (Q3) and Maximum, comprise what is referred to as the FiveNumber Summary. Each component of this summary represents a measure of position within a set of data. The first step in calculating values of the five-number
summary is to order the data from lowest to highest.
Quartiles divide a data set that has been ordered from smallest to largest
into four sections, each containing 25% of the data values. A term more
familiar might be percentile. For example, the 25th percentile, 50th
percentile, or 75th percentile.
Minimum
Maximum
440 481 482 483 483 514 514 554 554 554 562 612 623 631 638 664 671 677 690 707
Q1 :
Median (Q2 ):
Q3 :
The first quartile (Q1) is the value such that
25% of the values are lower than it, and 75%
of the values are higher than it. To find this
value apply the formula (n+1)/4 to obtain the
POSITION of the first quartile. Here the
formula yields position 5.25: [(20+1)/4 =
5.25]. Determine the value ¼ of the way
between the 5th value (483) and the 6th value
(514) by determining the difference between
these two numbers (514 – 483 = 31);
multiplying the difference by .25 (31 * .25 =
7.75); and adding this value to the smaller
number (483 + 7.75 = 490.75 = Q1).
Recall that the median in this data set is
found using the formula (n+1)/2 to obtain
the POSITION of the median (here the
position is (20 + 1)/2 = 10.5). Determining
the value half way between the 10th and 11th
values yields a Median of 558: [(554 +
562)/2 = 558]. Another name for the
median, is the second quartile (Q2) . It is
the value in the data set such that 50% of
the values are lower than it, and 50% of the
values are higher than it.
The third quartile (Q3) is the value such that
75% of the values are lower than it, and 25% of
the values are higher than it. To find this value
apply the formula [3(n+1)/4] to obtain the
POSITION of the third quartile. Here the
formula yields position 15.75: [(3(20+1))/4 =
15.75]. Determine the value ¾ of the way
between the 15th value (638) and the 16th value
(664) by determining the difference between
these two numbers (664 - 638 = 26); multiplying
the difference by .75 (26 * .75 = 19.5); and
adding this value to the smaller number (638 +
19.5 = 657.5 = Q3)
Q
1

n 1
4
The Interquartile Range is the difference between
the third quartile, and the first quartile. Here, the
interquartile range is 657.5 - 490.75 = 166.75
Q
IQR  Q  Q
3
1
2

n 1
2
Q
3

3(n  1)
4
The Data: The data set comes from Dudley’s Doughnuts. It is the amount of
flour (in pounds) used on each of 20 consecutive days in December 1999. Note that
the data have been ordered from smallest to largest in value.