Lauretta J. Fox
When statisticians study a group of measurements, they try to determine which measure is most representative of the group. The score about which most of the other scores tend to cluster is a
measure
of
central
tendency
. Three measures of central tendency are the mode, the median and the mean.
The
mode
of a set of numbers is the element that appears most frequently in the set. There can be more than one mode in a set of numbers. A set that has two modes is
bimodal
, and one that has three modes is
trimodal
. If no element of a set appears more often than any other element, the set has no mode. The mode is an important measure for business people. It tells them what items are most popular with consumers.
Example
1
:
Find the mode of the following set of numbers: 34, 26, 30, 34, 28, 32, 32, 34, 33, 31, 33, 30.
Solution
:
|
Element
|
Frequency
|
|
26
|
1
|
|
28
|
1
|
|
30
|
2
|
|
31
|
1
|
|
32
|
2
|
|
33
|
2
|
|
34
|
3
|
The number 34 occurs most frequently, hence 34 is the mode of the set.
Example
2
:
Find the mode of the following set of numbers: 13 17, 14, 20, 18
Solution
:
|
Element
|
Frequency
|
|
13
|
1
|
|
17
|
1
|
|
14
|
1
|
|
20
|
1
|
|
18
|
1
|
No number appears more than any other number in the set. The set has no mode.
Example
3
:
Find the mode of the following set of numbers: 1, 2, 2, 3, 4, 4, 5
Solution
:
|
Element
|
Frequency
|
|
1
|
1
|
|
2
|
2
|
|
3
|
14
|
|
4
|
2
|
|
5
|
1
|
The numbers 2 and 4 each appear twice. The set has two modes: 2 and 4.
Another measure of central tendency is the
median
. When the elements of a set of numbers have been arranged in ascending order, the number in the middle of the set is the median of the set. The median divides the set of data into two equal parts. On a cumulative frequency polygon the median is the 50th percentile. To determine which element of a set is the middle number, use the following formula:
Middle Number = (Total Number of Elements + 1); = 2
If the set contains an even number of elements, the median is the average of the two middle numbers.
Example
1
:
The weights of nine children are as follows: 99, 98, 73, 81, 79, 86, 90, 94, 71. Find the median weight.
Solution
:
Arrange the weights in order from lowest to highest: 71, 73, 79, 81, 86, 90, 94, 98, 99 (9 + 1) $dv$ 2 = 10 $dv$ 2 = 5 The fifth number of the set is the middle number. The median weight is 86.
Example
2
:
Ten students received the following scores on an examination: 96, 68, 78, 82, 87, 74, 80, 70, 86, 84. Find the median score.
Solution
:
Arrange the scores in ascending order: 68, 70, 74, 78, 80, 82, 84, 86, 87, 96.
(10 + 1) Ö 2 = 11 Ö 2 = 5.5
The two middle numbers of the set are the fifth and sixth numbers: 80 and 82.
(80 + 82) Ö 2 = 162 Ö 2 = 81
The median score is 81.
A third, and most widely used, measure of central tendency is the
arithmetic
mean
. The arithmetic mean is the average of a set of numbers. It is usually denoted by the symbol x. To calculate the arithmetic mean of a set of numbers, add the members of the set and divide the sum by the number of items in the set.
Example
:
Find the arithmetic mean of the following set of numbers: 25, 15, 20, 20, 10.
Solution
:
(25 + 15 + 20 + 30 + 10) Ö 5 = 100 Ö 5 = 20
The arithmetic mean of the set is 20.
Sometimes an item appears more than once in a set of measures. To find the arithmetic mean of a set of measures when some items occur several times, multiply each item in the set by Its frequency and divide the sum of these products by the total number of items in the set.
Example
:
Find the arithmetic mean of the following numbers: 28, 24, 22, 24, 26, 26, 22, 24, 22, 28, 30, 24.
Solution
:
|
Item
|
Frequency
|
Product
|
|
22
|
3
|
66
|
|
24
|
4
|
96
|
|
26
|
2
|
52
|
|
28
|
2
|
56
|
|
30
|
1
|
30
|
Sum of Products = 66 + 96 + 52 + 56 + 30 = 300
Total Number of Items = 3 + 4 + 2 + 2 + 1 = 12
Sum of Products $dv$ Total # of Items = 300 $dv$ 12 = 25
The arithmetic mean is 25.
When the data have been arranged in intervals in a frequency distribution, the arithmetic mean is obtained in the following manner:
-
1.) Multiply the midpoint of each interval by the frequency of the interval.
-
2.) Find the sum of the products obtained in step 1.
-
3.) Divide the sum obtained in step 2 by the total number of items in the distribution.
The formula used to find the arithmetic mean is:
n
x = 1/ni å1 xifi
|
x ; arithmetic mean
|
xi = midpoint of the interval
|
|
n = number of items in
|
fi = frequency of the interval
|
|
the distribution
|
= sum
|
Example
:
Find the arithmetic mean for the following distribution:
|
Scores
|
Midpoint
|
Frequency
|
xifi
|
|
96-100
|
98
|
3
|
294
|
|
91-95
|
93
|
4
|
372
|
|
86-90
|
88
|
8
|
684
|
|
81-85
|
83
|
7
|
581
|
|
76-80
|
78
|
6
|
468
|
|
71-75
|
73
|
5
|
365
|
|
66-70
|
68
|
3
|
204
|
|
61-65
|
63
|
2
|
126
|
|
56-60
|
58
|
2
|
116
|
Solution
n = 3 + 4 + 8+ 7+ 6+ 5+ 3+ 2 + 2 = 40
40
åxifi= 294 + 372 + 684 + 581 + 468 + 365+ i= 1 204 + 126 + 116 = 3210
40
x = 1/n å xifi 1/40 x 3210 = 3210/40 = 80.25
i = 1
The arithmetic mean of the distribution is 80.25.
Exercises
:
-
1.) Ten employees of a department store earn the following weekly wages: $200, $150, $160, $125, $160, $150, $180, $130, $170 $150
a.) Find the average weekly income.
b.) What is the median wage?
c.) Find the mode.
-
2.) Write mean, median, or mode to complete the sentence.
|
a.) 7, 13, 8, 5, 9, 12.
|
The
____
is 9.
|
|
b.) 6, 2, 4, 7, 6, 3.
|
The
____
is 6.
|
|
c.) 18, 10, 21, 17, 12.
|
The
____
is 17.
|
|
d.) 8, 3, 9, 4, 10, 14.
|
The
____
is 8.
|
|
e.) 13, 11, 8, 15, 9, 10.
|
The
____
is 10.5.
|
-
3.) Find the mean, the median and the mode for each set of numbers.
a.) 72, 68, 56, 65, 72, 56, 68.
b.) 13, 19, 12, 18, 24, 10.
c.) 125, 132, 120, 118, 128, 126, 120.
d.) 8, 4, 6, 4, 10, 4, 10.
-
4.) Find the arithmetic mean of the following numbers:
|
Number
|
Frequency
|
|
32
|
4
|
|
36
|
2
|
|
38
|
6
|
|
40
|
8
|
-
5.) The salaries of thirty people are listed below.
|
$12,500
|
$23,900
|
$18,750
|
$24,000
|
$$14,000
|
|
$18,750
|
$11,570
|
$25,000
|
$ 9,200
|
$15,000
|
|
$24,000
|
$22,000
|
$20,500
|
$12,500
|
$17,300
|
|
$10,980
|
$15,550
|
$18,750
|
$18,000
|
$16,200
|
|
$ 8,750
|
$12,500
|
$10 980
|
$13,000
|
$19,850
|
|
$32,000
|
$13,000
|
$22,000
|
$35,000
|
$21,000
|
a.) Arrange the salaries in intervals and make a frequency table for the set of data.
b.) What is the mode of the salaries?
c.) What is the median salary?
d.) What is the mean salary?
