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Measures of central tendency
In everyday life, we use descriptions like “typical”
or “average” to describe events and behavior. Your grade
point average is an example of this. It is the typical grade you have
received for all the courses you have completed. In statistics, an average,
like your grade point average, is referred to as a mean. It is the sum
of all the numbers in a set divided by the number of cases that were
added together. Two other measures of central tendency are also used.
These are the mode and the median. The mode is the value or attribute
that occurs most frequently. The median is the middle value in a set
of scores that are arranged from low to high. Ordinal, interval, or
ratio measurements can be ordered in this way. Table 17.6
shows all three measures of central tendency for the same distribution.
What is the “typical” size of household in this sample?
There are actually three answers to this question. The easiest one to
find is the mode.
Table 17.6. Illustration of mean,
median and mode
Size
of Households
(X) |
|
f |
|
fX |
|
Cumulative
frequency |
| |
2 |
3 |
|
6 |
|
3 |
|
| |
3 |
- |
|
- |
|
3 |
|
| |
4 |
5 |
|
20 |
|
8 |
|
| |
5 |
12 |
|
60 |
|
20 |
|
| |
6 |
17 |
|
102 |
|
37 |
|
| |
7 |
← Mean |
|
21 |
|
147 |
|
58 |
← Median |
| Mode = |
8 |
34 |
|
272 |
|
92 |
|
| |
9 |
29 |
|
261 |
|
121 |
|
| |
10 |
9 |
|
90 |
|
130 |
|
| |
11 |
4 |
|
44 |
|
134 |
|
| |
12 |
3 |
|
36 |
|
137 |
|
| |
13 |
1 |
|
13 |
|
138 |
|
| |
14 or more |
2 |
|
28 |
|
140 |
|
| N
|
1401 |
140 |
1,079=
∑fX |
Mode
Looking at the frequency column, the largest value
is 34 and this occurred for the household size of 8. Eight, therefore,
is the mode for this distribution. In a percentage distribution, the
mode is the value with the largest percentage; in a bar graph, it is
the highest bar; in a frequency polygon, it is the value which has the
highest point. A distribution, however, may have more than one mode.
For example, look at the frequencies for codes for five
responses to a questionnaire item:
1— Strongly agree 13
2 — Agree 28
3 — Uncertain 19
4 — Disagree 28
5 — Strongly disagree 9
This distribution has two modes — for “agree”
and for “disagree”. Each had an f of 28. When there are
two modes a distribution is said to be bimodal. The mode becomes less
meaningful as a measure of central tendency as the number of modes increases.
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