A set of data can contain many numbers. To help understand the data, you can let one number describe the set of data. This number is called a measure of central tendency because it represents the center, or middle, of the data. The most commonly used measures of central tendency are the mean, median, and mode.
The table below shows how much snack food Americans consume.
Example
1. Find the mean of the snack food data.
Alternative Solutions :
First, find the sum of the amounts consumed. Then
divide by the number of items of data. In this case there are 10 items of data.
Notice that the amount of potato chips consumed is far
greater than the amounts of other snacks consumed. Because the mean is an
average of several numbers, a single number that is so much greater or less
than the others can affect the mean a great deal. In such cases, the mean
becomes less representative of the values in a set of data.
The median is another measure of central tendency.
Example
2. Find
the median of the snack food data.
Alternative Solutions :
Arrange
the numbers in order from least to greatest.
Since
there is an even number of data items, the median is the mean of the two middle
values, 1.5 and 1.2. If there were an odd number of data items, the middle one
would be the median.
The median
is 1.35 pounds. The number of values that are greater than the median is the
same as the number of values that are less than the median.
A third measure of central tendency is the mode.
Sometimes a set of data has only one mode. In other cases,
every item in a data set occurs the same number of times. When this happens,
the set has no mode.
Example
Food Link
3.
Find
the mode of the snack food data.
Alternative Solutions :
Look
for the number that occurs most often.
0.4 0.4 0.6 0.8
1.2 1.5 1.5
2.5 4.9 6.7
In
this set, 0.4 and 1.5 each appear twice. So, the set of data has two modes, 0.4
pound and 1.5 pounds.
The snack food data has a mean of 2.05 pounds, a median of
1.35 pounds, and two modes of 0.4 pound and 1.5 pounds. As you can see, the
mean, median, and mode may not be the same value.
Measures of central tendency may not give an accurate
description of a set of data. Often, measures of variation are used to describe
the distribution of the data. A common measure of variation is the range.
To find the range of the snack food data, subtract the least
value of the data set from the greatest value.
The greatest value is 6.7.
The least value is 0.4.
So, the range of the data is 6.7 – 0.4 or 6.3.
Example
4. Find
the range of the data set {19, 21, 18, 17, 18, 22, 46}.
Alternative Solutions :
The
greatest value is 46. The least value is 17.
So, the
range is 46 – 17 or 29.
Example
Meteorology Link
5. Suppose
the temperature in a building is always 10°F cooler than the temperature
outside. The average outside temperature for two weeks is given below. Find the
mean, median, mode, and range for both the inside and outside temperatures.
Compare these values.
Alternative Solutions :
Note that
the mean, mode, and median of the outside and inside temperatures differ by
10°F, while the range is the same.
Data can be classified in several ways. Suppose a person
tracks the number of men and women who walk into a theater. These data are univariate because
there is only one variable being measured, gender. The data are also categorical because
they fit into one of two categories, man or woman. If the person also asks each
theatergoer’s age, the data would be bivariate because there are two quantities
being measured, gender and age. The age data are classified as measurement because
there are several values for age rather than just a few categories.
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