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In this lesson you will learn to represent data in multiple ways and compute ways to summarize that data. You will also explore potential sources of bias.
In this section, we’ll explore various aspects of data, from collection to representation and analysis. We won’t do it in that order, though, because it’s just more meaningful to first look at data and how it’s represented. Then you will be able to look at data more critically. That is, you can ask yourself questions about how the data was collected and whether it truly represents what its author says it does.
A distribution is an arrangement of values that gives information about the frequency with which they occur. A histogram is one representation of a distribution. A frequency table is another.
The center of a distribution of values is simply the middle of that distribution. A measure of central tendency is a way of defining the center of a distribution. Perhaps this sounds more complicated than it really is. Just think of a measure of central tendency as one way to summarize a set of (data) values.
In this section we’ll define and compute the three most common measures of central tendency: the mean, the median, and the mode.
The range is another measure that gives information about a set of data, but it is not a measure of central tendency. That’s because it doesn’t define the center of the distribution of the data. It’s actually called a measure of dispersion, because it describes the way that the data are dispersed.
The mean of a set of data is the arithmetic average. That’s the one you’re used to computing. It’s not called the average in statistics, because there are different kinds of averages.
To find the mean, simply add up the values in a set of data and divide by the number of values in the set.
The median of a set of data is the value in the middle of the set. To find the median, order the values in the set from least to greatest (or from greatest to least).
The mode of a set of data is the value that occurs most often. The mode of the set {4, 5, 3, 5, 4, 2, 1, 2, 4} is four, because four occurs more often than any other number in the set.
Unfortunately, it’s not as simple as it sounds. Sometimes a set may have more than one mode or no mode at all.
So how many modes are in the set {5, 4, 2, 4, 5, 2}? The answer is three. Every value in the set is also a mode of the set. Take some time to convince yourself that this is true.
The range of a set of data is the difference between the greatest and least values in the set. That makes the range positive. To find the range of a set of data, subtract the least value in the set from the greatest value.
You can see that the range gives a pretty poor summary of a set of data, because it relies on only two values in the set. What measure would be more meaningful for a set of salaries that included yours and Bill Gates’s?
For the set below, which measure is least?
{1, 8, 7, 3, 7, 6, 5, 6, 2}
Choice A is the correct answer. The mean of this set is equal to the sum of the values in the set, forty-five, divided by the number of values in the set, nine. So the mean is five. The median of the set is the middle number when the set is ordered. The ordered set is {1, 2, 3, 5, 6, 6, 7, 7, 8}, which has a median of six. The mode of the set is the value that occurs most often. The set has two modes: six and seven. The range of the set is the difference between the greatest and least values in the set. The range is eight minus one, or seven. The least of these measures is the mean, which is five.