Contributor: Lynn Ellis. Lesson ID: 13747
Learn about two measures of central tendency — mean and median — the measures of spread that go with them, and when to use each pair of numbers based on the shape of the data.
Look at this random shape.
Sometimes, the idea of a center is a little tricky.
The answer to these questions is that it depends.
Imagine that 11 people walk into the room. Your goal is to describe the collective heights of the people in a single number.
Here are their heights (in inches).
67, 64, 68, 69, 72, 67, 69, 68, 63, 66, 71
One way to describe the heights of these 11 people is with the mean.
The mean is the average of all of the numbers. We add all of these numbers together and divide by 11. Take a minute to do that yourself.
The mean of these heights is 67.6 inches (rounded to the nearest tenth).
This makes sense as a description of the heights of the group. The individual heights are all fairly close together (within 10 inches from shortest to tallest). 67.6 inches is close to the majority of the heights.
Another way to describe the heights of these 11 people is with the median.
The median is the center number when the numbers are in order from least to greatest. Put them in order and see which one is in the middle.
63, 64, 66, 67, 67, 68, 68, 69, 69, 71, 72
Now that they are in order, we can see that 68 inches is the middle number, so it is the median. Again, this makes sense as a description of the heights of the group. It is very close to the mean.
OK, now imagine that Tacko Fall, the tallest player in the NBA, walks into the room. Tacko Fall is 7'5" tall, so his height in inches is 89 inches.
Go ahead and calculate them for yourself.
The mean is now 69.4 inches. Tacko Fall's presence in the room increased the mean by almost 2 inches.
The median is the middle number. When you try to find the median yourself, you may have noticed no single number in the middle. When that happens, you take the average of the two middle numbers.
In this case, you would take the average of the 6th and 7th numbers in the ordered list. The median remains at 68. Tacko Fall's presence in the room does not change the median.
We call mean and median both measures of central tendency.
Return to the original question.
In the first case, with the 11 original people, both numbers provide good descriptions of the entire heights group.
In the second case, when Tacko Fall has joined the group, the median is a better description of the entire group because it is not overly influenced by one very tall man the way that he influences the mean.
Here is a visual of the first group of heights.
Notice that it is fairly symmetric.
While mean and median are excellent measures of central tendency for a symmetric data set, statisticians will normally use mean for symmetric data.
As you continue to study statistics, you will understand more about the power of using the mean for symmetric data.
Here is a visual of the second group of heights.
Notice that Tacko Fall's extreme height made the data skew right (the data trails out to the right.)
Median is the best measure of the central tendency for skewed data because the extreme individual data points do not influence it.
You have answered the first question about the center of the data, but identifying an entire set of data by a single number does not give enough information.
You could have a median or a mean of 68 when all the heights in the data set are 68 inches. You could also have a mean or a median of 68 when the shortest height is 50 inches, and the tallest height is 80 inches.
That is why you also need to consider how spread out the data is.
When you use mean for your measure of central tendency, you use standard deviation for your measure of spread. Standard deviation tells you the average distance from the mean of all data points.
When you use the median for your measure of central tendency, you use interquartile range for your measure of spread.
In the next section, you will practice identifying the best measures of central tendency and spread for various data sets.