Unit 8, Lesson 14
Comparing Mean and Median
Grade 6
14.1
Warm-up
Here are two dot plots. The first dot plot shows the heights of the first 22 U.S. presidents. The second dot plot shows the heights of the next 22 presidents.
<p>A dot plot from 162 to 194 by 2’s. Height in centimeters, labeled 1st-22nd presidents. The dot plot begins at 162, number of dots above each increment is 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 4, 1, 1, 0, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 0, 2, 1, 0, 0, 0 , 1, 0. </p>
<p>A dot plot from 162 to 194 by 2’s. Height in centimeters, labeled 23rd-44th presidents. The dot plot begins at 162, number of dots above each increment is 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 1, 3, 0, 3, 3, 0, 2, 0, 0, 3, 0, 0, 0, 1, 0, 0.</p>
Based on the two dot plots, decide if you agree or disagree with each of the following statements. Be prepared to explain your reasoning.
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The median height of the first 22 presidents is 178 centimeters.
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The mean height of the first 22 presidents is about 183 centimeters.
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A typical height for a president in the second group is about 182 centimeters.
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U.S. presidents have become taller over time.
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The heights of the first 22 presidents are more alike than the heights of the second 22 presidents.
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The MAD of the second data set is greater than the MAD of the first set.
Student Lesson Summary
Both the mean and the median are ways of measuring the center of a distribution. They tell us slightly different things, however.
The dot plot shows the number of stickers on 30 pages. The mean number of stickers is 21 (marked with a triangle). The median number of stickers is 20.5 (marked with a diamond).
<p>A dot plot for stickers on a page. The numbers 8 through 34, in increments of 2, are indicated. A diamond is indicated at 20.5 stickers and a triangle is indicated at 21 stickers. Data are as follows: 9 stickers, 1 dot; 10 stickers, 1 dot; 11 stickers, 2 dots; 12 stickers, 1 dot; 14 stickers, 1 dot; 16 stickers, 2 dots; 17 stickers, 1 dot; 18 stickers, 2 dots; 19 stickers, 1 dot; 20 stickers, 3 dots; 21 stickers, 1 dot; 22 stickers, 3 dots; 23 stickers, 1 dot; 24 stickers, 2 dots; 26 stickers, 2 dots; 28 stickers, 1 dot; 30 stickers, 1 dot; 32 stickers, 2 dots; 33 stickers, 1 dot; 34 stickers, 1 dot.</p>
The mean tells us that if the number of stickers were distributed so that each page has the same number, then each page would have 21. We could also think of 21 stickers as a balance point for the number of stickers on all of the pages in the set.
The median tells us that half of the pages have more than 20.5 stickers and half have less than 20.5 stickers. In this case, both the mean and the median could describe a typical number of stickers on a page because they are fairly close to each other and to most of the data points.
Here is a different set of 30 pages with stickers. It has the same mean as the first set, but the median is 23 stickers.
<p>A dot plot for “stickers on a page.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 stickers, and a diamond is indicated at 23 stickers. The data are as follows: 9 stickers, 1 dot; 10 stickers, 1 dot; 13 stickers, 1 dot; 14 stickers, 1 dot; 16 stickers, 1 dot; 17 stickers, 1 dot; 19 stickers, 1 dot; 20 stickers, 2 dots; 21 stickers, 2 dots; 22 stickers, 3 dots; 23 stickers, 6 dots; 24 stickers, 5 dots; 25 stickers, 4 dots; 26 stickers, 1 dot.</p>
In this case, the median is closer to where most of the data points are clustered and is therefore a better measure of center for this distribution. That is, it is a better description of the typical number of stickers on a page. The mean number of stickers is influenced (in this case, pulled down) by a handful of pages with very few stickers, so it is farther away from most data points.
In general, when a distribution is symmetrical or approximately symmetrical, the mean and median values are close. But when a distribution is not roughly symmetrical, the two values tend to be farther apart.
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