In this lesson, students review measures of center and variability from grade 6. They also work at deciding whether or not two distributions are very different from each other (MP3). This lesson introduces the idea of expressing the difference between the centers of two distributions as a multiple of a measure of variability as a way to help students make this determination (MP2). For the problems in this lesson, the populations under study are small and the data for the entire populations are known. In future lessons, students will revisit this calculation as a way to decide whether there is a meaningful difference between two populations given data from only a sample of each population.
- Calculate the mean and mean absolute deviation for a data set, and interpret (orally) these measures.
- Compare and contrast (orally and in writing) populations represented on dot plots in terms of their shape, center, spread, and visual overlap.
- Justify (in writing) whether two populations are “very different” based on the difference in their means expressed as a multiple of the mean absolute deviation.
Let’s compare two groups.
- I can calculate the difference between two means as a multiple of the mean absolute deviation.
- When looking at a pair of dot plots, I can determine whether the distributions are very different or have a lot of overlap.
The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.
To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. \(7+9+12+13+14=55\) and \(55 \div 5 = 11\).
mean absolute deviation (MAD)
The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11.
To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are.
\(4+2+1+2+3=12\) and \(12 \div 5 = 2.4\)
The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order.
For the data set 7, 9, 12, 13, 14, the median is 12.
For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. \(6+8=14\) and \(14 \div 2 = 7\).