7.SP.3 ~ Compare data sets using centers and variability of data distributions.

Key points! *Use the mean and mean absolute deviation (MAD) to describe symmetric distributions of data. *Use the median and the interquartile range (IQR) to describe skewed distributions of data.

Let’s review some terms you should know: Box Plot: A box plot is a graphical representation of statistical data based on the minimum, first quartile, median, third quartile, and maximum. Median: a measure of center; the middle number after the numbers have been ordered from low to high. Inter-quartile Range: the difference between the first and the third quartiles which comprise 50% of the data ~ a measure of variation

Mean: The "Mean" is computed by adding all of the numbers in the data together and dividing by the number elements contained in the data set. It is a measure of center. Example : Data Set = 2, 5, 9, 3, 5, 4, 7 Number of Elements in Data Set = 7 Mean = ( 2 + 5 + 9 + 7 + 5 + 4 + 3 ) / 7 = 5 Mean Absolute Deviation (MAD):  is the average distance between each data value and themean. It is a way to describe variation in a data set.

Skewed data ~

Skewed data ~

Let’s look at a problem: Jason wonders about the weights of his two favorite sports teams and if there is an overlap between them. In other words, would they have a lot of weights in common?

The weights of both teams.

The best way to compare this data is using the median and the IQR. Why?

Think about the difference between centers as a multiple of the IQRs. The difference between the medians ~ 200 – 175 = 25

The Inter-quartile ranges: Baseball ~ 217.5 – 190 = 27.5 Soccer ~ 185 – 154 = 31

The difference between medians: 25 and the inter-quartile ranges are both close to 30 Therefore ~ the difference in the medians ÷ the IQR is less than one ~ 25/30

But, what if the difference in the medians ÷ the IQRs was 2 or more? What does this mean? It means that the two types of players have much of their weights in common. But, what if the difference in the medians ÷ the IQRs was 2 or more?

It would mean that the two types of players have much less of their weights in common. Now, you try one!

So, what does this mean?

They have little data in common. This means the difference in the medians are more than 3 times the IQR, which shows there’s very little overlap in the two data sets. They have little data in common.

Remember ~ for the most number of values in common, you want the difference in the medians ÷ the IQR to be less than 2.

Summary: To compare data sets using the median and the inter-quartile range (IQR), you must find the difference between the medians of the data and divide it by the IQR of the data. If the quotient is less than 2, there is much data overlap. If the quotient is greater than 2, there is little data overlap.

Comparing data sets using the mean and MAD ~

Now you try one! 1.6 Express the difference in centers as a multiple of the MAD.

Let’s try another! The difference in inches between the mean heights of the poodles is _____ times the variation for either type. 5

Summary: To compare data sets using the mean and the mean absolute deviation (MAD), you must find the difference between the means of the data and divide it by the MAD of the data. If the quotient is less than 2, there is much data overlap. If the quotient is greater than 2, there is little data overlap.

Exit ticket: What is the median for each data set? b) What is the interquartile range of each data set? c) Express the distance between the medians as a multiple of the greater interquartile range.