Comparing Center & Variability Measurements of Two Data Sets

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  • 0:02 Two Score Sets
  • 0:47 Measures of Center
  • 3:41 Measures of Variability
  • 6:03 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Popelka-Brown
We'll learn how to compare two data sets based on their measures of center and variability. The measures of center include mean, median, and mode, and we'll use box plots to explore range and interquartile range.

Two Score Sets

You may not know this, but us teachers can be very competitive. Ms. Mathews is a colleague, and we compare student test scores to see who's doing a better job of teaching.

Here are Ms. Mathews' test scores:

0, 65, 72, 75, 78, 78, 82, 85, 94, and 98

And, here are mine:

15, 42, 50, 72, 72, 75, 78, 97, 98, and 99

It's hard to tell, just by looking at individual scores, who might be the better teacher. I have scores that are lower than Ms. Mathews', but I also have some that are higher.

Measures of Center

One way that we can compare our scores is to look at measures of center, which are values that usually represent the middle of a set of numbers. These values can give us an idea of how a ''normal'' student in our class performed on the test.

The first measure of center we'll examine is the mean of the scores, also called the average. To find Ms. Mathews' mean, we'll add all the scores then divide that sum by 10, since there are 10 scores.



As you can see, some students did better and some did worse, but the average score for Ms. Mathews' class is 72.7.

To find my average, we'll follow the same procedure:



From my average of 69.8, it looks like Ms. Mathews is the better teacher. Oh, well.

Using the median, which the number in the middle of an ordered number set, can help in some situations. In this case, we both have scores that are outliers, which are numbers that are so high or low that they don't really fit with the rest of the numbers. Ms. Mathews has an outlier of 0, and I have an outlier of 15. These scores are way lower than the others and bring our overall averages down. Maybe we both have a student who never studies, so we agree their scores aren't a true representation of how well we teach. Situations like this make the median a better choice for examining the middle of a data set.

To find the median of Ms. Mathews' scores, she puts them in order, from least to greatest (her scores were already in order, but they're not always given that way), then finds the number in the middle.


Because she has 10 scores, an even number, she has two scores in the middle. When this happens, you add the two scores and divide by 2. Since they're both the same number, the median is 78.

To find the median of my scores, I make sure they're in order, from least to greatest, then find the middle:


I have two different numbers in the middle, so I add them and divide by 2:



My median is 73.5. It looks like Ms. Mathews is still the better teacher.

Another measure of center is the mode, which is the number that occurs most often in a set. This one is easy to see. Ms. Mathews has two scores of 78. All other scores appear once, so her mode is 78. My mode is 72. Ms. Mathews is still winning. Darn!

Measures of Variability

I'm starting to feel bad about my teaching, so, instead of looking at measures of center, let's look at measures of variability, which show how spread out a set of numbers is. Maybe Ms. Mathews' middle scores are better, but I may still come out on top if we look at the scores' spread instead.

Examining measures of variability can be made easier by using a box plot, which is a graphic that shows the spread of a set of numbers. You can see the box plot for Ms. Mathews' scores on your screen right now:


A box plot is concentrated on a number line that contains the numbers in the data set. The box's lines extend out to the highest and lowest numbers in the set and make it easy to see the range, which is the difference between the highest and lowest scores. Ms. Mathews' range is 98 - 0 = 98.

The plot also divides the number set into four sections. The red portion of the line, the left half of the box, the right half of the box, and the green portion of the line each represent 25% of the scores in the set.

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