# Video: Determining the Measures of Central Tendency and Range Represented by a Given Number

Jacob collected the following data that represents the number of books his friends read last year. Which measure of data is represented by 9.5 books?

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### Video Transcript

Jacob collected the following data that represents the number of books his friends read last year. Which measure of data is represented by 9.5 books.

Let’s start this question by having a look at the data in the table that we’re given. The first value of one would represent a friend who had read one book last year. Five would represent a friend reading five books, and 12 would be a friend who’d read 12 books. We can see that there are some repeated values. For instance, 13 occurs twice. This would represent two friends who’d both read 13 books.

In the question, we’re asked to find a measure of data that would be equal to nine and a half books. A measure of data would be one of the mean, the mode, the median, or the range. And we can start by having a look at the mean of this data set.

The mean can be calculated by working out the sum of all the values divided by the number of values. So, to find the sum of all the numbers, we add together one, five, 12, six, two, seven, three, 14, 15, 13, 13, eight, two, 15, 11, 15, five, and 15. We can then divide it by 18 since there are 18 values in our data set. Evaluating the sum of our numbers then would give us 162 over 18, or 162 divided by 18, which we can calculate as nine, giving us a mean for this data set of nine.

Let’s now look at calculating the mode. The mode of a data set is the most common value. It is possible to have more than one mode. For example, if there were two number 12s and two number 14s, then we could say that the mode would be 12 and 14. To find the mode, we can simply inspect the data, but sometimes it can be helpful to write the numbers in order. So, to start our list, we can see that there is one number one. Then, we have two of the value two, one value of three, and two number fives. We can continue the list giving us six, seven, eight, 11, 12, 13, 13, 14, 15, 15, 15, and 15.

It’s always worthwhile at this point just checking that there are still 18 values in our ordered list. Looking at our list then, we can see that there are two of the value two, and two of the value five, and two of the value 13. However, there are four number 15s. This means that 15 is our most common value. And therefore, our mode is equal to 15.

Let’s now look at how we would find the median of the data set. The median is the number that’s halfway into the set when the values are written in order. Since there are 18 values in our data set, we could split our data set into two halves of nine values each. Therefore, the median must lie at this halfway point. So, our median is halfway between eight and 11. A quick way to calculate this is to add eight and 11 together and divide by two, which is equal to 19 over two. And that’s 9.5. So, our median here is 9.5.

So, finally, let’s look at how to calculate the range. The range of our data set would be the largest value take away the smallest value. In our data set, the largest value is 15, and the smallest is one. So, we can get the range by working out 15 take away one, giving us a range of 14.

So, all of our data measures have given us a measurement for the number of books read, a mean of nine books, a mode of 15 books, a median of 9.5 books, and a range of 14 books. We were asked which data measurement is represented by 9.5 books. And since there’s only one which gives the answer 9.5 books, our answer is the median.