### Video Transcript

In this video, we will learn how to find the range of a data set. We will begin by looking at a definition of the range and then answer a variety of questions.

The range of a data set is the difference between the largest and smallest values. We can therefore calculate the range by subtracting the smallest data value from the largest data value. The importance of calculating the range is it tells us how spread-out the data is. We will now look at some examples to practice finding and using the range.

The number of goals scored by 12 soccer players in a season are 13, 11, 12, five, five, nine, six, 11, eight, five, six, and 19. State whether the following statement is true or false. The range of the data is 14 goals.

We can calculate the range of any set of data by subtracting the smallest value from the largest value. Whilst we could find these values from the list by inspection, it is often useful to rewrite the data set in numerical order first. The smallest value in the data set is five. And there are three of these. There are two sixes in the data set. The next lowest value is eight. Continuing our list from smallest to largest, we have 9, 11, 11, 12, 13, and 19. The smallest value is equal to five and the largest value is 19. We can therefore calculate the range by subtracting five from 19. This is equal to 14. The statement in the question said that the range of the data is 14 goals, which is true.

Our next question involves working out the range from a set of data represented on a line plot.

The graph shows the weights, in kilograms, of emperor penguins at a zoo. Is the range of the weights 24 kilograms?

The range of any data set can be calculated by subtracting the smallest value from the largest value. As the weights on the line plot are already in order, we can see that the smallest value is 23 kilograms. The largest value is 49 kilograms. We can therefore calculate the range by subtracting 23 from 49. 40 minus 20 is equal to 20, and nine minus three is equal to six. Therefore, 49 minus 23 equals 26. The range of values of the emperor penguins is 26 kilograms. This means that the answer to the question “Is the range of the weights 24 kilograms?” is no.

Let the greatest element in a set be 445 and the range of the set be 254. What is the smallest element of this set?

We know that, in order to calculate the range of any set, we subtract the smallest element from the largest element. In this question, we are given the largest or greatest element and the range of the set and need to calculate the smallest element. We can do this by using the formula or by using the number line. Substituting in our values gives us 254 is equal to 445 minus 𝑥, where 𝑥 is the smallest element of the set. Adding 𝑥 to both sides of this equation gives us 𝑥 plus 254 is equal to 445. We can then subtract 254 from both sides of this equation to work out the value of 𝑥. 𝑥 is equal to 191 as 445 minus 254 is 191. We can therefore conclude that the smallest element of the set is 191.

As already mentioned, we could also have calculated this using a number line. We know that the greatest value is 445. The range is the difference between the greatest or largest value and the smallest value. In this case, we are told it is 254. The difference between the smallest and greatest value is 254, which means we can subtract this from 445 to calculate the smallest value. Once again, this gives us an answer of 191.

We will now look at a couple of more complicated problems to finish this video.

The following figure demonstrates the number of glasses of water a group of people consume per day. Describe how the range would change if an additional data value of one was added to the data set.

The graph tells us that eight people consume zero glasses of water per day. Five people consume one glass; two people consume two glasses; six people, three glasses; one person, four glasses; and, finally, seven people consume five glasses of water per day.

We know that, in order to calculate the range of a data set, we subtract the smallest value from the largest value. In this question, it will be the number of glasses. It doesn’t matter how many different people consume that number of glasses. The smallest value is therefore zero. As there’re some people, in this case, eight, that consume no glasses of water. The largest value is equal to five as this is the greatest number of glasses of water that anybody consumes. We can therefore calculate the range of the original data by subtracting zero from five. This is equal to five.

We are then told that there is an additional data value of one that is added to the data set. This means that there are now six people that consumed one glass of water per day. This does not impact the smallest or largest value though. So the new range is still equal to five minus zero. Adding the extra data value does not alter the range. We can therefore conclude that the range would remain unchanged at five.

Michael has the following data: Six, eight, 𝑘, eight, eight, and nine. If the range is three, which number could 𝑘 be? Is it (A) three, (B) four, (C) five, (D) six, or (E) 13?

We recall that we can calculate the range by subtracting the smallest value from the largest value. In this question, we will consider what the largest value and smallest values are when 𝑘 takes each of the five options. One of these options will have a range of three which will be the correct answer. When 𝑘 is equal to three, our list of values in ascending order are three, six, eight, eight, eight, and nine. As the largest value is nine and the smallest value is three, the range will be equal to nine minus three. As this is equal to six, option (A) is not correct.

When 𝑘 is equal to four, the smallest value is four and the largest value is nine. This time the range would be equal to nine minus four, which is equal to five. Once again, this is not correct. When 𝑘 is equal to five the smallest number is five. The largest number is still nine. The range in this case is equal to four, which once again is not correct. When 𝑘 is equal to six, we have two sixes. We could write these in either order. Our list is now six, six, eight, eight, eight, and nine. As the smallest number in this set of date of is six and the largest is nine, the range is equal to nine minus six. This is equal to three, which suggests that option (D) is correct.

We will check option (E) just to make sure. This time, 𝑘 is equal to 13. This means that the smallest number is six and the largest number is 13. The range is the difference between these values. 13 minus six is equal to seven. So this answer is also incorrect. This means that the correct answer is option (D). If the range is three, the number from the list that 𝑘 could be is six.

There are a few other numbers that were not one of the options that 𝑘 could be. As long as six remains the smallest number and nine remains the largest number, the range will always be three. This means that 𝑘 could be any one of the four integers, six, seven, eight, or nine. In this question, the only one of those that was listed as an option was six, which is why this is the only correct answer.

We will now summarize the key points from this video. The range of a data set is the difference between the largest and smallest values. We can therefore calculate the range of any data set by subtracting the smallest value from the largest value. The range of a set of data tells us how spread-out the data is. This means that adding any extra values to a data set doesn’t always change the range.

For example, let’s consider the data set four, seven, 10, 10, and 13. The smallest value here is four, and the largest value is 13. This means that the range is equal to nine as 13 minus four equals nine. Adding in any extra values between four and 13 inclusive will not affect the range. For example, if we added the number eight, the smallest number is still four and the largest number is 13. This means that the range is still nine.

It is also important to note that it doesn’t matter how many of each value we have. If we added any extra fours or extra 13s to this list, the range would still be nine. As well as using a list of data values, we can also calculate the range from a frequency table or graph.