By completing the table, find the range of the scores.
Let’s begin this question by having a look at the table and saying what it represents. In the first column, we have a number of scores denoted by the letter 𝑥. In the second column, we have frequency denoted by the letter 𝑓. And frequency is the number of times something happens.
So, in our first row, the score of three happened six times. In the third column, we have 𝑓 dot 𝑥. That means that we take our frequency, 𝑓, and we multiply it by the score, 𝑥. And we do this for every row of the data. In the bottom of the table, we have a total row. So, that would be the total of the frequency and the total of the 𝑓 dot 𝑥 column. So, let’s see which of the missing values we can fill in.
In the first row, we have a missing entry in the 𝑓 dot 𝑥 column. Since we know that 𝑓 dot 𝑥 is found by multiplying the 𝑓, the frequency, by the score, we could fill in the missing value here. As we have a frequency of six and a score of three, this will give us six times three is 18 which we can fill into the table. In the next row, we have a missing 𝑓 value, which means that we have a missing value in the calculation five times what equals 25. And since five times five is 25, we can fill in the missing value, five.
In the next row, we have two missing values of the 𝑓 and the 𝑓 dot 𝑥 value. Since we’re trying to calculate nine times what equals what, we don’t have enough information to fill in either of these blanks. So, let’s try something else. What we’re looking for in the table is a place in either a row or a column where there’s just one blank value. So, if we look at our frequency column, we can see that there is one missing value in this column. In this case, we have a total of 25. So, that means that we need to calculate six plus five plus what plus three equals 25. So, that will give us 14 plus what equals 25. And since we can work out that 14 plus 11 is 25, we can then fill in our missing value into the table.
Now, we can look again at our third row. Since we know that our frequency we’ve just worked out is 11, that means that our missing value in the 𝑓 dot 𝑥 column must be equal to nine times 11, which will give us a value of 99. So, if we look at our next row, we can see that there are two missing values. And we know that we can’t get an answer when there’s two blanks.
So, let’s consider the column 𝑓 dot 𝑥. Now, we can see that there is one missing value in the column. Since we know that 187 is the total, that means we have the calculation 18 plus 25 plus 99 plus what equals 187. We can simplify the values on our left-hand side, giving us 142 plus what equals 187. And we can work out our missing value, 45, by calculating 187 subtract 142. So, we have one missing value left in our table. Remembering that we’re multiplying along a row, this means that we must have what times three equals 45. Since dividing 45 by three will give us 15, our missing value for score must be 15.
Going back to the original question then, we were asked to find the range of the scores. We can remember that the range is the difference between the highest and the lowest values, which we can calculate as the highest value subtract the lowest value. Looking at the table, it would be very easy to think that the highest value is 187 and the lowest value is three. But this would be wrong. We’re asked to find the range of the scores and not the range of everything in the table. So, let’s see how we can do this.
In this case, we’re considering just the values in the column score. If we had a value in the total, we wouldn’t want to include that. Here, we can see that the highest value is 15 and the lowest value is three. So, we’re going to calculate 15 minus three, which we can simplify, giving us that the range of the scores is 12.