The table shows information about the total number of goals scored by 54 football
teams during a season. Identify the modal class interval.
The data has been presented in a grouped frequency table. We don’t know the exact number of goals scored by each team, but we know the interval
in which it lies. Remember the mode of a set of data is the most common value. As all of the classes in the grouped frequency table have the same width, the modal
class interval be the one with the highest frequency.
Looking at the table, we can see that the highest number of teams, which represents
the frequency, is 19. This corresponds to the class interval 39 is less than or equal to 𝑔, the number of
goals, is less than 52. This is the modal class interval.
Calculate an estimate for the mean number of goals scored per team.
We can only find an estimate of the mean number of goals. We can’t calculate it exactly as we don’t know the original data values. They’ve been grouped into classes. We need to recall the method for estimating the mean from a grouped frequency
We start by finding the midpoint of each interval. We don’t know the original data values. So the midpoint of each interval is our best guess for the values with the least
error on average. To find each midpoint, we add together the two endpoints of the interval and halve
So for the first interval, the midpoint is zero plus 13 over two, which is 6.5. For the second interval, the midpoint is 13 plus 26 over two, which is 19.5. We find the midpoints of the other three intervals in the exact same way, giving
32.5, 45.5, and 58.5.
These remember are our estimates of the mean values within each class interval. We then multiply each midpoint by its frequency to give an estimate of the sum of the
values in that interval. For the first interval, this is 6.5 multiplied by nine which gives 58.5. For the second interval, it’s 19.5 multiplied by 13 which gives 253.5. We multiply the midpoint by the frequency for the final three intervals in the same
way giving 227.5, 864.5, and 351.
Remember these values give an estimate of the number of goals scored in each
interval. We then add together the values in this final column to give an estimate of the total
number of goals. This gives 1755.
To find the mean of a set of data, we add together all the pieces of data and then
divide by how many there are. We found our estimate of the sum of all the pieces of data 1755. And we’re told in the question that the number of football teams is 54. So the number of pieces of data is 54. We could also have found this by summing up the frequency column in the table. Our calculation for our estimate of the mean then is 1755 divided by 54, which gives
32.5. We don’t have to round this value to the nearest whole number. Even though the number of goals itself is an integer, the mean doesn’t have to
Be really careful here. A really common mistake is to divide by the number of classes in the frequency table,
rather than the number of pieces of data. If we’d done that, then our answer would’ve been 1755 divided by five, which gives
351. This clearly can’t be the correct answer.
From a logical point of view alone, 351 is an incredibly high estimate for the mean
number of goals scored per team. Looking at the table though, we can see that the greatest number of goals covered in
the table is 65. The mean number of goals can’t possibly be higher than this. So 351 is clearly incorrect. However, it is a very common mistake to make.
Our value of 32.5 is within the range of the data in the table and it’s also fairly
central. So it makes sense.
Our estimate of the mean number of goals scored per team is 32.5.