The cooking club conducted a survey
about how many hot meals each member cooked in July. One of the frequencies is
missing. The cooking club uses midpoints to
calculate an estimate for the mean number of meals each member cooked in July. They find that the estimate of the
mean is 15. Work out the missing frequency.
To find the mean of a set of data,
we find the sum of all of the data values and divide it by how many data values
there are or the total frequency. However, in this question, we
haven’t been given the original data values. Instead, they’ve been presented in
a grouped frequency table. We just know that, for example,
there were 15 members of the cooking club who cooked between zero and six meals in
For this reason, we can’t work out
the exact sum of the data values. We can only estimate it. And this is why the cooking club
were only able to work out an estimate for their mean. We’re told that the cooking club
uses midpoints to calculate their estimate. And what we’ll do is go through the
process that they would have used.
We can use the letter 𝑥 to
represent the missing frequency. The cooking club have already
calculated the midpoints for us. These are found by averaging the
endpoints of each interval. These midpoints give a best
estimate for each data value within an interval with the smallest error on
The next step in estimating a mean
from a grouped frequency table is to multiply the midpoint of each interval by its
frequency. Remember the midpoint gives the
best estimate for each data value. So the midpoint multiplied by the
frequency gives the best estimate for the total of the data values in each
For the first interval, we have
three multiplied by 15 which is 45. For the next interval, 10
multiplied by 29 which is 290. The values for the third and fifth
intervals can be found in the same way. For the interval with the missing
frequency, we multiply the midpoint 24 by the frequency which we’ve now called
𝑥. We can, therefore, express this as
Now, we have our best estimate of
the sum of the data values in each interval. So our next step is to find an
estimate of the sum of all the data values which we do by adding the five values in
the final column. We have 45 plus 290 plus 680 plus
24𝑥 plus 248. We can simplify this expression by
combining all the numbers to give 24𝑥 plus 1263.
Next, we need an expression for the
total frequency which we find by summing the five values in the frequency column: 15
plus 29 plus 40 plus 𝑥 plus eight. We can again simplify this by
grouping the numbers to give 𝑥 plus 92.
Remember that we were told what the
cooking club’s estimate of the mean was. It’s 15. So we can set this expression equal
to 15. And it gives an equation that we
now need to solve in order to find the value of 𝑥, the missing frequency.
Here is that equation we just
worked out: 24𝑥 plus 1263 over 𝑥 plus 92 equals 15, where 𝑥 represents the
missing frequency. The first step in solving this
equation is to eliminate the denominator on the left-hand side which we can do by
multiplying both sides by 𝑥 plus 92. On the left, this will cancel out
the factor of 𝑥 plus 92 in the denominator. And on the right, we have a bracket
which we need to expand.
15 multiplied by 𝑥 is 15𝑥 and 15
multiplied by 92 is 1380. We have 𝑥s on both sides of this
equation. So we want to group them all on one
side. And we want to group them on the
side which has the larger number of 𝑥 to start off with. That’s the left-hand side. So we’ll subtract 15𝑥 from each
side to eliminate the 15𝑥 on the right of the equation, which gives nine 𝑥 plus
1263 equals 1380.
Next, we subtract 1263 from each
side to eliminate the positive 1263 on the left, giving nine 𝑥 equals 117. The final step is to divide both
sides of this equation by nine, giving 𝑥 equals 13.
Remember that 𝑥 represents the
missing frequency. So it needs to be an integer
value. As we’ve got a whole number, an
answer of 13, we can be reasonably confident that our answer is correct.