The table below shows the amount of
water consumed by 200 cyclists in a race. Work out an estimate for the mean
amount of water consumed.
To find the mean of a set of data,
we find the sum of all of the values in the dataset and divide this by how many
values there are. However, here we haven’t been given
all of the original data values. Instead, the data has been
presented in a grouped frequency table.
We know, for example, that there
are three cyclists who consumed more than zero litres of water and less than or
equal to 0.5 litres. But we don’t know the exact values
for each of these cyclists within this interval. This is why the question tells us
to work out an estimate for the mean. Because if we don’t know the
original data values, we can only work out an estimate of their sum.
There’s a process that we need to
follow for working out the best estimate of the sum. First, we find the midpoint of each
class interval by finding the average of its two end points. This is essentially giving us our
best guess for each data value with the least error.
So for each interval, we add
together the endpoints and then divide by two. For the first interval then, we
have a half multiplied by zero plus 0.5 which is 0.25. For the second interval, we have a
half of 0.5 plus one which is 0.75 and so on. Our midpoints are 0.25, 0.75, 1.25,
1.75, and 2.25.
Next, we multiply the midpoint of
each interval by the frequency for that interval. Remember the midpoint was giving us
the best estimate for each individual data value in an interval. So by multiplying it by the
frequency — that’s the number of values in that interval — we’re getting our best
estimate for the sum of the values in each interval.
For the first interval, we have
0.25 multiplied by three which is 0.75, then 0.75 multiplied by 34 which is 25.5,
next 1.25 multiplied by 59 which is 73.75, and so on. The final two values in this column
are 140 and 54.
So we now have an estimate for the
sum of the values in each interval. And the last step is to find an
estimate for the sum of all the values. So we find the total of the five
values in this final column. The sum of these five values is
So now, we’re able to calculate our
estimate of the mean. We’ve adapted our formula slightly
so that we’re dividing the estimated sum of all the values by the total number of
values. Our estimated sum of all the values
is 294 and the total number of values is the same as the total number of cyclists in
the race is 200. So we have 294 over 200. Dividing 294 by 200 gives 1.47
We need to include units with our
answer, which looking back at the data in the table, we can see our litres. So we’ve worked out an estimate for
the mean amount of water consumed is 1.47 litres.