Video Transcript
The frequency distribution shows the weights of 30 children. Estimate the mean weight recorded.
The data for the weights of these 30 children has been presented in a grouped
frequency table. The groups are given as open intervals: six dash, 10 dash, and so on. This means that the first group contains all the children whose weight was greater
than or equal to six kilograms but less than 10 kilograms, because that’s the
starting point for the next group. We don’t know the exact weights of any of the children. But instead we know how many of the children are in each interval.
We are asked to estimate the mean weight recorded. We can only estimate this value because we don’t know the exact data. In general, the mean of a data set is found by dividing the sum of all the data
values by how many values there are. We know that the total number of values in this data set is 30. But we can’t find the sum of the data values because we don’t know the individual
values themselves.
Instead, we have to estimate this by first finding a single value that is
representative of each of the groups in the data. The best value to use is the midpoint of each group. We’ll add a new row to the table in which to record these midpoints. We calculate the midpoint of each group by finding the average of the upper and lower
class boundaries.
For the first group, which contains all the children whose weight was greater than or
equal to six kilograms but less than 10 kilograms, the midpoint is six plus 10 over
two. That’s 16 over two, which is eight. The midpoint for the next group, which contains all the children whose weight was
greater than or equal to 10 kilograms but less than 14 kilograms, is 10 plus 14 over
two, which is 12.
We can then calculate the remaining midpoints in the same way, each time finding the
average of the lower and upper class boundaries. For the final class, which is given as 30 dash, we have to make an assumption about
its upper boundary. We assume that this class has the same width as the previous class. In fact, every class has a width of four. So we assume this class also has a width of four, and hence its upper boundary is
34. The final midpoint is therefore 30 plus 34 over two, which is 32.
We’ve now found a representative value for the data within each class. We can then estimate the total weight of each group of children by multiplying the
midpoint of each group by the frequency for that group. For example, in the first group, there are six children whose weight is each
approximately eight kilograms. So their overall weight is approximately eight times six, which is 48 kilograms. The remaining values in this row of the table are 60, 80, 60, 144, 84, and 64.
We can then find an estimate for the sum of all the data values, which in this case
is the total weight of all the children, by summing the values in the final row of
the table. We can then tweak the formula for the mean. If we use the letter 𝐹 to represent the frequencies, 𝑥 to represent the midpoints,
and 𝐹𝑥 to represent the products, the estimated mean is equal to the total of all
the frequencies multiplied by the midpoints divided by the total frequency. This corresponds to dividing our estimate of the total weight by the total number of
children.
Using the values from our table, that’s 540 over 30, which is 18. This value makes sense because it’s within the range of the data and also fairly
central. Our estimate of the mean weight recorded is 18 kilograms.
