### Video Transcript

Consider the following frequency
distribution. Complete the table by finding the
midpoints of each class.

This data set has been presented in
a grouped frequency distribution. The classes are given as open
intervals: zero dash, five dash, ten dash, and so on, up to 25 dash.

To determine the midpoint of each
interval, which would be required if we wanted to estimate the mean of the data, we
first need to determine the upper boundary for each class. We assume that there are no gaps in
the data. So the upper boundary for each
class is the lower boundary for the previous one. The upper boundary for the first
class is therefore five. We could express this as a
double-sided inequality: 𝑥 is greater than or equal to zero and strictly less than
five.

Using a weak inequality at the
lower end of the class and a strict inequality at the upper end ensures that the
classes have no gaps but also don’t overlap. Alternatively, we could add the
upper boundary of each class into the table, remembering that when we write, for
example, five to 10, we mean greater than or equal to five but strictly less than
10. Then, to calculate the midpoint of
each class, we take the mean of the upper and lower boundaries for that class.

For example, we can see that for
the two midpoints that are already filled in, 7.5 is equal to five plus 10 over two,
and 17.5 is equal to 15 plus 20 over two. The midpoint for the first class is
therefore zero plus five over two, which is 2.5. For the third class, it’s 10 plus
15 over two, which is 12.5. And for the fifth class, it’s 20
plus 25 over two, which is 22.5.

There’s one class we haven’t
mentioned yet, which is the final one. When determining the upper boundary
for this class, we need to assume it has the same width as the previous class. In fact, all classes in this
frequency distribution have the same width of five. So, we assume that the upper
boundary for the final class is 30, and then the midpoint is calculated as 25 plus
30 over two, which is 27.5.

The four midpoints needed to
complete the table are 2.5, 12.5, 22.5, and 27.5.