Video Transcript
In this video, we will learn how to
estimate the mean using grouped frequency tables. Let’s begin by recapping what a
grouped frequency table is. A grouped frequency table is a
frequency table with data organized into smaller groups, often referred to as sets
or classes. These are often useful when we are
working with large data sets or data sets with a large range of values. As a result, a grouped frequency
table is a manageable way of representing the data. However, the disadvantage of
grouped frequency tables is that we cannot extract the original data values from
them.
Let’s consider the table shown,
which represents the marks that students received in an examination. The groups are given as open
intervals: zero dash, 10 dash, 20 dash, and 30 dash. This means that the data values in
the first group can be considered as zero or more but less than 10. Its boundary values would be zero
and 10. We observe that one student
achieved a mark in this interval. As we don’t know the exact mark, it
becomes more difficult when performing any statistical calculations based on a
grouped frequency table.
We recall that we can calculate the
mean of a data set by dividing the sum of the data values by the total number of
data values. If we have the individual scores
listed as shown, then the mean is easy to calculate. We find the sum of all the scores,
which is equal to 360, and divide this by 15, as there are 15 scores in total. This would give us a mean mark
equal to 24. In our grouped frequency table
however, this is not possible. So instead, we find an estimate for
the mean. We begin by adding two rows to our
table. The first for the midpoint of each
class interval. This midpoint can be calculated by
adding the grouped boundary values for our first column, zero and 10, and then
dividing by two. This is equal to five.
For the second class interval, we
add the boundaries of 10 and 20 and then divide by two. This is equal to 30 divided by two,
giving us a midpoint of 15. The next column has a midpoint of
25, as 20 plus 30 divided by two is 25. Assuming that each class has equal
width, the upper boundary of our last group is 40 and the midpoint of 30 and 40 is
35.
Our next step is to multiply the
frequency in each column by the midpoint. This will give us an estimate for
the total number of marks in that group. One multiplied by five is five. Five multiplied by 15 is 75. Repeating this for the final two
columns, we have values of 125 and 140. Our next step is to find the sum of
the second and fourth rows. One plus five plus five plus four
is equal to 15. So there were 15 students in total
that took the examination. Adding the four numbers in the
bottom row gives us 345.
We are now in a position to
calculate an estimate for the mean. We divide the total of the
frequency multiplied by the midpoint by the total of the frequency. In this example, we have 345
divided by 15. And this is equal to 23. We will now consider an example in
a different context.
In an extract of a book, the number
of words per sentence was counted. Find the missing numbers in the
following table. Use the previous table to calculate
an estimate for the mean number of words. Give your answer to two decimal
places.
In this question, we are given a
grouped frequency table that lists the number of words per sentence in a book along
with their frequencies. 15 sentences had between one and
seven words inclusive. 20 sentences had between eight and
14 words. 45 sentences had between 15 and 21
words. And we also have the frequencies
for between 22 and 28 words, 29 and 35 words, and 36 and 42 words. The third column of our table
corresponds to the midpoints. To calculate these values, we add
the two boundary values in each group and divide by two. For example, in the first row, we
add one and seven to give us eight, and dividing this by two gives us four. This is the midpoint of the group
one to seven.
We need to calculate the midpoint
for the second group in our table, where the number of words are between eight and
14 inclusive. The midpoint is therefore equal to
eight plus 14 divided by two. This simplifies to 22 divided by
two, which is equal to 11. The final column of our table
corresponds to the product of the frequency and the midpoint. In the first row, 15 multiplied by
four is equal to 60. We need to calculate the frequency
of 20 multiplied by the midpoint of 11. And this is equal to 220. The two missing numbers in the
table are 11 and 220.
The second part of this question
wants us to calculate an estimate for the mean number of words. And we recall that an estimate for
the mean can be found by dividing the total of the frequency multiplied by midpoint
column by the total frequency. Adding an extra row to our table,
we need to find the totals of the second and fourth columns. Adding the frequencies gives us a
total of 160. And adding the numbers in the last
column for the frequencies multiplied by the midpoints gives us 3391. Our estimate for the mean is
therefore equal to 3391 divided by 160. This is equal to 21.19375. And as we are asked to give our
answer to two decimal places, this is equal to 21.19. This is an estimate for the mean
number of words per sentence in the book.
Before looking at another example,
let’s summarize the steps we need to take in order to calculate an estimate for the
mean of a grouped frequency table.
This can be done in four simple
steps. Firstly, we find the midpoint 𝑥 of
each group in the table by adding the boundary values and dividing by two. Secondly, we multiply the midpoints
by their frequencies 𝐹 of the corresponding classes to give 𝐹𝑥. It is often useful to add
additional rows or columns to the frequency table to record these. Next, we find the sum of 𝐹𝑥,
known as the total of 𝐹𝑥. Finally, we divide this sum by the
total frequency known as total 𝐹. This can be summarized as
follows. The estimate for the mean is equal
to the total of 𝐹𝑥 divided by the total of 𝐹. Let’s now consider an example where
we apply this.
The following table shows the
salaries 𝑠 of employees in a certain company, given in Egyptian pounds. Estimate the mean salary in
Egyptian pounds, giving your answer approximated to two decimal places.
The given table is a grouped
frequency table. The first column tells us that five
employees earned a salary greater than or equal to 1000 Egyptian pounds and less
than 3000 Egyptian pounds. The second column tells us that
eight employees earned a salary greater than or equal to 3000 Egyptian pounds and
less than 5000 Egyptian pounds. There were three employees and five
employees in the third and fourth groups, respectively. As we are not told the exact
salaries of any of the employees, we cannot determine the exact mean. Instead, we need to find an
estimate for the mean.
To do this, we follow a four-step
process, the first of which is to find the midpoint of each group, which we will
call 𝑥. To find the midpoint of each group,
we add the boundary values and divide by two. This means that in the first
column, we need to add 1000 and 3000 and then divide this answer by two. This is equal to 2000. So the midpoint of 1000 and 3000
Egyptian pounds is 2000 Egyptian pounds. Repeating this for the second
column, we see that the midpoint of 3000 and 5000 is 4000. Likewise, the midpoint of 5000 and
7000 is 6000, and the midpoint of 7000 and 9000 is 8000.
Our next step is to multiply the
midpoints by the frequencies 𝐹, which in this case is the number of employees in
each group. In the first column, we multiply
2000 by five and add the answer to the bottom row of our table, which we have
labeled 𝐹𝑥. 2000 multiplied by five is
10000. Next, we multiply 4000 by eight,
giving us 32000. 6000 multiplied by three is
18000. And finally, 8000 multiplied by
five is 40000.
Our next step is to find the total
of 𝐹, the number of employees, and the total of 𝐹𝑥. This is because the estimate for
the mean is calculated by dividing the total of 𝐹𝑥 by the total of 𝐹. The sum of five, eight, three, and
five is 21. Adding 10000, 32000, 18000, and
40000 gives us 100000. We can therefore calculate an
estimate for the mean by dividing 100000 by 21. This is equal to 4761.9047 and so
on. We are asked to give our answer to
two decimal places. And we can therefore conclude that
an estimate for the mean salary at the company in Egyptian pounds is 4761 pounds
90.
We will now summarize the key
points from this video.
We saw in this video that a grouped
frequency table organizes data into smaller groups called classes. Since we cannot determine the
individual data values from a grouped frequency table, we cannot calculate the exact
mean. Instead, we calculate an estimate
for the mean. This is done following a four-step
process.
Firstly, we find the midpoint 𝑥 of
each group in the table by adding the boundary values and dividing by two. Secondly, we multiply the midpoints
by the frequencies. We then find the sum of these
values and finally divide this sum by the total frequency. This can be summarized as the
estimate for the mean is equal to the total of 𝐹𝑥 divided by the total of 𝐹,
where 𝑥 is the midpoint of each group and 𝐹 is the corresponding frequency. Finally, we saw in our examples
that it is useful to add rows or columns to the frequency table in order to record
these values.
