### Video Transcript

In this video, we will learn how to
use grouped frequency tables to identify the modal class. And we’ll see how we can estimate
the mode by using a histogram.

Let’s begin by reminding ourselves
of what a grouped frequency table is and in general how we calculate the mode of a
data set. A grouped frequency table is a
frequency table with data organized into smaller groups or classes. For example, let’s say that the
speeds in kilometers per hour of vehicles along a road were recorded into this
grouped frequency table. Notice how the data is organized
into these groups. The first group of 30 dash would
indicate vehicles which had a speed of 30 kilometers per hour up to but not
including 40 kilometers per hour, which is the lower boundary of the next group. The frequency of five tells us that
five vehicles had a speed of 30 kilometers per hour up to but not including 40
kilometers per hour.

The advantage of grouped frequency
tables is that it’s easier to analyze large data sets and those which have a wide
range of values. However, it does mean that we need
a different approach for performing different statistical methods, for example,
finding the averages. Here, we look specifically at how
we can find the mode of a grouped frequency table.

Recall that the mode is the most
common or frequently occurring value or values, because we can have more than one
mode. Let’s consider this ordered data
set. We have one, one, two, three,
three, five, five, six, seven, seven, seven, seven, and eight. The value which occurs the most is
seven, and so seven would be the mode of this data set. We could even represent this
information as a frequency table. In this data set, there are two
ones, one two, two threes, zero fours, two fives, one six, four sevens, and one
eight.

To find the mode by using this
table, we look for the value or values with the highest frequency. In this case, the highest frequency
is four. And that means that the most
commonly occurring value is seven. So we have recapped how we can find
the mode from a data set, how we can find the mode from an ungrouped frequency
table. So now let’s see how we can use a
grouped frequency table.

If we return to this grouped
frequency table regarding the speeds of vehicles, we can identify the class which
has the highest frequency. This highest frequency is 35, and
that occurs in the class 50 dash. We can identify that this class is
the modal class. We say that the modal class of a
grouped frequency distribution is the class or classes with the highest
frequency. Notice that just like the mode,
when we deal with modal class, there may be more than one.

We will now look at some
examples. And in the first one, we will
identify the modal class.

For the given frequency
distribution, what is the modal class?

If we have a look at the table we
are given, we could observe that in this frequency distribution, we have
classes. For example, the first class zero
dash would indicate values which are zero or greater up to but not including 10. In this question, we need to find
the modal class. The modal class is like the
mode. The modal class of a grouped
frequency distribution is the class or classes with the highest frequency. We can identify that the highest
frequency here is 12, and this occurs in the class 30 dash. It is this class which will be the
answer for the modal class.

We must be very careful because
it’s a very common mistake to give the answer of 12. But the value of 12 means that 12
individual data points exist in the class of 30 dash. And this is higher than the number
of data points in the other classes. The modal class is 30 dash.

As previously mentioned, we can’t
find the exact mode from a grouped frequency distribution. But we will now consider how we can
find a graphical estimate for the mode in a grouped frequency distribution, which we
do by using a histogram. Let’s see the steps that we need to
take in order to do this.

Using the data from the previous
question, we need to start with an actual histogram. Therefore, the first step if we
don’t have a histogram is to draw it. We also need to identify the modal
class, which is the class with the highest bar. We then draw a straight line
connecting the top-left corner of the tallest bar to the top-left corner of the bar
representing the frequency of the following class. We then do the same thing from the
top-right corner. We draw a straight line connecting
the top-right corner of the tallest bar to the top-right corner of the bar
representing the frequency of the class immediately before. Then, from the point of
intersection of these two lines, we draw a vertical line down to the 𝑥-axis. This value is the estimate for the
mode. In this example, we could say that
an estimate for the mode is 36.

We’ll now see how we can apply
these steps in the following example.

For the given histogram, which of
the following is the best estimate of the mode? Option (A) 12, option (B) 40,
option (C) 50, option (D) 30, or option (E) 44.

We can recall that the modal class
of a grouped frequency distribution is the class or classes with the highest
frequency. When we are using a histogram with
classes of equal width, the class with the highest frequency is the class with the
highest bar in the histogram. We can therefore identify that the
modal class here is that of the class 40 dash. The values in this class will be 40
or greater but less than 50, which is the boundary of the next class.

But of course, in this question, we
need to find an estimate for the mode rather than simply the modal class. We can understand that the mode
will lie within the modal class, which will be values from 40 up to but not
including 50. The estimates in option (A) and (D)
can therefore not be correct. To find an estimate for the mode
using the histogram, we can apply the steps we need to take to estimate the mode
graphically.

Once we have identified the modal
class, we draw a straight line connecting the top-left corner of this tallest bar to
the top-left corner of the rectangle representing the frequency of the following
class. Next, we draw a straight line
connecting the top-right corner of the tallest bar to the top-right corner of the
rectangle representing the frequency of the class immediately before. Then, from the point of
intersection of these lines, we draw a vertical line down to the 𝑥-axis. This point on the 𝑥-axis
represents an estimate for the mode. We can see that this line lies
slightly to the left of the midpoint of 40 and 50. Therefore, we can give the answer
that an estimate for the mode must be the value of 44.

In each of the following two
questions, we will need to draw a histogram first before using it to find an
estimate for the mode.

The table represents the time taken
by some people to travel to work. Calculate an estimation for the
modal number of minutes the people take to travel to work.

The table that we are given showing
the time taken for people to travel to work is in the form of a grouped frequency
table. For example, 10 people take a time
of 4.5 minutes or more up to but not including 9.5 minutes to get to work. We are asked to calculate an
estimate for the mode, and the mode is the most common value or values. It’s not possible to extract the
mode from a grouped data set. The only thing we can do from a
grouped frequency table is recognize the modal class, which is the class with the
highest frequency. As the class 9.5 minutes or more
has the highest frequency of 15, then this is the modal class.

So, from the table, we have
identified the modal class, and we can further identify an estimate for the mode by
drawing a histogram. There are some important points to
note when drawing a histogram. When we have a histogram, we don’t
have bars with spaces in between them. Instead, we have a continuous
𝑥-axis representing the variable, which will be time in minutes here. The rectangular bars representing
the frequencies have their vertical line segments on the upper and lower boundaries
of each class.

So, once we have our histogram, we
identify the modal class, which is the class with the highest bar. We then draw two lines from each of
the top corners of this modal class bar to the adjacent corners of the classes on
each side, which will look something like this. We then draw a vertical line from
the point of intersection to the 𝑥-axis. We can use the grid lines to help
us identify this estimate for the mode. Therefore, we can give the answer
that an estimate for the modal number of minutes the people took to travel to work
is 11.5 minutes.

We will now see one final
example.

Some seeds are planted and the
height of the resulting plants, in centimeters, are measured after six weeks. Using the given table, calculate an
estimated mode for the height of the plants in centimeters.

This table gives us the results of
the heights of plants, which are measured after six weeks of growing. The heights of these plants are
given as groups, with the first group being plants which are one centimeter or more
up to but not including seven centimeters, which is the lower boundary of the next
group. Here, we are asked to work out an
estimate for the mode. We can’t calculate an exact mode
from a grouped frequency, but we can calculate an estimate for the mode by drawing a
histogram.

Once we have drawn our histogram,
we identify the modal class. That’s the class with the tallest
bar. This will be the class 19 dash,
which we can also see in the table has the highest frequency. We then draw a straight line
connecting the top-left corner of the tallest bar to the top-left corner of the
rectangle representing the frequency of the following class. We then draw a second straight line
connecting the top-right corner of this tallest bar to the top-right corner of the
rectangle representing the frequency of the class immediately before. Finally, we draw a vertical line
from the intersection point of these two lines to the 𝑥-axis. This line meets the 𝑥-axis just
before 20. And if we use the smaller minor
grid lines, we can identify that this would be at the point 19.5. We can say then that an estimated
mode for the height of the plants is 19.5 centimeters.

We can now summarize the key points
of this video. The mode is the most commonly or
frequently occurring value or values. This is sometimes called the modal
value. The modal class of a grouped
frequency distribution is the class or classes with the highest frequency. We cannot extract an exact mode
from a grouped frequency table; we can only estimate it. By using a histogram with equal
class widths, we identify the tallest bar, which is the modal class. We draw two straight lines from
each of the top corners of the modal class bar to the adjacent corners of the class
bars on each side. We then draw a vertical line from
the intersection point of these two lines to the 𝑥-axis. And this gives us an estimate for
the mode.