Video Transcript
The following frequency table shows the range of weights of watermelons produced in a
farm and the number of watermelons for each weight range. Which of the following is the estimated value of the mode for such data? Option (A), 15.00; option (B), 15.67; option (C), 1500; option (D), 1666.67; or
option (E), 2000.
In this problem, we are given the weights of watermelons on a farm, with the data
given in the form of a grouped frequency table and a histogram. So if we look at the table and consider a watermelon weighing 1650 grams, then this
weight would appear in the class given as 1500 dash, as this class represents
weights which are 1500 grams or more up to weight which is less than 2000 grams, as
this is the lower boundary of the following class.
We are asked to determine an estimate for the mode, which we can recall is the most
common, or frequently occurring, value. We will not be able to find an exact mode because the data is given as a grouped
frequency distribution. And we don’t have all the original values of the weights. So, the first step here is to find the modal class, which is the class or classes
with the highest frequency. We can do this either from the table or the histogram.
Using the table, we can identify that the highest frequency is 25. This is in the class 1500 dash, so this is the modal class. The mode will therefore be greater than or equal to 1500 grams but less than 2000
grams. Note that we could also have determined this from the histogram, because the modal
class is the class that has the highest frequency, which will be the class with the
tallest bar in the histogram. Now, we are given some answer options to help us work out the estimated value of the
mode. We can certainly eliminate answer options (A) and (B) because these are not in the
range allowed.
So how do we determine which of the other options is correct? Well, given a histogram of a grouped frequency distribution, we could follow a fixed
method to find an estimate for the mode, using the bar representing the modal
class. First, we 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. Then, 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. Finally, we draw a vertical line from the point of intersection of these lines down
to the 𝑥-axis. This value is the estimate for the mode.
We can observe that this point on the 𝑥-axis is approximately one-third of the width
of the bar. Calculating one-third of the class width of 500 would give us 166.6 recurring. And adding this to the lower boundary of the class of 1500 would give us 1666.6
recurring, which would be an estimate for the mode.
Therefore, the correct answer was that given in option (D), 1666.67, since this is an
approximated value of the estimate of the mode that we calculated. In context, this means that the most common weight of the watermelons produced on the
farm can be estimated as 1666.67 grams.
