# Question Video: Finding an Estimate for the Mode of a Grouped Frequency Distribution Mathematics

The 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? [A] 15.00 [B] 15.67 [C] 1,500 [D] 1,666.67 [E] 2,000

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### 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.