Question Video: Estimating the Mode of a Grouped Frequency Distribution Mathematics

Video Transcript

The following frequency table shows the range of electricity consumed by houses in a city. Using the given histogram, what is the estimated value of the mode?

Here we are given a table and a histogram representing the same grouped data on energy consumption. We need to find an estimate for the mode, which we can recall is the value with the highest frequency. And so, the first thing we need to do is identify the modal class, that is, the class or classes with the highest frequency. And we can use either the table or the histogram to help us.

Using the table, the highest frequency, that’s the largest number of houses using a certain number of kilowatt hours per month of energy, is 120 houses. That belongs to the class 600 dash. 120 houses used 600 or more kilowatt hours per month of energy, up to, but not including, 700 kilowatt hours per month. So, the class 600 dash is the modal class. But we can also determine this from the histogram. The highest bar in the histogram is in the class 600 dash. We can see that the electricity consumption of this group is between 600 and 700 kilowatt hours per month. So, this indicates the modal class.

Next then, we use the histogram to estimate the mode, and we do this by following a number of steps. First, we draw a line from the left corner of the tallest bar to the left corner of the rectangle representing the frequency of the following class, which is the class of energy consumption between 700 and 800 kilowatt hours per month. Then, we draw a line from the right corner of the tallest bar to the right corner of the rectangle representing the frequency of the class immediately before. That’s the class between five and six hundred kilowatt hours per month of energy. And finally, we draw a vertical line from the point of intersection of these two lines down to the 𝑥-axis. This value gives us the estimate for the mode.

The answer is therefore 650, which means that the most common amount of energy consumed by houses in the city is 650 kilowatt hours per month.

As an aside, in this question, the value of 650 is exactly in the middle of the group. This arises from the fact that the two groups on either side of the modal class have the same frequency. And so the bars in the histogram are the same height for these equal class widths. However, it will not always be the case that the estimate for the mode will be in the middle of the class. So, we must take care to get an accurate estimate for the mode by drawing the lines from the bar on either side of the modal class on the histogram, as we did here.