# Question Video: Estimating the Mode for a Grouped Data Set Mathematics

Some seeds are planted and the height of the resulting plants, in cm, are measured after 6 weeks. Using the given table, calculate an estimated mode for the height of the plants in cm.

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### Video Transcript

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. Option (A) 10, option (B) 15, option (C) 19.5, option (D) 45, or option (E) 47.

Let’s begin by considering the given table, which records the heights of the plants in centimeters. For example, the height of plants in the first group are those that are one centimeter or more up to but not including seven centimeters. We can say that the upper boundary of the first class is seven centimeters because seven centimeters is the lower boundary of the following class.

Now we need to calculate an estimate for the modal height of the plants. We recall that the mode is the most commonly occurring value or values, that is, the value or values with the highest frequency. The reason that this is an estimate is because the data is grouped, so we won’t be able to find the exact mode.

Now the way that we can find this estimate for the mode is by identifying the modal class and drawing a histogram. So here is a histogram we could create to represent the data. And there are a few points to note about the drawing of a histogram. For example, we can look at the first two groups, with the first having boundaries of one centimeter and seven centimeters and the second group having boundaries of seven centimeters and 13 centimeters. There are no gaps between the bars in the histogram, because this is continuous data that we are representing.

The height of the bars represents the frequency of each group, which in the context of this question is the number of plants that have heights which fall into the given groupings of height ranges. So the tallest bar is the group with the highest frequency. This is the bar that has plants of height 19 centimeters or more up to a height less than 25 centimeters.

And we’ve already been thinking about highest frequency: it’s the mode. And so we can identify that this group, 19 dash, is the modal class, either by using the histogram to observe that it’s the class with the highest bar or by using the table, where this group has the highest frequency value of 47.

Sometimes in a question, we may just be asked to find the modal class, and the class 19 dash would be the answer. However, here we need to find a single value that would be an estimate for the mode. And so we’re going to draw some extra lines on the histogram to do this.

The first thing to do is to find the top-left corner of the modal class, which is the tallest bar. We then draw a line from this point to the top-left corner of the bar representing the frequency of the following class. Then we do the same with the top-right corner of the modal class. This time, we join this point at the top-right corner of the bar to the top-right corner of the bar representing the frequency of the class immediately before it. We then must find the point of intersection of these two new lines that we have just drawn. And finally, we draw a vertical line from their point of intersection to the 𝑥-axis.

Reading the 𝑥-axis at this point, we can see that this is at the value 19.5. This value of 19.5, which is answer option (C), is the estimate for the mode. In the context of the question, that means that we have estimated that the most common height of the plants is 19.5 centimeters.