Question Video: Estimating the Mode from a Histogram Mathematics

The speeds, in kilometers per hour, of cars driving on a road were recorded in the table and are represented in the histogram. Which of the following is the best estimation for the modal speed of the cars? [A] 55 km/h [B] 70 km/h [C] 80 km/h [D] 83 km/h [E] 90 km/h

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Video Transcript

The speeds, in kilometers per hour, of cars driving on a road were recorded in the table below and are represented in the histogram. Which of the following is the best estimation for the modal speed of the cars? Option (A) 55 kilometers per hour. Option (B) 70 kilometers per hour. Option (C) 80 kilometers per hour. Option (D) 83 kilometers per hour. Or option (E) 90 kilometers per hour.

In this question, we are considering the various speeds of cars that were recorded. And the table shows this data in different groups, or classes, of speeds. For example, the second group has speeds that are 60 kilometers per hour up to speeds that are less than 70 kilometers per hour. So if a car was traveling at 66 kilometers per hour, it would be recorded in this group. And it’s the frequency which tells us how many cars travel at each of the speed groupings. So we can see that 15 cars had speeds in this second group.

Now we need to find an estimation for the modal speed, which is another way of asking for an estimate for the mode. We can recall that the mode is the most commonly occurring value or values. Or we can think of this as the value or values with the highest frequency. If we look at the table, we can see that the highest frequency is 25. This is the frequency of the class 80 is less than or equal to 𝑥 is less than 90.

So we can in fact say that this is the modal class. That would mean that the most common speeds are in the group of speeds which are greater than or equal to 80 kilometers per hour and less than 90 kilometers per hour. But we aren’t simply asked to find the modal class here. We want an estimate for the mode. And the mode will lie within the modal class.

Note that we must be careful not to confuse the words “estimation” or “estimate” here with guessing. We say that this is an estimate simply because this is grouped data, and we don’t know the original individual speeds of every car. But there is a mathematical process to finding the estimate for the mode of data such as this. And that process involves using the histogram. Notice that if we aren’t given a histogram, we would have to draw one ourselves.

In this process we will especially be considering the bar of the modal class. And if we hadn’t already identified this modal class, then we can also do that directly from the histogram, since it’s the tallest bar. We begin by drawing a line like this from the top-left corner of the tallest bar to the top-left corner of the bar representing the frequency of the following class. And we do the same on the other side. We draw a line from 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 we draw a vertical line from the point of intersection of these lines to the 𝑥-axis. This value on the 𝑥-axis represents the estimate for the mode.

And so we can give the answer that an estimate for the modal speed of the cars is 83 kilometers per hour, which was the answer given in option (D).