Lesson Video: Grouped Frequency Tables: Estimating the Mode | Nagwa Lesson Video: Grouped Frequency Tables: Estimating the Mode | Nagwa

Lesson Video: Grouped Frequency Tables: Estimating the Mode Mathematics • Second Year of Preparatory School

In this video, we will learn how to use grouped frequency tables to identify the modal class and estimate the mode using a histogram.

13:58

Video Transcript

In this video, we will learn how to use grouped frequency tables to identify the modal class. And we’ll see how we can estimate the mode by using a histogram.

Let’s begin by reminding ourselves of what a grouped frequency table is and in general how we calculate the mode of a data set. A grouped frequency table is a frequency table with data organized into smaller groups or classes. For example, let’s say that the speeds in kilometers per hour of vehicles along a road were recorded into this grouped frequency table. Notice how the data is organized into these groups. The first group of 30 dash would indicate vehicles which had a speed of 30 kilometers per hour up to but not including 40 kilometers per hour, which is the lower boundary of the next group. The frequency of five tells us that five vehicles had a speed of 30 kilometers per hour up to but not including 40 kilometers per hour.

The advantage of grouped frequency tables is that it’s easier to analyze large data sets and those which have a wide range of values. However, it does mean that we need a different approach for performing different statistical methods, for example, finding the averages. Here, we look specifically at how we can find the mode of a grouped frequency table.

Recall that the mode is the most common or frequently occurring value or values, because we can have more than one mode. Let’s consider this ordered data set. We have one, one, two, three, three, five, five, six, seven, seven, seven, seven, and eight. The value which occurs the most is seven, and so seven would be the mode of this data set. We could even represent this information as a frequency table. In this data set, there are two ones, one two, two threes, zero fours, two fives, one six, four sevens, and one eight.

To find the mode by using this table, we look for the value or values with the highest frequency. In this case, the highest frequency is four. And that means that the most commonly occurring value is seven. So we have recapped how we can find the mode from a data set, how we can find the mode from an ungrouped frequency table. So now let’s see how we can use a grouped frequency table.

If we return to this grouped frequency table regarding the speeds of vehicles, we can identify the class which has the highest frequency. This highest frequency is 35, and that occurs in the class 50 dash. We can identify that this class is the modal class. We say that the modal class of a grouped frequency distribution is the class or classes with the highest frequency. Notice that just like the mode, when we deal with modal class, there may be more than one.

We will now look at some examples. And in the first one, we will identify the modal class.

For the given frequency distribution, what is the modal class?

If we have a look at the table we are given, we could observe that in this frequency distribution, we have classes. For example, the first class zero dash would indicate values which are zero or greater up to but not including 10. In this question, we need to find the modal class. The modal class is like the mode. The modal class of a grouped frequency distribution is the class or classes with the highest frequency. We can identify that the highest frequency here is 12, and this occurs in the class 30 dash. It is this class which will be the answer for the modal class.

We must be very careful because it’s a very common mistake to give the answer of 12. But the value of 12 means that 12 individual data points exist in the class of 30 dash. And this is higher than the number of data points in the other classes. The modal class is 30 dash.

As previously mentioned, we can’t find the exact mode from a grouped frequency distribution. But we will now consider how we can find a graphical estimate for the mode in a grouped frequency distribution, which we do by using a histogram. Let’s see the steps that we need to take in order to do this.

Using the data from the previous question, we need to start with an actual histogram. Therefore, the first step if we don’t have a histogram is to draw it. We also need to identify the modal class, which is the class with the highest bar. We then 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. We then do the same thing from the top-right corner. 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. Then, from the point of intersection of these two lines, we draw a vertical line down to the 𝑥-axis. This value is the estimate for the mode. In this example, we could say that an estimate for the mode is 36.

We’ll now see how we can apply these steps in the following example.

For the given histogram, which of the following is the best estimate of the mode? Option (A) 12, option (B) 40, option (C) 50, option (D) 30, or option (E) 44.

We can recall that the modal class of a grouped frequency distribution is the class or classes with the highest frequency. When we are using a histogram with classes of equal width, the class with the highest frequency is the class with the highest bar in the histogram. We can therefore identify that the modal class here is that of the class 40 dash. The values in this class will be 40 or greater but less than 50, which is the boundary of the next class.

But of course, in this question, we need to find an estimate for the mode rather than simply the modal class. We can understand that the mode will lie within the modal class, which will be values from 40 up to but not including 50. The estimates in option (A) and (D) can therefore not be correct. To find an estimate for the mode using the histogram, we can apply the steps we need to take to estimate the mode graphically.

Once we have identified the modal class, we draw a straight line connecting the top-left corner of this tallest bar to the top-left corner of the rectangle representing the frequency of the following class. Next, we draw a straight line connecting the top-right corner of the tallest bar to the top-right corner of the rectangle representing the frequency of the class immediately before. Then, from the point of intersection of these lines, we draw a vertical line down to the 𝑥-axis. This point on the 𝑥-axis represents an estimate for the mode. We can see that this line lies slightly to the left of the midpoint of 40 and 50. Therefore, we can give the answer that an estimate for the mode must be the value of 44.

In each of the following two questions, we will need to draw a histogram first before using it to find an estimate for the mode.

The table represents the time taken by some people to travel to work. Calculate an estimation for the modal number of minutes the people take to travel to work.

The table that we are given showing the time taken for people to travel to work is in the form of a grouped frequency table. For example, 10 people take a time of 4.5 minutes or more up to but not including 9.5 minutes to get to work. We are asked to calculate an estimate for the mode, and the mode is the most common value or values. It’s not possible to extract the mode from a grouped data set. The only thing we can do from a grouped frequency table is recognize the modal class, which is the class with the highest frequency. As the class 9.5 minutes or more has the highest frequency of 15, then this is the modal class.

So, from the table, we have identified the modal class, and we can further identify an estimate for the mode by drawing a histogram. There are some important points to note when drawing a histogram. When we have a histogram, we don’t have bars with spaces in between them. Instead, we have a continuous 𝑥-axis representing the variable, which will be time in minutes here. The rectangular bars representing the frequencies have their vertical line segments on the upper and lower boundaries of each class.

So, once we have our histogram, we identify the modal class, which is the class with the highest bar. We then draw two lines from each of the top corners of this modal class bar to the adjacent corners of the classes on each side, which will look something like this. We then draw a vertical line from the point of intersection to the 𝑥-axis. We can use the grid lines to help us identify this estimate for the mode. Therefore, we can give the answer that an estimate for the modal number of minutes the people took to travel to work is 11.5 minutes.

We will now see one final example.

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.

This table gives us the results of the heights of plants, which are measured after six weeks of growing. The heights of these plants are given as groups, with the first group being plants which are one centimeter or more up to but not including seven centimeters, which is the lower boundary of the next group. Here, we are asked to work out an estimate for the mode. We can’t calculate an exact mode from a grouped frequency, but we can calculate an estimate for the mode by drawing a histogram.

Once we have drawn our histogram, we identify the modal class. That’s the class with the tallest bar. This will be the class 19 dash, which we can also see in the table has the highest frequency. We then draw a straight line connecting the top-left corner of the tallest bar to the top-left corner of the rectangle representing the frequency of the following class. We then draw a second straight line connecting the top-right corner of this tallest bar to the top-right corner of the rectangle representing the frequency of the class immediately before. Finally, we draw a vertical line from the intersection point of these two lines to the 𝑥-axis. This line meets the 𝑥-axis just before 20. And if we use the smaller minor grid lines, we can identify that this would be at the point 19.5. We can say then that an estimated mode for the height of the plants is 19.5 centimeters.

We can now summarize the key points of this video. The mode is the most commonly or frequently occurring value or values. This is sometimes called the modal value. The modal class of a grouped frequency distribution is the class or classes with the highest frequency. We cannot extract an exact mode from a grouped frequency table; we can only estimate it. By using a histogram with equal class widths, we identify the tallest bar, which is the modal class. We draw two straight lines from each of the top corners of the modal class bar to the adjacent corners of the class bars on each side. We then draw a vertical line from the intersection point of these two lines to the 𝑥-axis. And this gives us an estimate for the mode.

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