Video: The Mode of a Data Set

In this video, we will learn how to find and interpret the mode of a data set.

12:09

Video Transcript

In this video, we will learn how to find and interpret the mode of a data set.

The mode is an example of a measure of center or measure of central tendency. If we have a set of data, we can find a single number that represents the whole data set or give us some information about typical values by finding a measure of center. Examples of this are the mean, median, and mode. In this video, we will only discuss the mode. We will begin by looking at a definition.

The mode of a data set is the value which appears most often. If two or more values appear most often, we can have more than one mode. Finally, if all of the values appear the same number of times, there is no mode in the data set. We will now look at some examples where we calculate the mode.

The following data points represent the number of goals scored by a player in 10 consecutive matches: two, one, two, zero, one, two, one, four, four, and two. What was the mode score?

We know that the mode of any data set is the value that appears the most often. In this question, we want to find the number of goals that were scored most often in the 10 matches. We could do this by inspection, just by looking at the data. However, it is often easier to do so using a frequency table or tally chart. In our frequency table, we have two rows: the number of goals and the frequency.

As the least number of goals the player scored was zero and the highest was four, we have the integer values zero, one, two, three, and four. The player scored zero goals once in the 10 matches, as zero only appeared once in the list. They scored one goal three times, two goals four times. They didn’t score three goals at all, as there are no threes in the list. Finally, they scored four goals on two occasions. As there were 10 matches in total, our frequency row needs to add up to 10. One plus three is four. Adding four gives us eight, and adding two gives us 10. This is a quick check that insures we’ve used every item in the data set.

As the mode is the value that appears most often, we need to find the value with the highest frequency. As four is the highest frequency and this corresponds to two goals, we can say that the mode score was two goals. The most common number of goals scored by the player in the 10 matches was two.

We will now look at a question from a larger data set.

The table shows the number of books that 30 students read in a year. Find the mode number of books read.

We know that the mode of a data set is the value which appears the most often. In this question, we need to find the most common number of books read out of the 30 students. One way to do this is to set up a frequency table. Our frequency table in this question would have two rows: the number of books and the frequency. The least number of books read was one and the most was 10. Therefore, this row contains the integers from one to 10.

There were two students that read one book in the year. Two students also read two books. There were three students that read three books. Four students read four books. Two of the students read five books. There were three students that read six books. Three students read seven books. Eight read eight books, two read nine books, and one student read 10 books. As there were 30 students in total, we need to check that our frequencies sum to 30. This is a quick check that ensures we have used each item from the data set.

As the mode is the number that appears most often, we are looking for the highest frequency; this is eight. The frequency of eight corresponds to eight books. Therefore, the mode number of books read by the 30 students is eight. It is important in this question that our answer is given in terms of books read.

Our next question involves finding the mode of nonnumerical data.

A bookstore sold 11 books by Haruki Murakami, two books by Henry Thoreau, and six books by Carl Jung. Determine the mode for this data.

We know that the mode of any data set is the value or item that appears most often. In this question, we have 11 books by Haruki Murakami. We have two books by Henry Thoreau. And finally, we have six books by Carl Jung. We are looking for the book that appears the most. As 11 is greater than two and six, 11 is the highest frequency. As this value corresponds to the books of Haruki Murakami, we can say that he is the mode for this data. The mode will be the author with the highest number of books.

As mentioned at the start of this video, some data sets have more than one mode. This is one such question.

Find the mode of the values four, seven, two, eight, nine, three, four, two, four, eight, six, and eight.

The mode of a data set is the value that appears most often. We note however that there can be more than one mode if more than one value appears most often. We can begin this question by setting up a frequency table containing the value and the frequency. The lowest value from our list was two, and the highest value was nine. We will, therefore, include the integer values from two to nine.

The number two appeared twice in the list. Therefore, it has a frequency of two. There was one three in the list. The number four appeared three times. There were no fives in the list. So, the frequency of five is zero. There was one six in the list. The number seven also appeared once. The number eight was in the list three times. And finally, there was one nine in our list of values.

As there were 12 values altogether, we need to ensure that our frequencies sum to 12. As the frequencies two, one, three, zero, one, one, three, and one do indeed sum to 12, this is a quick check to ensure we have used each of the values from our list. As the mode is the value that appears most often, we are looking for the highest frequency. In this case, this is equal to three and corresponds to the values four and eight. As both four and eight appear three times in the list, the mode of this data set is four and eight.

In this question, we have a situation where there is more than one mode.

Our final question involves calculating the missing value in a data set when the mode is given.

Elizabeth has the following data: three, six, four, five, and 𝑚. If the mode is six, find the value of 𝑚.

We know that the mode of any set of data is the value that appears most often. In this set of data, Elizabeth has the numbers three, six, four, five, and the missing number 𝑚. We are also told that the mode is six. This means that the most popular, or most common, number is six. As three, four, five, and six all appear once at present, the only way that the mode can be six is if the missing number 𝑚 is six. The set of data three, six, four, five, and six will have a mode of six. It is usually very straightforward to find a missing number in a data set when given the mode.

We will finish this video by summarizing the key points. The mode is an example of measure of center, otherwise known as a measure of central tendency. The mode of a set of data is the value which appears the most. In any specific data set, there could be one mode, more than one mode, or no mode at all. The set of data four, seven, six, seven, and five has one mode. This is equal to seven as seven occurs most frequently. The set of data five, eight, four, eight, and four has more than one mode. In fact, there are two modes: four and eight, as these occur more frequently than any other numbers.

Finally, the set of numbers five, seven, six, two, and nine has no mode. This is because each of the numbers occurs just once. None of them occur more frequently than any other number. If we are given a large set of data, it is often useful to draw a frequency table or tally chart to display the data first.

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