Video: Mean, Median, and Mode

In this video, we will learn how to find the measures of central tendency like the mean, median, and mode.

17:30

Video Transcript

In this video, we will learn how to find three measures of central tendency, or average. They are the mean, the median, and the mode. We will begin by defining all three terms. We will then look at some questions where we need to calculate the mean, median, and mode from a data set.

We will begin by defining the mean. To calculate the mean of a numerical data set, we add up all the data values and then divide the total by the number of values in the data set. For example, if we had the set of numbers four, three, seven, six, and five, we would firstly add all the values to give us a total of 25. As there were five values, the mean would be 25 divided by five, which is equal to five.

Let’s now consider the median. To work out the median of a data set, we follow two steps. Firstly, we organize the data according to size in either ascending or descending order. Secondly, we count the values. If the number of values is an odd number, then the median is the middle value. If the number of values is even, then the median is the mean of the two values at the center of the ordered data set.

Consider the set of numbers four, nine, eight, six, and three. Putting these numbers in ascending order from smallest to largest gives us three, four, six, eight, and nine. As there are five numbers in total, the third number in ascending order is the median, in this case, six. If our data set had had an extra number 10, then our list in ascending order would be three, four, six, eight, nine, 10. This time, our data set has an even number of values, so there are two middle numbers, six and eight. The median can be calculated by finding the midpoint of these two values, or the mean of these two values. The answer, in this case, is seven.

Finally, let’s consider the definition of the mode. The mode is the most commonly or frequently occurring value or values. We sometimes call this the modal value. If we consider the data set four, seven, six, seven, eight, and two, we notice that the number seven appears twice. Therefore, this is the mode or modal value. A data set may have one mode, more than one mode, or no mode at all. If the data set has two modes, we say it is bimodal.

A data set can only have one mean and one median. Whilst there are many different kinds of mean, the one that we are using here, and the one most commonly used, is called the arithmetic mean. We will now look at a question where we will determine the mean, median, and mode for a data set.

The scores earned on a math test are 86, 80, 76, 68, 73, 85, 74, 70, 71, and 70. Find the mean, median, and mode for the set of data.

We recall that in order to calculate the mean from a data set, we firstly need to add all the values. In this question, this is the sum of the scores in the test. We then need to divide this total by the number of values, in this case, 10. The sum of the scores is 753. 753 divided by 10 is 75.3. This means that the mean of the scores is 75.3 marks.

We can calculate the median by finding the middle value. Before doing this, we must order the numbers from smallest to largest or largest to smallest. In ascending order, the scores are 68, 70, 70, 71, 73, 74, 76, 80, 85, and 86. As they are an even number of values, in this case, 10, there will be two middle numbers. These are 73 and 74. The median will be the value halfway between these two numbers, or the mean of the two numbers. This is equal to 73.5. Therefore, the median score is 73.5.

The mode is the most frequently occurring value or values in our data set. In this question, eight of the values appear once, and 70 appears twice. As 70 occurs more often than any other data value, the mode of the scores is 70. This set of data has a mean of 75.3, a median of 73.5, and a mode of 70.

We will now look at a question where we can find the mean, median, and mode from a bar graph.

The bar chart shows the yearly membership of a robotics club from 2001 to 2005. Find the mean, median, mode, and range of the data.

We can see from the bar chart that we have five values, 49, 31, 31, 29, and 50. These are the number of members in each year from 2001 to 2005. The first part of this question asks us to calculate the mean. To find the mean of a set of values, we firstly find the sum of the values. In this case, we add 49, 31, 31, 29, and 50. We then divide this total by the number of values, which in this case is five. The sum of the values is 190. We need to divide this by five. This is equal to 38. So, the mean number of members is 38.

The second part of the question asks us to find the median value. In order to do this, we firstly list the values in ascending or descending order. The median is the middle value. And as there are five values here, the median will be the third value. The median number of members in the club is 31.

Next, we need to work out the mode. This is the most common or most frequently occurring number. The numbers 49, 29, and 50 appear once, whereas the number 31 appears twice. This means that 31 is the mode of the data set.

Finally, we’re asked to calculate the range. This is the highest value minus the lowest value. As the highest value is 50 and lowest value 29, we need to subtract 29 from 50. This is equal to 21. The mean, median, mode, and range of members in the robotics club are 38, 31, 31, and 21, respectively.

We will now look at a question where we need to select a data set with a given mode and median.

Which of the following sets of data has a mode of 48 and a median of 20. Is it A) 48, 21, 11, 48, 20, 17? B) 21, 48, 19, 48, 17, 11? C) 47, 47, 11, 48, 20, 17? D) 10, 16, 19, 21, 47, 47? Or E) 20, 48, 48, 11, 11, 19?

We recall that the mode is the most frequently occurring value. Therefore, we need to find a set of data where 48 is the most common or most frequently occurring number. 48 does not occur in set D. Therefore, this cannot be the correct answer. Whilst there is a 48 in set C, there are two 47s. Therefore, the mode of set C is 47. We can, therefore, rule this out as the correct answer.

Set E has two 48s, but it also has two 11s. This means that it has two modes, or it’s bimodal. It has a mode of 11 and 48. This means that option E is also incorrect. Both option A and option B have two 48s. This is the most frequently occurring value in both data sets. This means that the mode of both of these is 48.

Let’s now consider our second piece of information. The median of the data set has to be 20. We know that the median is the middle value once the numbers are in ascending or descending order. Once we have put both of these data sets in order, we notice that they have an even number of values, in this case, six. This means that there are two middle members, in set A, 20 and 21 and in set B, 19 and 21.

The median can be calculated by finding the mean of these two values. This is the same as finding the midpoint of the two values. The mean of 20 and 21 is 20.5. And the mean of 19 and 21 is 20. This means that set A has a median of 20.5 and set B has a median of 20. We can, therefore, rule out set A. The set of data that has a mode of 48 and a median of 20 is set B, 21, 48, 19, 48, 17, and 11.

Our next question will look at what happens to the mean, median, and mode when removing a data value.

Last month, Daniel scored 82, 61, 86, and 82 in his English quizzes. If his lowest score was to be dropped, which of the following would increase. Is it A) mean, B) median, or C) mode?

In order to answer this question, we can calculate the mean, median, and mode before the score was dropped and after the score was dropped. In order to calculate the mean from a data set, we firstly add all the values, in this case, 82, 61, 86, and 82. We then divide this total by the number of values we have, in this case, four. The total, or sum, of Daniel’s scores is 311. Dividing this by four gives us a mean of 77.75.

Daniel’s lowest score is being dropped. This is 61. In order to calculate the mean after this has been dropped, we add 82, 86, and 82 and then divide by three. This is equal to 84.6 recurring. When Daniel’s lowest score is dropped, his mean has increased from 77.75 to 84.6 recurring. This suggests that mean is the correct answer. It is, however, worth checking whether the median and mode have increased or decreased.

In order to work out the median, we order the values from smallest to largest or largest to smallest. We then find the middle number. As there are an even number of values, there will be two middle numbers. As these are both equal to 82, the median before the lowest score was dropped is 82. After the lowest score is dropped, Daniel has values of 82, 82, and 86. Once again, the median is equal to 82. Daniel’s median score has, therefore, not increased.

The mode is the most frequently occurring value. In both cases, before and after the lowest score was dropped, the mode was 82. This is because the score 82 occurred more often than any other score. We can, therefore, conclude that when Daniel’s lowest score was dropped, the median and mode remain the same, but the mean increased.

Our final question will look at creating a data set given its range, median, and modes.

A set of four numbers has a range of seven, a median of 13, and a mode of 16. Given that the highest number is also the mode, what are the four numbers?

We’re told that there are four numbers in a data set. Let’s consider them in ascending order from left to right. We are told that the mode is 16 and that the highest number is also the mode. As the mode is the most frequently occurring or most common number, we must have two 16s in the last two boxes.

We are told that the median is equal to 13. As the median is the middle value, and there are two middle numbers, each of these must be equidistant from 13. The two numbers in the middle of our data set are 10 and 16 as these have a median of 13. We are told that the range is equal to seven. And this is the difference between the largest and smallest number. 16 minus seven is equal to nine. Therefore, the smallest number is nine. The set of four numbers is nine, 10, 16, and 16.

Let’s now look at some of the key points from this video. We can calculate the mean, median, mode, and range from a data set. A data set can only have one mean and one median. There can, however, be one mode, more than one mode, or no mode at all. There are many different kinds of mean, but the one that we have used, which is most common, is called the arithmetic mean. In this video, we have calculated averages from a data set. We can also do this from discreet and grouped frequency tables.

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