Lesson Video: The Mean of a Data Set | Nagwa Lesson Video: The Mean of a Data Set | Nagwa

Lesson Video: The Mean of a Data Set Mathematics

In this video, we will learn how to calculate the mean of a set of whole numbers or decimals.

14:25

Video Transcript

In this video, we will learn how to calculate the mean of a data set and how to find a missing number in a set, given the other data and the mean.

Let’s begin by recalling what the mean of a set is. The mean is a measure of center; that is, it’s a typical value in the data set. There are other measures of center. The mode and the median would also give us a typical value for the set. In this video, we’ll be focusing on the mean. So, in order to actually calculate the mean, we begin by adding all the data values and then we divide by the number of data values. We can think of this in the form of a formula as the mean equals the sum of the data values over the total number of data values.

So, for example, if we had a data set with five numbers, we would add them together to get the numerator, the sum of these data values, and then divide by five as that’s the total number of values in our set. We’ll now have a look at some questions. And in the first question, we’ll find the mean of several data sets.

Which of the following data sets has a mean of 59? Option (A) 11, 50, 21, 72, 48. Option (B) 52, 76, 23, 50, 15. Option (C) 90, 64, 49, 13, 50. Option (D) 74, 79, 27, 59, 56. Option (E) 81, 34, 85, 21, 76.

In this question, we need to work out which data set has a mean of 59. We can recall that to find the mean, we find the sum of the data values and then divide by the total number of data values. When we find the answer to this as a 59, then we find a data set with a mean of 59.

So, if we begin by looking at the data set in option (A), we’ll start by adding up the data values. So, on our numerator, we’ll have 11 plus 50 plus 21 plus 72 plus 48. And as we have five numbers in our data set, then we’ll be dividing the sum of these values by five. We can simplify this then to 202 over five. We can evaluate this by using a calculator or by changing it into a mixed number, giving us a mean for the set in A as 40.4. And so, option (A) does not have a mean of 59.

We can find the mean of set (B) by applying the same principle. We’re adding together the values in our data set. And as we have five numbers, we’ll be dividing by five. The sum of our values equates to 216. So, we work out 216 over five. This answer of 43.2 means that option (B) does not have a mean of 59.

The mean of option (C) can be found by adding our five values and dividing by five. And so, this time, the mean is 53.2, but still not a mean of 59.

In option (D), the sum of our values is 295. So, we calculate 295 over five, which gives us a mean of 59. And so, we found a data set that has a mean of 59.

We can check our last data set just to make sure we haven’t got another set with a mean of 59. This time, we’re going to use a different method first. We can notice in option (D) that when we found a data set that did have a mean of 59, the total of those numbers was 295. Because we know that five multiplied by 59 would give us 295. In the data set of (E) that we’re going to check, we know that there are also five numbers in the data set. So, a quicker check to see if there’s a mean of 59 would be to check if the numbers add up to 295. So, if we add 81, 34, 85, 21, and 76, we would get 297 not 295. So, we know that the mean of this set will not be 59.

If we wanted to go about this in the longer way by finding the mean, we would add these values and divide by five. So, 297 divided by five is 59.4. It would, of course, round to 59. But it isn’t the exact value we were looking for. So, we can discount option (E). And our answer is option (D) 74, 79, 27, 59, and 56 would be a data set with the mean of 59.

In the next question, we’ll look at what happens when we’re given the mean, but we have to find another piece of information.

In the soccer world cup, a team scored 32 total goals with a mean of two goals per game. How many games did they play in total?

In this question, we know that a soccer team scored 32 goals in total over a number of games. And the mean number of goals is two goals per game. We need to work out how many games they actually played. In this situation, we could write that the mean goals per game is equal to the total number of goals over the total number of games.

We can fill in the fact that we’re told that the mean goals per game is two, the total number of goals is 32, and the number of games is what we need to work out. Let’s call this value 𝑦. We now need to solve to find the value of 𝑦. So, if we multiply both sides of our equation by 𝑦, we would have two 𝑦 equals 32. To find 𝑦 then, we divide both sides of our equation by two, giving us the answer 16. And so, the number of games they played in total is 16.

We can check our answer by thinking that if we know that there are 32 goals in total and we’re saying that they played 16 games. Then the mean number of goals would be two. And so, we must have been correct that there are 16 games played in total.

In the next question, we’ll work out how to find a missing value in a data set, given the mean.

Given the mean of the values eight, 22, 4, 12, and 𝑥 is 15, find the value of 𝑥.

In this question, we’re given a data set which includes an unknown value 𝑥. We’re told that the mean of these values is 15 and we need to find the unknown value of 𝑥. The mean of a data set is found by calculating the sum of the data values and dividing by the total number of data values. So, we can start by adding our values eight plus 22 plus four plus 12 plus 𝑥. And as we know that there are five data values, then we would be dividing by five.

Notice that even though we don’t know what 𝑥 is yet, we still include it as one of the values. We’ve been given that the mean is equal to 15. So, we can put this on the left-hand side of our equation. At the minute, we don’t know the value of 𝑥, but we can go ahead and add together the other values in our set. We can then simplify our calculation as 15 equals 46 plus 𝑥 over five. We now need to rearrange this equation in order to find the value of 𝑥.

Multiplying both sides of this equation by five gives us 75 equals 46 plus 𝑥. Subtracting 46 then from both sides of our equation, we have 75 minus 46 equals 𝑥. 29 equals 𝑥 then. And so, our answer is that 𝑥 equals 29. We can check our answer by substituting the value of 𝑥 equals 29, summing with the rest of the values, dividing by five, and checking that we get a value of 15. And thus confirming the value 𝑥 equals 29.

We’ll now look at two story problems involving the use of the mean.

The ages of the people at a gathering were 49, 27, 37, 44, 34, 36, 19, 24, 23, 40, 20, 21, and 43. When one person joined the gathering, the mean age became 31. How old was the person who joined?

We’re told in this question that there are a number of people at a gathering. We’re also given the ages of these people. We’re told that another person joins the gathering, but we don’t know their age. We’re told, however, that the mean of everybody’s age is 31. So, how do we go about working out the age of this new person from being given the mean?

We can recall that the mean is equal to the sum of the data values over the total number of data values. In this context, the data values here are the ages. And the total number of data values would be the total number of people at the gathering. We can then fill into this formula the information that we’re given. We’re told that the mean is 31. And then to find the sum of the ages, we would add all the ages that we’re given. We don’t know this new arrival’s age. But if we call it 𝑥, then we can also include it in our data sum.

Counting all the people at this gathering and including the person that we don’t know their age would give us a value of 14 on the denominator. Adding together all our values on the numerator, we get 417 plus our unknown value of 𝑥 over 14. Rearranging, we multiply both sides of our equation by 14, giving us 434 equals 417 plus 𝑥. Subtracting 417 from both sides of our equation, we’ll have 434 subtract 417 equals 𝑥, giving us that 𝑥 is equal to 17. As we defined the age of this new person to be 𝑥, then that means that they must be 17 years old.

In order to get a C in math, William must have an average score of 70 or more. His test scores so far are 63.3, 85, 79.73, and 70.57. Calculate the minimum score William needs to get on his final test to earn a C.

In this question, we’re looking at William’s test scores in math. We’re told that he has four results so far and he has a final test. What score does he need to achieve in his final test in order to get a C? And to get a C, we’re told that his average score must be 70 or more. So, what exactly do we mean by average here?

The word average is actually not a very specific word. Sometimes, it means one of mean, mode, or median and, sometimes, it just refers to the mean. In the situation of tests, like we see here, we don’t want the mode as that would be the most common one. And we don’t want the median, as that would be the middle value. We do in fact want to calculate the mean.

In this situation, we could write that the mean score is equal to the sum of the test scores over the number of tests. Let’s begin by finding the sum of the test scores. The unknown test score that we want to find out we can define as any variable. But let’s call it 𝑥 here. As we’re considering this test that he hasn’t sat yet, then the total number of tests will be five. The mean score that we’re interested in here is 70. But in fact, he needs to get a mean that is greater than or equal to 70.

And so, the sum of William’s test scores over the total number of tests, which is five, must be greater than or equal to 70, where 70 is less than or equal to the sum of the scores over five. Simplifying our numerator, we have 70 is less than or equal to 298.6 plus 𝑥 over five. We can then multiply both sides of this inequality by five, giving us that 350 is less than or equal to 298.6 plus 𝑥. We then subtract 298.6 from both sides of this inequality, giving us the value 51.4 is less than or equal to 𝑥.

This means, therefore, that our unknown score of 𝑥 must be greater than or equal to 51.4. We can see that this score is lower than his other scores. So, let’s have a check of our results. Let’s say that we wanted to calculate the mean of just the four tests that William’s already taken. We would add together his test results and then divide by four, since we’re looking at just four tests. We’ve already worked out that the sum of these results is 298.6. And so, dividing by four would give the value of 74.64.

So, what does this mean? Well, William is already performing above the average that he needs, above 70. So, even in his final test, he can score lower than this, but still get a mean of 70 or more. We can give our answer for the minimum score that William needs to get in order to earn a C as 51.4.

We can now summarize the key points of this video. We saw, firstly, that the mean is a measure of center of a data set. In order to calculate the mean, we add the data values and divide by the number of data values. We can also write this as the mean equals the sum of the data values divided by the total number of data values. And finally, as we saw in a number of questions, we can use this formula to find a missing data value, given the other data values and the mean.

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