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.
