The table shows information about
the heights of 10 sunflowers. The following statements are about
the mean and range of the actual heights of the sunflowers. Tick the correct box for each
statement. The mean could be less than 40
centimetres. True or false? The mean must be more than 40
centimetres. True or false? The mean could be as high as 80
centimetres. True or false?
And there are three further
statements about the range that we’ll consider in a moment. Let’s begin by recalling what we
actually know about the mean of a data set. To find the mean from a list of
numbers, we add together all of the values and we divide by how many values there
are. In this case, we could add together
the heights of the sunflowers and then divide this number by 10, the total number of
To answer this question, we’ll
consider what would need to be the case for each of these statements to be true. What would need to be true for the
mean to be less than 40 centimetres? Well, when we add together the
total heights of the sunflowers and divide that by 10, we would need that number to
be less than 40.
We can form a statement about the
total height of all our sunflowers by multiplying both sides of our inequality by
10. 40 multiplied by 10 is 400. So we can see that the total height
of our sunflowers must be less than 400 centimetres. Let’s think about what would need
to be the case for this statement to be true.
Well, imagine each of our
sunflowers has a height right at the bottom of their class. We have two sunflowers that have a
height a little bit over zero and less than or equal to 40 centimetres. We’ll say that these two sunflowers
have a height of just over zero centimetres. The next four sunflowers have a
height of just over 40 centimetres. Remember 40 is the lower limit of
this class interval. And the last four sunflowers have a
height a little bit bigger than 80 centimetres.
Adding together these values and we
get 480. This means that the total height of
our sunflowers must be greater than 480 or 480 centimetres. We said though that for our first
statement to be correct, the total height of the sunflowers must be less than 400
centimetres. But we’ve just shown that the total
height has to be greater than 480. So this statement must be
false. The mean cannot be less than 40
centimetres. This also means that the mean must
be more than 40 centimetres. This next statement is true.
Let’s now consider the final
statement about the mean: it could be as high as 80 centimetres. We’re going to perform a similar
process to the one we used when we were trying to establish whether the mean could
be less than 40 centimetres. We’re now going to consider the
absolute greatest height of our sunflowers.
The first two sunflowers could be
as tall as 40 centimetres. The next four could be as high as
80 centimetres and the last four could be as tall as 120 centimetres. Adding together these values and
the maximum possible height is 880 centimetres. If this was the case, dividing 880
by 10 to find the mean gives us a value of 88 or 88 centimetres. So in fact, yes, the mean could be
as high as 80 centimetres. This statement is true. In fact, it could be as high as 88
Let’s clear some space and consider
the next three statements. These three statements are the
range must be more than 40 centimetres, true or false? The range could be less than 40
centimetres. True or false? And the range could be over 100
centimetres. True or false?
Remember to find the range of a set
of numbers, we subtract the smallest value from the largest value. This gives us a measure of spread,
of how spread out the data is. We’ll begin by working out the
largest possible value for the range. To find the largest possible value
for the range, we’ll need to subtract the smallest possible height of a sunflower
from the largest possible height of a sunflower.
The largest possible height is 120
centimetres and the smallest possible height is a little bit over zero. We’ll use zero for clarity. And 120 minus zero is 120. In fact, the range will never quite
be 120 centimetres and that’s because the smallest possible height isn’t quite
zero. So we can say that the maximum
range of our heights of sunflowers will be less than 120 or 120 centimetres.
To find the absolute smallest value
for our range, we’re going to subtract the largest smallest height from the smallest
largest height. This time, that’s 80 minus 40 which
is 40. And in fact, because the smallest
largest value is a little bit over 80, we can say that the smallest possible value
for our range will be greater than 40 centimetres. And we now have enough information
to be able to decide which of these statements is true.
We’ve said that our range
absolutely has to be greater than 40 centimetres. So this first statement is
correct. And we can see that the converse is
false. It cannot be less than 40
centimetres. And since the absolute maximum
value of the range is just a little bit under 120 centimetres, we can see that the
range could indeed be over 100 centimetres. This final statement is true.
So we have the range must be more
than 40 centimetres to be true, the range could be less than 40 centimetres to be
false, and the range could be over 100 centimetres to also be true.