### Video Transcript

In this video, we will learn how to
find the range of a data set. We will begin by looking at a
definition of the range and then answer a variety of questions.

The range of a data set is the
difference between the largest and smallest values. We can therefore calculate the
range by subtracting the smallest data value from the largest data value. The importance of calculating the
range is it tells us how spread-out the data is. We will now look at some examples
to practice finding and using the range.

The number of goals scored by
12 soccer players in a season are 13, 11, 12, five, five, nine, six, 11, eight,
five, six, and 19. State whether the following
statement is true or false. The range of the data is 14
goals.

We can calculate the range of
any set of data by subtracting the smallest value from the largest value. Whilst we could find these
values from the list by inspection, it is often useful to rewrite the data set
in numerical order first. The smallest value in the data
set is five. And there are three of
these. There are two sixes in the data
set. The next lowest value is
eight. Continuing our list from
smallest to largest, we have 9, 11, 11, 12, 13, and 19. The smallest value is equal to
five and the largest value is 19. We can therefore calculate the
range by subtracting five from 19. This is equal to 14. The statement in the question
said that the range of the data is 14 goals, which is true.

Our next question involves working
out the range from a set of data represented on a line plot.

The graph shows the weights, in
kilograms, of emperor penguins at a zoo. Is the range of the weights 24
kilograms?

The range of any data set can
be calculated by subtracting the smallest value from the largest value. As the weights on the line plot
are already in order, we can see that the smallest value is 23 kilograms. The largest value is 49
kilograms. We can therefore calculate the
range by subtracting 23 from 49. 40 minus 20 is equal to 20, and
nine minus three is equal to six. Therefore, 49 minus 23 equals
26. The range of values of the
emperor penguins is 26 kilograms. This means that the answer to
the question โIs the range of the weights 24 kilograms?โ is no.

Let the greatest element in a
set be 445 and the range of the set be 254. What is the smallest element of
this set?

We know that, in order to
calculate the range of any set, we subtract the smallest element from the
largest element. In this question, we are given
the largest or greatest element and the range of the set and need to calculate
the smallest element. We can do this by using the
formula or by using the number line. Substituting in our values
gives us 254 is equal to 445 minus ๐ฅ, where ๐ฅ is the smallest element of the
set. Adding ๐ฅ to both sides of this
equation gives us ๐ฅ plus 254 is equal to 445. We can then subtract 254 from
both sides of this equation to work out the value of ๐ฅ. ๐ฅ is equal to 191 as 445 minus
254 is 191. We can therefore conclude that
the smallest element of the set is 191.

As already mentioned, we could
also have calculated this using a number line. We know that the greatest value
is 445. The range is the difference
between the greatest or largest value and the smallest value. In this case, we are told it is
254. The difference between the
smallest and greatest value is 254, which means we can subtract this from 445 to
calculate the smallest value. Once again, this gives us an
answer of 191.

We will now look at a couple of
more complicated problems to finish this video.

The following figure
demonstrates the number of glasses of water a group of people consume per
day. Describe how the range would
change if an additional data value of one was added to the data set.

The graph tells us that eight
people consume zero glasses of water per day. Five people consume one glass;
two people consume two glasses; six people, three glasses; one person, four
glasses; and, finally, seven people consume five glasses of water per day.

We know that, in order to
calculate the range of a data set, we subtract the smallest value from the
largest value. In this question, it will be
the number of glasses. It doesnโt matter how many
different people consume that number of glasses. The smallest value is therefore
zero. As thereโre some people, in
this case, eight, that consume no glasses of water. The largest value is equal to
five as this is the greatest number of glasses of water that anybody
consumes. We can therefore calculate the
range of the original data by subtracting zero from five. This is equal to five.

We are then told that there is
an additional data value of one that is added to the data set. This means that there are now
six people that consumed one glass of water per day. This does not impact the
smallest or largest value though. So the new range is still equal
to five minus zero. Adding the extra data value
does not alter the range. We can therefore conclude that
the range would remain unchanged at five.

Michael has the following data:
Six, eight, ๐, eight, eight, and nine. If the range is three, which
number could ๐ be? Is it (A) three, (B) four, (C)
five, (D) six, or (E) 13?

We recall that we can calculate
the range by subtracting the smallest value from the largest value. In this question, we will
consider what the largest value and smallest values are when ๐ takes each of
the five options. One of these options will have
a range of three which will be the correct answer. When ๐ is equal to three, our
list of values in ascending order are three, six, eight, eight, eight, and
nine. As the largest value is nine
and the smallest value is three, the range will be equal to nine minus
three. As this is equal to six, option
(A) is not correct.

When ๐ is equal to four, the
smallest value is four and the largest value is nine. This time the range would be
equal to nine minus four, which is equal to five. Once again, this is not
correct. When ๐ is equal to five the
smallest number is five. The largest number is still
nine. The range in this case is equal
to four, which once again is not correct. When ๐ is equal to six, we
have two sixes. We could write these in either
order. Our list is now six, six,
eight, eight, eight, and nine. As the smallest number in this
set of date of is six and the largest is nine, the range is equal to nine minus
six. This is equal to three, which
suggests that option (D) is correct.

We will check option (E) just
to make sure. This time, ๐ is equal to
13. This means that the smallest
number is six and the largest number is 13. The range is the difference
between these values. 13 minus six is equal to
seven. So this answer is also
incorrect. This means that the correct
answer is option (D). If the range is three, the
number from the list that ๐ could be is six.

There are a few other numbers
that were not one of the options that ๐ could be. As long as six remains the
smallest number and nine remains the largest number, the range will always be
three. This means that ๐ could be any
one of the four integers, six, seven, eight, or nine. In this question, the only one
of those that was listed as an option was six, which is why this is the only
correct answer.

We will now summarize the key
points from this video. The range of a data set is the
difference between the largest and smallest values. We can therefore calculate the
range of any data set by subtracting the smallest value from the largest value. The range of a set of data tells us
how spread-out the data is. This means that adding any extra
values to a data set doesnโt always change the range.

For example, letโs consider the
data set four, seven, 10, 10, and 13. The smallest value here is four,
and the largest value is 13. This means that the range is equal
to nine as 13 minus four equals nine. Adding in any extra values between
four and 13 inclusive will not affect the range. For example, if we added the number
eight, the smallest number is still four and the largest number is 13. This means that the range is still
nine.

It is also important to note that
it doesnโt matter how many of each value we have. If we added any extra fours or
extra 13s to this list, the range would still be nine. As well as using a list of data
values, we can also calculate the range from a frequency table or graph.