### Video Transcript

In this video, we will learn how to
find and interpret the median of a data set. The median is a type of
average. And we will begin by defining
it. The median is an example of a
measure of center or a measure of central tendency. The mode and mean are also measures
of center, although in this video we will only discuss the median. The median is a single number which
gives us some information about the typical values in a data set. Specifically, the median of a set
of data represents the middle value. Half of the data is above the
median and half of the data is below the median. We will begin by looking at some
examples where we calculate the median of a small data set.

Find the median of the values six,
eight, 16, six, and 19.

The median is the middle value of a
set of data. In order to calculate the median of
a small set of data, we follow two simple steps. Firstly, we put the numbers in
ascending order. In this question, the order will be
six, six, eight, 16, and 19. Our second step is to find the
middle number or value. With a small data set, we can do
this by crossing off numbers from either end. We begin by crossing off six and
19, the smallest and largest numbers. We then cross off six and 16. We are left with one number in the
middle. Therefore, the median of the set of
values is eight.

An alternative way of finding the
median after putting the numbers in ascending order is to use a little rule or
formula. The median position can be
calculated using the formula ๐ plus one divided by two, where ๐ is the number of
values. In this question, we have five
numbers. Therefore, the median position can
be calculated by adding five to one and then dividing by two. This is equal to three. The median number will therefore be
the third number in the list. Counting the numbers from left to
right in ascending order, we see that the third number is eight. Therefore, the median is eight.

Find the median of the values 13,
five, nine, 10, two, and 15.

We can calculate the median of any
data set by following two steps. Firstly, we put the numbers in
ascending order. In this question, our six numbers
in ascending order are two, five, nine, 10, 13, and 15. The median is the middle
number. Therefore, we need to find the
middle value from our list. One way to do this is to cross off
a number from each end of the list. We cross off the highest number,
15, and the lowest number, two. We then cross off the next highest
and next lowest, 13 and five. We are now left with two middle
values, nine and 10.

To find the median in this case, we
find the number that is halfway between the middle values. This can be calculated by adding
the two middle values and then dividing by two. Nine plus 10 is equal to 19, and
dividing this by two gives us 9.5. The median of the set of six values
is therefore equal to 9.5. Half of our values must be above
this, in this case, 10, 13, and 15. And half of the values must be
below 9.5, nine, five, and two.

An alternative method here wouldโve
been to have found the median position. We do this using the formula ๐
plus one divided by two, where ๐ is the number of values. In this question, we had six
values. We need to add six to one and
divide by two. This is equal to 3.5. The median will therefore lie
between the third and fourth value. As the third value was equal to
nine and the fourth value 10, once again, we have proved that the median was
9.5.

What is the median of the following
numbers: 11, 11, eight, eight, nine, and nine?

The median of any data set is the
middle number. We can calculate this using two
steps. Firstly, we put the numbers in
ascending order. In this question, we have two
eights, followed by two nines, followed by two 11s. For a small data set like this, we
can then find the middle number by crossing off one number from either end, firstly,
the eight and the 11. We can then cross off the second
eight and the second 11. We are left with two middle
numbers. Normally, when there are two middle
values, we have to calculate the number that is halfway between them. In this case, since both the middle
numbers are the same, this value is the median. The median of the six numbers 11,
11, eight, eight, nine, and nine is nine.

The next question weโll look at is
more complicated as there is a missing number, but we are given the median.

Jennifer has the following data:
10, eight, seven, nine, and ๐. If the median is eight, which
number could ๐ be? Is it (A) seven, (B) 8.5, (C) nine,
(D) 9.5, or (E) 10?

The median represents the middle of
the data. This means that half of the data
values are above it and half are below it. We can start by putting the data in
order and thinking about how we can work out the number ๐. Writing the four values that we
know in order gives us seven, eight, nine, and 10. We are told that the median is
eight. As the median is the middle of five
values, it must be the third number when listed in order from lowest to highest. There must be two values above
eight and two values below eight. For the number to be written to the
left of eight, it must be less than or equal to eight.

If ๐ was any integer value less
than seven, it would be the first number in the list. Positive integers here could be
one, two, three, four, five, and six. None of these are options in the
question, though. The missing number could also be
seven as both the sevens would be written to the left-hand side of eight as they are
less than eight. Out of our five options, the
correct answer is option (A), seven. ๐ could not be 8.5, nine, 9.5, or
10 as all of these are greater than eight and would be on the right-hand side of
eight.

It is possible that the missing
number couldโve been eight. If this was the case, seven would
be the first number in our list. We would then have two eights. The third number would still be
eight, which is the median. Whilst the value of ๐ could be any
number less than or equal to eight, the correct answer in this case is seven.

The final question that weโll look
at is more complicated as it involves a large data set.

Find the median of the set of data
represented in this line plot.

The number line here goes from two
to 14. And the number of crosses
represents how many of each value we have. There are four crosses above the
number four. Therefore, we have four fours. There are six crosses above the
number five, so we have six fives. We have two sixes and two
sevens. Continuing this, we have six
eights, three nines, two 10s, three 11s, three 12s, six 13s, and three 14s. We know that the median is the
middle value when the numbers are in ascending order.

One way to answer this question
would be to write all the numbers out from smallest to largest. We would write the number four four
times. We would write the number five six
times. There would be two sixes and two
sevens. The list would continue as shown
all the way up to three 14s. There are a total here of 40
values. We could find the middle number by
crossing off one from each end, firstly, the highest number and the lowest
number. We repeat this by crossing another
four and another 14. Crossing off the next 10 smallest
numbers and next 10 largest numbers would leave us with the numbers from seven to
11. We could continue this process
until weโre left with two middle values, eight and nine.

When there are an even number of
values in total, there will always be two middle values. We can then find the median by
finding the number that is halfway between them. We do this in this case by adding
eight and nine and then dividing the answer by two. Eight plus nine is equal to 17, and
half of this is 8.5. Clearly, 8.5 is halfway between
eight and nine. Therefore, this is the median of
the set of data.

An alternative method here would be
to calculate the median position first. The median position can be
calculated using the formula ๐ plus one divided by two, where ๐ is the total
number of values. 40 plus one is equal to 41. And dividing this by two gives us
20.5. As 20.5 lies between the integers
20 and 21, we know that the median will be halfway between the 20th and 21st
value.

To find the 20th and 21st values,
we can work out the running total or cumulative frequency. We do this by adding the number of
values we have. Four plus six is equal to 10. Adding another two gives us 12. This means that 12 values are six
or lower. Adding another two gives us 14, and
adding six gives us 20. This means that there are 20 values
that are eight or less. As there are 40 values in total,
there must therefore be 20 values that are nine or greater. The 20th value is equal to eight,
and the 21st value is equal to nine. Once again, finding the midpoint of
these two values gives us a median of 8.5. This method is useful when we have
a large number of values as it saves writing out the whole data set.

We will now summarize the key
points from this video on the median of a set of data. In order to calculate the median
from a data set, our first step is to write the data in ascending order from least
to greatest. Next, we need to find the middle
value or values. If there are an odd number of data
values, there will be exactly one value in the middle. However, if there are an even
number of data values, there will be two values in the middle. Our third and final step is to find
the median.

If there are an odd number of data
values, the median is the middle number. If there are an even number of data
values, the median is halfway between the two middle numbers. Finally, if we have a large data
set, we can calculate the median position using the formula ๐ plus one divided by
two, where ๐ is the total number of data values.