Video Transcript
From the following cumulative
frequency graph that represents the masses of some balls that have different
colors, find an estimate for the median. Option (A) 1.7 kilograms,
option (B) 2.1 kilograms, option (C) 2.7 kilograms, option (D) 2.9 kilograms, or
option (E) 3.1 kilograms.
Let’s begin by noting that
cumulative frequency is a running total of the frequencies. And the median of a set of
values is the middle value. Now in this problem, we are
considering a number of balls that have different masses. If we had all the masses of the
balls, we could order them from lightest to heaviest or heaviest to
lightest. Then the median value would be
the mass of the ball at the middle position. But even if we did have the
original masses of each individual ball, instead of having to list all the
masses of them in order, we could use the cumulative frequency graph to help
us.
This first reading on the chart
with an 𝑥-value of a mass of one kilogram and a 𝑦-value of two for cumulative
frequency means that there are two balls with a mass less than one kilogram. And the next value of seven
balls at the two-kilogram mark doesn’t mean that seven balls have a mass of two
kilograms. It means that seven balls have
a mass that is less than two kilograms, and this includes the two balls that
have a mass less than one kilogram.
So how many balls are there in
total in the problem? Well, by looking at the highest
point on the cumulative frequency graph, we can see that 15 balls have a mass
less than five kilograms. So there are 15 balls in
total. So if we had the 15 balls laid
out in order of lightest to heaviest, we need to work out the median
position. And the position of the median
is at half of the total frequency. Half of 15 is 7.5.
Then to find the median using
the graph, we draw a horizontal line from the 𝑦-axis at the median position of
7.5 like this until this line meets the curve. Next, we draw a vertical line
downwards from this point to the 𝑥-axis. It is this point on the 𝑥-axis
that gives us the estimate for the median.
Reading the axis carefully, we
can give the answer that an estimate for the median mass of the balls is 2.1
kilograms, which was the answer given in option (B).
We should be careful not to
make a very common mistake when using cumulative frequency graphs to find the
median. This comes from incorrectly
taking half of the value on the 𝑥-axis. Half of the total possible
masses of five kilograms is 2.5 kilograms, but this is not an estimate for the
median. Neither should we draw a line
upwards from this point to the curve and then read the value from the
𝑦-axis. If we did do this, it would
only tell us that approximately 9.7 balls had a mass less than 2.5 kilograms,
but it wouldn’t tell us anything about the median. And therefore, we should take
care to calculate half of the final cumulative frequency to find the median
position and then draw a line from this point to correctly estimate the
median.
