Video Transcript
The cost, in dollars, of cans
of soda in different places is recorded in the table below. Determine an estimate for the
median cost of soda approximated to the nearest hundredth.
In this problem, we are given
the cost of soda as a grouped frequency table. If we consider the first cost
column, this has zero dollars and a hyphen. So, the values in this group
will be the costs that are greater than or equal to zero dollars but less than
50 cents, because that’s the lower boundary of the next class. And the frequency of one means
that one can of soda was in this cost boundary.
Because we don’t know the cost
of every individual can of soda, we can only calculate an estimate for the
median, which is the middle value when the costs are ordered from least to
greatest or greatest to least.
One way in which we can
estimate the median is by drawing a cumulative frequency diagram. We can recall that the
cumulative frequency, or ascending cumulative frequency, of a value 𝑎 indicates
the frequency of values that are less than 𝑎. Now the best way to record the
cumulative frequencies is by adding to the table or perhaps even drawing a new
table.
So let’s draw a new table. And this time, instead of
frequency on the second row, we have cumulative frequency. The first group in the original
table is values that are zero dollars or more but less than 50 cents. However, it is common to
include a starting cumulative frequency of zero. In this context, we are saying
that there were no cans of soda sold for less than zero dollars. The second group in the new
table would be values less than 50 cents. And then we can continue to
create new group headings for the cost in the same way up until the amount of
two dollars.
However, let’s think about what
this group represents. The values in this final group
are two dollars or more. This group doesn’t have an
upper boundary. So we don’t know what values
these are less than. However, in grouped frequency
distributions like this, we can assume that the class widths are all the same,
so in this case 50 cents. We can say that the values in
this group would be two dollars or more and less than two dollars and 50
cents. So in the table we are
creating, the cumulative frequency of the final group would be values less than
two dollars and 50 cents.
Now let’s work out the values
for the cumulative frequencies of each class. The first nonzero cumulative
frequency comes from the first value in the original table. There was one can of soda that
had a cost less than 50 cents. The next cumulative frequency
is for costs less than one dollar. We know that there were six
cans from 50 cents or more up to one dollar. But this one can costing less
than 50 cents is also less than one dollar. So the cumulative frequency is
the sum of these, which is seven.
For the next cost of less than
one dollar 50, we can do the same. 15 cans cost one dollar or more
but less than one dollar 50. So by adding 15 to the previous
cumulative frequency of seven, we know that 22 cans cost less than one dollar
50. And we can find the two
remaining cumulative frequencies by adding 21 and then seven to give us values
of 43 and 50.
Now remember that we’ve done
this so that we can plot a cumulative frequency diagram. The coordinates that we plot on
the graph will have 𝑥-coordinates of the less than values and the
𝑦-coordinates of their respective cumulative frequencies. We then just need to take care
when drawing the grid that we have enough space to include all the values.
So here, we have the points
plotted and joined with a smooth curve. And then we can find an
estimate for the median. Since the highest value in the
cumulative frequency is 50, we know that there were 50 cans of soda. And we can find the median
position by dividing the total frequency, that’s the total number of cans, by
two. Half of 50 is 25, so that means
that the median cost of a can of soda would be the cost of the 25th can. And we can use the graph to
determine an estimate for the cost of the 25th can.
We draw a horizontal line from
25 on the 𝑦-axis until it meets the curve and then draw a vertical line
downwards from this point to the 𝑥-axis. Reading from the graph, the
value for the cost is 1.60. So the answer for the median
cost of a can of soda is one dollar and 60 cents.
