Cumulative frequency diagrams

What cumulative frequency means Extended

The cumulative frequency of a value is the number of data items that are less than or equal to that value. It is a running total of the frequencies.

Grouped continuous data, such as times or masses, is usually given in classes like $10<t⩽20$. Once the data is grouped we no longer know the exact values, only how many fall in each class. What we do know for certain is how many values are at most 10, at most 20, at most 30, and so on: each of those is the total of the frequencies up to and including that class.

That is why a cumulative frequency is always attached to the upper bound of a class. The cumulative frequency for $t⩽20$ counts every value in the classes $0<t⩽10$ and $10<t⩽20$, and only when we reach the top of the second class can we be sure every one of those values has been counted.

A cumulative frequency diagram turns these running totals into a curve. From the curve you can estimate how many values lie below any value you choose, and, working the other way, which value has a given number of data items below it. That second idea is what gives us the median, the quartiles and the percentiles.

Building a cumulative frequency table Extended

To turn a grouped frequency table into a cumulative frequency table:

1. Write each class as "less than or equal to its upper bound": $t⩽10$, $t⩽20$, and so on.
2. The first cumulative frequency is the first frequency.
3. Each later cumulative frequency is the one before plus the frequency of the new class.
4. Check: the last cumulative frequency must equal the total number of data items. If it does not, an addition has gone wrong.

Cumulative frequencies can never go down. Each one is at least as big as the one before, because you are only ever adding more data. A table whose running totals fall has a mistake in it.

Drawing the cumulative frequency curve Extended

Put the variable (time, mass, length) on the horizontal axis and cumulative frequency on the vertical axis, which goes from 0 up to the total.

Plot each cumulative frequency at the upper bound of its class: for the commuters, $(10,4)$, $(20,15)$, $(30,34)$ and so on. Not at the midpoint, and not at the lower bound. The reason is the one above: only at the top of the class do we know that all of those values have been counted.

Also plot the starting point: at the lower bound of the first class the cumulative frequency is 0, because no commuter took 0 minutes or less. So the curve starts at $(0,0)$.

The syllabus asks for the plotted points to be clearly marked, for example as small crosses ($\times$), and joined with a smooth curve. Draw it freehand through every cross in one smooth sweep, not with a ruler from point to point. A smooth curve reflects the assumption that the values are spread out evenly within each class, rather than bunched up at the class boundaries.

Most cumulative frequency curves are a stretched S shape: shallow at first, steep in the middle where most of the data is, then levelling off at the total. A steep section means a lot of data in a narrow range of values.

Median, quartiles and percentiles from the curve Extended

If the $n$ data items were put in order, the median would be the value halfway along, the lower quartile a quarter of the way along, and the upper quartile three quarters of the way along. On a cumulative frequency curve, "a quarter of the way along" means "where the cumulative frequency is a quarter of the total". So for $n$ data items:

• median: read at cumulative frequency $\frac{1}{2}n$;
• lower quartile (LQ): read at $\frac{1}{4}n$;
• upper quartile (UQ): read at $\frac{3}{4}n$;
• the $p$th percentile: read at $p\%$ of $n$. The 90th percentile, for example, is read at $0.9n$: 90% of the data is at or below it.

The median is the 50th percentile, the lower quartile the 25th and the upper quartile the 75th.

To read a value: find the cumulative frequency on the vertical axis, go across to the curve, then down to the horizontal axis and read the value there. Draw these lines on the graph in the exam. They show the examiner your method, and a method mark can be awarded even if the reading is slightly out.

With a small set of ungrouped data you may have met the median as the $\frac{n+1}{2}$th value. On a cumulative frequency curve the data is grouped, every reading is an estimate, and the curve is continuous, so use $\frac{1}{2}n$, $\frac{1}{4}n$ and $\frac{3}{4}n$. These are what the mark schemes expect.

All of these are estimates, for two reasons: the original values were grouped, so we do not know them exactly, and the curve between the plotted points is drawn by eye.

The interquartile range Extended

$$\text{interquartile range}=\text{upper quartile}-\text{lower quartile}$$

The interquartile range (IQR) is the width of the middle half of the data. A quarter of the data lies below the lower quartile and a quarter above the upper quartile, so half lies between them.

The IQR measures spread, or consistency. A small IQR means the middle half of the data is tightly bunched; a large IQR means it is spread out. Unlike the range, it is not affected by a few extreme values, because it ignores the bottom quarter and the top quarter of the data entirely.

The IQR is a difference of two readings, so it is always a positive number, in the same units as the data. Do not give the two quartiles as your answer: subtract them.

Comparing two distributions Extended

To compare two sets of data from their cumulative frequency curves, make two comparisons:

1. An average: compare the medians. The higher median means the values are generally higher.
2. A spread: compare the interquartile ranges. The smaller IQR means the values are more consistent.

Always say what the comparison means in the context, not just which number is bigger. "The median time for Route A is 4 minutes less than for Route B, so journeys on Route A are generally quicker" earns the mark; "Route A has a lower median" alone may not.

On a single graph, the curve that is further to the right has the higher values, and the steeper curve has the data more bunched together.

Reading the other way: how many values? Extended

A cumulative frequency curve also answers questions such as "how many commuters took 45 minutes or less?". This time start on the horizontal axis: go up from 45 to the curve, then across to the cumulative frequency axis and read the number.

Three kinds of question come from that one reading:

• at most a value: the reading itself;
• more than a value: the total minus the reading;
• between two values: read at both and subtract the smaller reading from the larger.

A question may ask for a percentage instead: divide the number by the total and multiply by 100.

Questions of this type are often dressed up as a claim to test, such as "at least 80% of patients are seen within 25 minutes". Find the relevant number or percentage from the curve, compare it with the claim, and say clearly whether the claim is supported.