Cumulative Frequency Diagrams

From a frequency table to a cumulative frequency table

A cumulative frequency is a running total. Start at the top of the table and keep adding: the cumulative frequency against a class is the number of values that fall in that class or in any class before it. If 8 students took up to 10 minutes and another 8 took between 10 and 20 minutes, then the cumulative frequency at 20 minutes is $8+8=16$ - sixteen students took 20 minutes or less.

Cambridge writes the classes against an upper boundary, so a cumulative frequency table reads $t\le 10$, $t\le 20$, $t\le 30$ and so on. The last cumulative frequency is always the total number of values, which makes it a free check: if your final figure is not the total given in the question, an addition has gone wrong somewhere above it.

Each entry in the table is a statement about everything so far, not about one class. The 52 written against $t\le 40$ means 52 students took 40 minutes or less; it does not mean that 52 students took between 30 and 40 minutes. To get a single class frequency back out of the table, subtract one cumulative frequency from the next.

Plotting the cumulative frequency curve

A cumulative frequency diagram plots the cumulative frequency against the upper class boundary of each class, never the midpoint. For the class $20<t\le 30$ the point is at $t=30$, because the cumulative frequency of 36 counts everyone who took 30 minutes or less - and that is a statement about 30, not about 25.

The first point is not at the origin. No values are counted until the end of the first class, so the curve begins at the first upper boundary, which here is $(10,8)$. Some books start at $(0,0)$ when the classes begin at 0 and no value can be negative; if the classes start above 0, start at the first upper boundary instead.

Plot each point as a small cross and join them with a single smooth curve, drawn freehand in one flowing line. Cambridge's own note is that the points should be clearly marked, for example as small crosses, and joined with a smooth curve. The curve rises the whole way across and flattens out as it approaches the total at the right-hand end.

• Horizontal axis: the variable itself - time in minutes, mass in grams - scaled to the last upper class boundary
• Vertical axis: cumulative frequency, from 0 up to the total number of values
• The class $20<t\le 30$ with cumulative frequency 36 is plotted at $(30,36)$, not at $(25,36)$
• The last point is always (last boundary, total): here $(60,80)$

Reading the median off the curve

The median is the middle value in order. With $n$ values it sits at position $\frac{n}{2}$, so for the 80 students it is the 40th value. Find 40 on the cumulative frequency axis, move horizontally across to the curve, then drop vertically to the horizontal axis and read the value there. For this data the curve gives about 32.5 minutes.

The answer is an estimate read from a graph, so a small range of readings is accepted. Mark schemes for this topic allow roughly $\pm 1$ of the value the curve gives, which is why 32 and 33 both score as well as 32.5.

Draw the two guide lines on the diagram even when you can see the answer. A reading taken at $\frac{n}{2}$ is usually worth a mark on its own, and a bare number with no lines drawn can lose it. The guide lines also make it obvious to an examiner that a slightly different reading is a reading rather than a guess.

• Median position: $\frac{n}{2}$
• Lower quartile position: $\frac{n}{4}$
• Upper quartile position: $\frac{3n}{4}$
• For $n=80$: the median is the 40th value, the lower quartile the 20th and the upper quartile the 60th

The quartiles and the interquartile range

The lower quartile is the value one quarter of the way up the ordered list, at position $\frac{n}{4}$, and the upper quartile is three quarters of the way up, at position $\frac{3n}{4}$. For the 80 students those are the 20th and 60th values, which the curve turns into about 22 minutes and about 45 minutes.

The interquartile range is the upper quartile minus the lower quartile: $45-22=23$ minutes. It measures the spread of the middle half of the data, so it takes no notice of the extreme values at either end.

That is exactly why the interquartile range is the more useful measure when the data contains one unusually large or unusually small value. A single student who took 58 minutes stretches the range out to nearly an hour, while the interquartile range barely moves, because that student sits above the upper quartile and is not part of the middle half.

How many values lie above or below a given value

Reading from a value on the horizontal axis up to the curve and across to the vertical axis gives the number of values at or below that value. The number above it is the total minus that reading, because every value is either at or below the value or above it.

For the 80 students, the curve passes through $(40,52)$, so 52 students took 40 minutes or less and $80-52=28$ took more than 40 minutes. When the value you are asked about is one of the class boundaries, the curve gives it exactly and there is no estimation to do at all.

If the value sits inside a class, read the curve where the value cuts it and expect an answer within about 1 or 2 of the exact figure. Do not try to find a class frequency and add it on: the question is about a single point on the horizontal axis, and the curve answers it directly.

• Number who took 40 minutes or less: read the curve at 40, which gives 52
• Number who took more than 40 minutes: $80-52=28$
• Number who took 30 minutes or less: 36, straight from a plotted point
• Number who took more than 30 minutes: $80-36=44$

Percentiles, and comparing two distributions

A percentile is the value below which a given percentage of the data lies. The median is the 50th percentile, the lower quartile the 25th and the upper quartile the 75th. To read the 90th percentile of the 80 students, work out $\frac{90}{100}\times 80=72$ first, find 72 on the cumulative frequency axis, move across to the curve and read down.

Two cumulative frequency curves drawn on the same axes can be compared straight off the page. The curve that sits further to the right has larger values, so its median is higher. The curve that rises more steeply through the middle has less spread, so its interquartile range is smaller.

When a question asks which set of data is more consistent, or more reliable, or less variable, the answer is the one with the smaller interquartile range - and the reason has to be given in those words. Naming the group earns one mark out of two; saying that its interquartile range is smaller, so the middle half of its values is closer together, earns both.