Mathematics, three units · Grade 11

How do you find an average from a frequency table rather than from a list?

Multiply each value by its frequency, add all those products, and divide by the sum of the frequencies — not by the number of rows in the table. If a mark of 80 appears 7 times it contributes 560 to the total and 7 to the count. Dividing by the row count is the commonest error on this item.

Learning objectives

Frequency and relative frequency

Frequency is how many times a value appears. Relative frequency is that frequency divided by the total number of observations, written as a fraction, a decimal or a percentage.

One quick check is always worth making: the relative frequencies must add to one, or to a hundred percent. If they do not, something was misread or miscalculated, and it is better to find that out here.

The modal value is the one with the highest frequency. With grouped data we speak of the modal group, which is the tallest rectangle on the histogram.

Histograms from grouped data

When there are many distinct values they are collected into intervals — 0 to 10, 10 to 20 and so on — and a rectangle is drawn for each, its height being the frequency.

On a histogram the rectangles touch, because the horizontal axis is continuous. On an ordinary bar chart, where the categories are separate, there are gaps between them. That is the distinction, and it is not cosmetic.

A histogram gives you the modal group and the shape of the spread, but not the exact values. That is the price of grouping, which is why an average computed from grouped data is always an approximation taken from the midpoint of each interval.

A graph that misleads without lying

The last item on the paper sometimes hands you a graph and asks whether it misleads. The answer is nearly always in the vertical axis.

An axis that does not start at zero magnifies small differences until they look dramatic: sales rising from 98 to 102 look like a doubling if the axis begins at 97. The data are correct and the picture lies.

Two other common devices: intervals of unequal width on a histogram, which distort the ratio between rectangles, and a three-dimensional drawing in which the volume grows far faster than the number. In your answer, name exactly what the designer did rather than merely saying the graph misleads.

Worked examples

  1. A table shows: 70 appears 4 times, 80 appears 7 times, 90 appears 9 times. Find the mean.

    1. Multiply: 70 times 4 is 280
    2. 80 times 7 is 560, and 90 times 9 is 810
    3. Sum of products: 1,650
    4. Sum of frequencies: 4 plus 7 plus 9, which is 20

    Answer: 1,650 over 20, which is 82.5

  2. In that same table, what is the relative frequency of the mark 80?

    1. Its frequency is 7
    2. The total number of observations is 20
    3. 7 over 20

    Answer: 0.35, that is 35 percent

  3. A sales chart shows bars of 98, 100 and 102, with the axis starting at 97. How does it mislead?

    1. The real difference from first to last is 4 out of about 100, roughly 4 percent
    2. On the drawing, the first bar stands one unit above the axis and the last stands five
    3. The ratio the eye sees is five to one

    Answer: The axis does not start at zero, so a 4 percent change looks like a fivefold one

Common mistakes

Dividing by the number of rows instead of the sum of the frequencies
Each row stands for several observations, not one. Dividing by the row count treats a mark that appeared once and a mark that appeared ten times as though they carried equal weight.
Drawing a histogram with gaps between the rectangles
The horizontal axis of a histogram is continuous, so the intervals border one another. Gaps turn it into a bar chart, which describes separate categories rather than intervals.
Writing that the graph misleads without saying how
The mark is for the explanation. A complete answer points at the axis that does not start at zero, the unequal interval widths or the three-dimensional volume, and says what that does to the reader.

What to remember

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