Mathematics, three units · Grade 11

Two classes share an average — what else do you need to compare them?

The spread. The average says where the centre of the marks sits, but not how far from it they lie. A small standard deviation means a uniform class whose marks cluster near the average; a large one means a split class, with high achievers and failures, whose average describes nobody in it.

Learning objectives

Mean and median

The mean is the sum of the values divided by how many there are. The median is the middle value once they are sorted from smallest to largest, and with an even count it is the average of the two middle ones.

When the data are symmetric the two sit close together. When there is one extreme value the mean is dragged towards it while the median barely moves — which is exactly what makes them two different measures rather than two routes to the same answer.

This is why a median salary describes a typical person far better than a mean salary. One chief executive on the list lifts the mean for the whole department and leaves the median untouched.

What the standard deviation adds

The standard deviation measures spread: how far the values sit from the mean, on average. It is in the same units as the data, so marks give a deviation in points.

A standard deviation of zero means everybody scored exactly the same. The larger it grows the more scattered the class is, and the less the average describes any actual person.

On a three-unit paper you are almost never asked to compute a standard deviation by hand. What is asked is to compare two given deviations and explain in words what the difference means — and that is where the marks are.

Finding a missing value

When the mean and the count are given, the total is known: mean times count. That is the line which opens almost every question of this kind.

From there it is algebra. If 20 pupils average 74, the total is 1,480. If nineteen of the marks are known, the difference is the missing one.

The same tool answers "what do I need on the last exam": write the total required for the target average, subtract what has already been accumulated, and what remains is the mark needed. If it comes out above 100, the answer is that it cannot be done, and that is a legitimate answer.

Worked examples

  1. Five marks: 60, 70, 80, 90, 100. Find the mean and the median.

    1. Sum: 60 plus 70 plus 80 plus 90 plus 100, which is 400
    2. Mean: 400 over 5
    3. The data are already sorted, and the middle one is the third

    Answer: The mean is 80 and the median is 80 — equal, because the data are symmetric

  2. Change the top mark from 100 to 200. What happens to each measure?

    1. New sum: 500, so the mean is 100
    2. The order has not changed, and the middle value is still 80

    Answer: The mean jumps to 100 and the median stays at 80 — that is the effect of an outlier

  3. 20 pupils average 74. A pupil scoring 95 joins. What is the new average?

    1. Previous total: 74 times 20, which is 1,480
    2. New total: 1,480 plus 95, which is 1,575
    3. Divide by 21

    Answer: The new average is 75

Common mistakes

Taking a median without sorting the data
The median is defined on an ordered list. Picking the middle item of an unsorted one gives an arbitrary number that measures nothing.
Averaging two averages by adding and halving
That works only when the groups are the same size. A class of 30 and a class of 10 need a weighted mean — all the marks together, divided by 40.
Reading a large standard deviation as high marks
Standard deviation measures spread, not level. A weak uniform class and a weak split class can share a low average and differ completely in deviation.

What to remember

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