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CDS Mathematics · Formula sheet

Averages formulas

5 formulas and 6 common traps for CDS Mathematics Averages, grouped by subtopic.

Full notes with worked examples

Working Through the Total

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Sum = count × mean

Total

sum=n×xˉ\text{sum} = n \times \bar{x}

Correcting a wrong entry

Corrected mean

xˉnew=xˉ+right−wrongn\bar{x}_{\text{new}} = \bar{x} + \dfrac{\text{right} - \text{wrong}}{n}

Common traps

A shared item is counted twice

When 'the first six' and 'the last six' of eleven items overlap, their totals together count the sixth item twice. Subtract the grand total to isolate it.

The new average, or the old one?

'Increasing his average by 66' gives an equation in the OLD average; the question often asks for the NEW one. Add the 66 back before answering.

Keep the signs

One misreading can raise the total while another lowers it: 9595 read as 5959 adds 3636, 2525 read as 5252 takes away 2727. The net is +9+9, not 6363.

Weighted and Combined Averages

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The combined average

Combined average

xˉ=n1xˉ1+n2xˉ2n1+n2\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}

Ratio of group sizes

Alligation

n1n2=xˉ2−xˉxˉ−xˉ1\dfrac{n_1}{n_2} = \dfrac{\bar x_2 - \bar x}{\bar x - \bar x_1}

Common traps

Not the plain average of the averages

Averages of 140140 and 152152 combine to 146146 only when the groups are equal. With more students in the taller class, the answer lies above 146146.

The ratio is inverted

The group CLOSER to the combined average is the BIGGER one. With means 6565 and 7070 and a combined 6767, the distances are 22 and 33, so the sizes are 3:23 : 2, not 2:32 : 3.

Averages of Consecutive Numbers

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The middle term

Equally spaced list

average=first+last2\text{average} = \dfrac{\text{first} + \text{last}}{2}

Common traps

Only for equally spaced lists

The first ten composite numbers 4,6,8,9,10,12,14,15,16,184, 6, 8, 9, 10, 12, 14, 15, 16, 18 are not equally spaced; first plus last over two gives 1111, but the true average is 11.211.2.

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