CDS Mathematics · Averages

Working Through the Total

An average hides a total: sum = number of items × mean, and almost every average question is solved on the totals, not the means.

All Averages notes CDS Mathematics strategy

Why this matters

Twenty PYQs, two of them HARD. Adding a member, dropping one, a sixth item shared by two groups, the largest reading a week can hold, a misread mark: turn every mean into a total, do the arithmetic on totals, and divide once at the end.

Concept 1 of 2: Sum = count × mean

Averages cannot be added or subtracted, but totals can. Multiply each mean by its count, combine the totals the way the story does, then divide by the new count.

Definition

Total $= n \times \bar x$ .
An item added: new item $=$ new total $-$ old total.
Two groups that share an item: that item $=$ (sum of the group totals) $-$ (grand total).
The largest possible single value: set all the others to their smallest allowed value.
Every item raised by $c$ : the mean rises by $c$ .

Total

\text{sum} = n \times \bar{x}

Worked example

The mean of

8

numbers is

12

. One more number is added and the mean becomes

13

. Find it.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q51Easy

Example 1 · Averages · Sum and Mean

The mean of 5 numbers is 15. If one more number is included, the mean of the 6 numbers becomes 17. What is the included number ?

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

6

' gives an equation in the OLD average; the question often asks for the NEW one. Add the

6

back before answering.

Drill 14 more on sum = count × mean

Concept 2 of 2: Correcting a wrong entry

A misread value changed the total by (wrong

-

right). Fix the total, divide by the same count, and the mean moves by that difference over

n

Definition

Correct total $=$ recorded total $-$ wrong values $+$ right values.
The mean changes by $\dfrac{\text{net correction}}{n}$ .
Several corrections: add their effects, keeping signs.

Corrected mean

\bar{x}_{\text{new}} = \bar{x} + \dfrac{\text{right} - \text{wrong}}{n}

Worked example

The mean of

25

marks is

40

. A

73

was entered as

37

. Find the correct mean.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q90Easy

Example 2 · Averages · Sum and Mean

The mean of 20 observations is 17. On checking it was found that the two observations were wrongly copied as 3 and 6. If wrong observations are replaced by correct values 8 and 9, then what is the correct mean ?

Keep the signs

One misreading can raise the total while another lowers it:

95

read as

59

adds

36

25

read as

52

takes away

27

. The net is

+9

, not

63

Drill 4 more on correcting a wrong entry

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Sum = count × mean
Total
$\text{sum} = n \times \bar{x}$
Correcting a wrong entry
Corrected mean
$\bar{x}_{\text{new}} = \bar{x} + \dfrac{\text{right} - \text{wrong}}{n}$

Watch out for (3)

A shared item is counted twice
→ Sum = count × mean
The new average, or the old one?
→ Sum = count × mean
Keep the signs
→ Correcting a wrong entry

Test yourself on Averages

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Averages chapter test Drill all 20 questions