Probability & Statistics

8 methods · 21 questions

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Means do not add, but absolute totals do. Whenever a question gives the mean of a group, immediately multiply by the count to find the total. Once everything is an absolute total, you can add, subtract, and equate them safely.

Trigger: A question modifying a group (adding, removing, merging) or giving algebraic constraints, using means instead of totals.

Instances:
(i) adding/removing items: new total = old total $\pm$ added items;
(ii) merging groups: combined total = sum of the totals of the sub-groups, which you divide by the total count for the new mean;
(iii) algebraic constraints: mean of $n$ items $= x$ immediately becomes sum $= n x$ .

Linked questions (3)

NSAA 2018 Section 1, Part A (Mathematics questions) · Q7

The mean of

n

numbers is

p

The mean of two of these numbers is

q

The mean of the remaining numbers is 10

Which of the following is a correct expression for

n

in terms of

p

and

q

A $\frac{2 ( q - 10 )}{( p - 10 )}$
B $\frac{2 ( q - 10 )}{( 10 - p )}$
C $\frac{2 ( q - 10 )}{( p + 10 )}$
D $\frac{2 ( 10 - q )}{( p + 10 )}$
E $\frac{2 ( 10 + q )}{( p - 10 )}$
F $\frac{2 ( 10 + q )}{( 10 - p )}$

Worked solution

Answer:A

A mean is a total divided by a count, so the total of all

n

numbers is

n p

.

Two of them have mean

q

, so their total is

2 q

. The remaining

n - 2

numbers have mean

10

, so their total is

10 (n - 2)

.

The two parts make up the whole:

n p = 2 q + 10 (n - 2) ⟹ n p - 10 n = 2 q - 20 ⟹ n (p - 10) = 2 (q - 10) .

Dividing by

p - 10

n = \frac{2 ( q - 10 )}{p - 10} .

The answer is A.

NSAA 2016 Section 1, Part A (Mathematics questions) · Q4

The mean age of the twenty members of a running club is exactly 28.

The mean age increases by exactly 2 years when two new members join.

What is the mean age of the two new members?

A20 years
B22 years
C30 years
D40 years
E50 years
F52 years

Worked solution

Answer:E

Work with totals rather than with means, because totals add and means do not.

Twenty members of mean age exactly

28

have a combined age of

20 \times 28 = 560

.

Once the two new members have joined there are

22

people, and their mean age is

28 + 2 = 30

, so their combined age is

22 \times 30 = 660

.

The two newcomers therefore account for

660 - 560 = 100

years between them, and their mean age is

\frac{100}{2} = 50.

The answer is E.

NSAA 2023 Section 1, Part A (Mathematics questions) · Q4

Six numbers in increasing order are:

2, x, x, y, 23.5, 27.5

The mean of the six numbers is 12.

The median of the six numbers is 7.25.

What is the mode of the six numbers?

A $4.5$
B $4.75$
C $5$
D $9.625$
E $10$
F $11.75$

Worked solution

Answer:A

The mean is 12, so the six numbers total

6 \times 12 = 72

2 + 2 x + y + 23.5 + 27.5 = 72, that is 2 x + y = 19.

With six values in order the median is the mean of the third and fourth:

\frac{x + y}{2} = 7.25, that is x + y = 14.5.

Subtracting the second equation from the first eliminates

y

and gives

x = 4.5

, and then

y = 10

.

The list is

2, 4.5, 4.5, 10, 23.5, 27.5

, which is indeed in increasing order. Only

4.5

appears more than once, so it is the mode.

The answer is A.