A set of n pieces of data has mean x ¯ and standard deviation S . Another set of 2 n pieces of data has mean x ¯ and standard deviation 1 2 S . Find the standard deviation of all these pieces of data together in terms of S .

Rearranging Σ x 2 N - x ¯ 2 = SD 2 gives Σ x 2 = N SD 2 + x ¯ 2 , where N is number of data values. For the n + 2 n = 3 n values in the two datasets together: Sum of squares Σ x 2 = n S 2 + x ¯ 2 + 2 n 1 4 S 2 + x ¯ 2 = 3 n 2 S 2 + 3 n x ¯ 2 Sum of values Σ x = n x ¯ + 2 n x ¯ = 3 n x ¯ We know that, SD ( x ) = Σ x 2 N - Σ x N 2 , where N is total number of data values. Thus, SD = 3 n 2 S 2 + 3 n x ¯ 2 3 n - 3 n x ¯ 3 n 2 = 1 2 S 2 + x ¯ 2 - x ¯ 2 = S 2 or S 2 2 .

EASY

AS and A Level

IMPORTANT

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An author has written $15$ children's books. The first eight books that she wrote contained between $240$ and $250$ pages each. The next six books contained between $180$ and $190$ pages each. Correct to $1$ decimal place, the standard deviation of the number of pages in the $15$ books together is $31.2 .$

Important Questions on Measures of Variation

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3C>Q 10

A set of

n

pieces of data has mean

\bar{x}

and standard deviation

S .

Another set of

2 n

pieces of data has mean

\bar{x}

and standard deviation

\frac{1}{2} S .

Find the standard deviation of all these pieces of data together in terms of

S .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3D>Q 1

Two years ago, the standard deviation of the masses of a group of men and a group of women were

8 kg

and

6 kg,

respectively. Today, all the men are

5 kg

heavier and all the women are

3 kg

lighter. Find the standard deviation for each group today.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3D>Q 2

Twenty readings of $y$ are summarised by the totals $Σ {(y - 5)}^{2} = 890$ and $Σ (y - 5) = 130 .$ Find the standard deviation of $y .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3D>Q 3

The amounts of rainfall,

r mm,

at a certain location were recorded on

365

consecutive days and are summarised by

Σ {(r - 3)}^{2} = 9950

and

Σ (r - 3) = 1795.8

. Calculate the mean daily rainfall and the value of

Σ r^{2} .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3D>Q 4

Exactly

20

years ago, the mean age of a group of boys was

15.7

years and the sum of the squares of their ages was

16000

. If the sum of the squares of their ages has increased by

8224

in this

20

-year period, find the number of boys in the group.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3D>Q 5

Readings from a device, denoted by

y,

are such that

Σ {(y - 3)}^{2} = 2775, Σ y = 105

and the standard deviation of

y

13 .

Find the number of readings that were taken.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3D>Q 6

Mei measured the heights of her classmates and, after correctly analysing her data, she found the mean and standard deviation to be

163.8 cm

and

7.6 cm .

Decide whether or not these measures are valid, given the fact that Mei measured all the heights from the end of the tape measure, which is exactly

1.2 cm

from the zero mark.
Explain your answers.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3D>Q 7

A transport company runs $21$ coaches between two cities every week. In the past, the mean and variance of the journey times were $4$ hours $35$ minutes and $53.29$ minutes?

What would be the mean and standard deviation of the times if all the coaches departed $10$ minutes later and arrived $5$ minutes earlier than in the past?
Are there any situations in which achieving this might actually be possible?

An author has written 15 children's books. The first eight books that she wrote contained between 240 and 250 pages each. The next six books contained between 180 and 190 pages each. Correct to 1 decimal place, the standard deviation of the number of pages in the 15 books together is 31.2.

Important Questions on Measures of Variation

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