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 .

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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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 $S .$

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 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?

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 8

During a sale, a boy bought six pairs of jeans, each with leg length

x cm .

He also bought four pairs of pants, each with leg length

(x - 2) cm .

The boy is quite short, so his father removed

4 cm

from the length of each trouser leg. Find the variance of the leg lengths after his father made the alterations.

A set of n pieces of data has mean x¯ and standard deviation S. Another set of 2n pieces of data has mean x¯ and standard deviation 12S. Find the standard deviation of all these pieces of data together in terms of S.

Important Questions on Measures of Variation

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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 $S .$