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

Earn 100

The five values in a dataset have a sum of $250$ and standard deviation of $15 .$ A sixth value is added to the dataset, such that the mean is now $40 .$ Find the variance of the six values in the dataset.

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 8

A group of

10

friends played a mini-golf competition. Eight of the friends tied for second place, each with a score of

34,

and the other two friends tied for first place. Find the winning score, given that the standard deviation of the scores of all

10

friends was

= \frac{13625 + (- 10)^{2}}{6} - 40^{2} = 687.5

and that the lowest score in golf wins.

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 9

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 .

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.

The five values in a dataset have a sum of 250 and standard deviation of 15. A sixth value is added to the dataset, such that the mean is now 40. Find the variance of the six values in the dataset.

Important Questions on Measures of Variation

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The five values in a dataset have a sum of $250$ and standard deviation of $15 .$ A sixth value is added to the dataset, such that the mean is now $40 .$ Find the variance of the six values in the dataset.