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#### Question 1:

Calculate the mean for the following distribution:

 x: 5 6 7 8 9 f: 4 8 14 11 3

The given distribution in tabulated form is

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N andrespectively. Therefore,

Hence the mean is

#### Question 2:

Find the mean of the following data:

 x: 19 21 23 25 27 29 31 f: 13 15 16 18 16 15 13

The given distribution in tabulated form is

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N andrespectively. Therefore,

Hence the mean is

#### Question 3:

Find the mean of the following distribution:

 x: 10 12 20 25 35 f: 3 10 15 7 5

The given distribution in tabulated form is

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing. Find the sum of all entries in the second and third column to obtain N andrespectively.

Find the sum of all entries in the second and third column to obtain N and respectively. Therefore,

Hence the mean is

#### Question 4:

Five coins were simultaneously tossed 1000 times and at each toss the number of heads were observed. The number of tosses during which 0, 1, 2, 3,4 and 5 heads were obtained are shown in the table below. Find the mean number of heads per toss.

 No. of heads per toss No. of tosses 0 1 2 3 4 5 38 144 342 287 164 25 Total 1000

The given data can be tabulated in the form

The row of x denotes the number of heads per toss and the row of f denotes the number of tosses.

Prepare the following frequency table of which the first column consists of the number of heads and the second column the number of tosses (frequencies). Multiply the frequency of each row with the corresponding number of heads to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N (already given in the question) and respectively. Therefore,

Hence the mean number of heads per toss is

#### Question 5:

The mean of the following data is 20.6. Find the value of p.

 x: 10 15 p 25 35 f: 3 10 25 7 5

The given distribution in tabulated form is

We have to find the value of p using the information that the mean of the distribution is 20.6.

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N andrespectively. Therefore,

The mean is

Hence, we have

#### Question 6:

If the mean of the following data is 15, find p.

 x: 5 10 15 20 25 f: 6 p 6 10 5

The given data in tabulated form is

We have to find the value of p using the information that the mean of the data is 15.

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N andrespectively. Therefore,

The mean is

Hence, we have

#### Question 7:

Find the value of p for the following distribution whose mean is 16.6.

 x: 8 12 15 p 20 25 30 f: 12 16 20 24 16 8 4

The given distribution in tabulated form is

We have to find the value of p using the information that the mean of the distribution is 16.6.

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N andrespectively. Therefore,

The mean is

Hence, we have

#### Question 8:

Find the missing value of p for the following distribution whose mean is 12.58.

 x: 5 8 10 12 p 20 25 f: 2 5 8 22 7 4 2

The given distribution in tabulated form is

We have to find the value of p using the information that the mean of the distribution is 12.58.

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N and respectively. Therefore,

The mean is

Hence, we have

#### Question 9:

Find the missing frequency (p) for the following distribution whose mean is 7.68.

 x: 3 5 7 9 11 13 f: 6 8 15 p 8 4

The given distribution in tabulated form is

We have to find the value of p using the information that the mean of the distribution is 7.68.

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N and respectively. Therefore,

The mean is

Hence, we have

#### Question 10:

Find the value of p, if the mean of the following distribution is 20.

 x: 15 17 19 20+p 23 f: 2 3 4 5p 6

The given data in tabulated form is

We have to find the value of p using the information that the mean of the data is 20.

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N and respectively. Therefore,

The mean is

Hence, we have

If p is negative then the 4th frequency becomes negative. But frequency can’t be negative. Hence the possible value of p is 1

#### Question 11:

Candidates of four schools appear in a mathematics test. The data were as follows:

 Schools No. of Candidates Average Score I II III IV 60 48 Not available 40 75 80 55 50

If the average score of the candidates of all the four schools is 66, find the number of candidates that appeared from school III.

Let the number of candidates appeared from school III is.

Then the given data can be tabulated as

The row of x denotes the average scores and the row of f denotes the number of candidates. We have to find the value of p using the information that the average score of all the four schools is 66.

Prepare the following frequency table of which the first column consists of the average scores and the second column the number of candidate (frequencies). Multiply the frequency of each row with the corresponding average scores to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N and respectively. Therefore,

The mean is

Hence, we have

Hence the number of candidates appeared from school III is.

#### Question 12:

Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 50.

 x: 10 30 50 70 90 f: 17 f1 32 f2 19

Total 120.

The given distribution in tabulated form is

We have to find the value of missing frequencies f1 and f2, using the information that the mean of the distribution is 50 and the total frequency is 120.

Prepare the following frequency table of which the first column consists of the values of the variate and the second column the corresponding frequencies. Multiply the frequency of each row with the corresponding values of variable to obtain the third column containing.

Find the sum of all entries in the second and third column to obtain N and respectively. Therefore,

The mean is

Hence, we have two equations

Adding them we get

Putting the value of f1 in the second equation we get

#### Question 1:

Find the median of the following data (1-8)

83, 37, 70, 29, 45, 63, 41, 70, 34, 54

The given data is 83, 37, 70, 29, 45, 63, 41, 70, 34 and 54.

Arranging the given data in ascending order, we have

29, 34, 37, 41, 45, 54, 63, 70, 70, 83

Here, the number of observation, which is an even number.

Hence, the median is

#### Question 2:

133, 73, 89, 108, 94, 104, 94, 85, 100, 120

The given data is 133, 73, 89, 108, 94, 104, 94, 85, 100 and 120.

Arranging the given data in ascending order, we have

73, 85, 89, 94, 94, 100, 104, 108, 120, 133

Here, the number of observation, which is an even number.

Hence, the median is

#### Question 3:

31, 38, 27, 28, 36, 25, 35, 40

The given data is 31, 38, 27, 28, 36, 25, 35 and 40.

Arranging the given data in ascending order, we have

25, 27, 28, 31, 35, 36, 38, 40

Here, the number of observation, which is an even number.

Hence, the median is

#### Question 4:

15, 6, 16, 8, 22, 21, 9, 18, 25

The given data is 15, 6, 16, 8, 22, 21, 9, 18, and 25.

Arranging the given data in ascending order, we have

6, 8, 9, 15, 16, 18, 21, 22, 25

Here, the number of observation, which is an odd number.

Hence, the median is

#### Question 5:

41, 43, 127, 99, 71, 92, 71, 58, 57

The given data is 41, 43, 127, 99, 71, 92, 71, 58 and 57.

Arranging the given data in ascending order, we have

41, 43, 57, 58, 71, 71, 92, 99, 127

Here, the number of observation, which is an odd number.

Hence, the median is

#### Question 6:

25, 34, 31, 23, 22, 26, 35, 29, 20, 32

The given data is 25, 34, 31, 23, 22, 26, 35, 29, 20 and 32.

Arranging the given data in ascending order, we have

20, 22, 23, 25, 26, 29, 31, 32, 34, 35

Here, the number of observation, which is an even number.

Hence, the median is

#### Question 7:

12, 17, 3, 14, 5, 8, 7, 15

The given data is 12, 17, 3, 14, 5, 8, 7 and 15.

Arranging the given data in ascending order, we have

3, 5, 7, 8, 12, 14, 15, 17

Here, the number of observation, which is an even number.

Hence, the median is

#### Question 8:

92, 35, 67, 85, 72, 81, 56, 51, 42, 69

The given data is 92, 35, 67, 85, 72, 81, 56, 51, 42 and 69.

Arranging the given data in ascending order, we have

35, 42, 51, 56, 67, 69, 72, 81, 85, 92

Here, the number of observation, which is an even number.

Hence, the median is

#### Question 9:

Numbers 50, 42, 35, 2x + 10, 2x − 8, 12, 11, 8 are written in descending order and their median is 25, find x.

The given data in descending order is

50, 42, 35, 2x+10, 2x-8, 12, 11, 8

Here, the number of observation, which is an even number.

Hence, the median is

But, it is given that the median is 25. Hence, we have

#### Question 10:

Find the median of the following observations: 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33. If 92 is replaced by 99 and 41 by 43 in the above data, find the new median?

The given data is 46, 64, 87, 41, 58, 77, 35, 90, 55, 92 and 33.

Arranging the given data in ascending order, we have

33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92

Here, the number of observation, which is an odd number.

Hence, the median is

If 92 and 41 are replaced by 99 and 43 respectively, then the given data becomes 46, 64, 87, 43, 58, 77, 35, 90, 55, 99 and 33.

Arranging the new data in ascending order, we have

33, 35, 43, 46, 55, 58, 64, 77, 87, 90, 99

Here, the number of observation, which is an odd number.

Hence, the new median is

#### Question 11:

Find the median of the following data: 41, 43, 127, 99, 61, 92, 71, 58, 57 If 58 is replaced by 85, what will be the new median

The given data is 41, 43, 127, 99, 61, 92, 71, 58 and 57.

Arranging the given data in ascending order, we have

41, 43, 57, 58, 61, 71, 92, 99, 127

Here, the number of observation, which is an odd number.

Hence, the median is

Ifis replaced by 85, then the given data become 41, 43, 127, 99, 61, 92, 71, 85 and 57.

Arranging the new data in ascending order, we have

41, 43, 57, 61, 71, 85, 92, 99, 127

Here, the number of observation, which is an odd number.

Hence, the new median is

#### Question 12:

The weights (in kg) of 15 students are : 31, 35, 27, 29, 32, 43, 37, 41, 34, 28, 36, 44, 45, 42, 30. Find the median. If the weight 44 kg is replaced by 46 kg and 27 kg by 25 kg, find the new median.

The weights (in kg) of 15 students are 31, 35, 27, 29, 32, 43, 37, 41, 34, 28, 36, 44, 45, 42 and 30.

Arranging them in ascending order, we have

27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 44, 45

Here, the number of observation, which is an odd number.

Hence, the median is

If the weights 44kg and 27kg are replaced by 46kg and 25kg respectively, then the given data becomes 31, 35, 25, 29, 32, 43, 37, 41, 34, 28, 36, 46, 45, 42 and 30.

Arranging the new data in ascending order, we have

25, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 45, 46

Here, the number of observation, which is an odd number.

Hence, the new median is

#### Question 13:

The following observations have been arranged in ascending order. If the median of the data is 63, find the value of x:

23, 32, 48, 50, x, x+2, 72, 78, 84, 95

The given data in ascending order is

29, 32, 48, 50, x, x+2, 72, 78, 84, 95

Here, the number of observation, which is an even number.

Hence, the median is

But, it is given that the median is 63. Hence, we have

#### Question 1:

Find out the mode of the following marks obtained by 15 students in a class:

Marks: 4, 6, 5, 7, 9, 8, 10, 4, 7, 6, 5, 9, 8, 7, 7.

Given that the marks obtained by 15 students are 4, 6, 5, 7, 9, 8, 10, 4, 7, 6, 5, 9, 8, 7 and 7.

Make the following frequency table.

Since the value 7 occurs maximum number of times, that is, 4. Hence, the mode value is 7.

#### Question 2:

Find the mode from the following data:

125, 175, 225, 125, 225, 175, 325, 125, 375, 225, 125

The given data is 125, 175, 225, 125, 225, 175, 325, 125, 375, 225, and 125.

Make the following frequency table.

Since the value 125 occurs maximum number of times, that is, 4. Hence, the mode value is

#### Question 3:

Find the mode for the following series:

7.5, 7.3, 7.2, 7.2, 7.4, 7.7, 7.7, 7.5, 7.3, 7.2, 7.6, 7.2

The given series is 7.5, 7.3, 7.2, 7.2, 7.4, 7.7, 7.7, 7.5, 7.3, 7.2, 7.6 and 7.2.

Make the following frequency table.

Since the value 7.2 occurs maximum number of times, that is, 4. Hence, the modal value is.

#### Question 4:

Find the mode of the following data in each case:

(i) 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18

(ii) 7, 9, 12, 13, 7, 12, 15, 7, 12, 7, 25, 18, 7

(i) The given data is 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14 and 18.

Make the following frequency table.

Here occurs maximum number of times (that is 4). Hence, the modal value is 14.

(ii) The given data is 7, 9, 12, 13, 7, 12, 15, 7, 12, 7, 25, 18 and 7.

Make the following frequency table.

Since the value 7 occurs maximum number of times, that is, 5. Hence, the modal value is 7.

#### Question 5:

The demand of different shirt sizes, as obtained by a survey, is given below:

 Size: 38 39 40 41 42 43 44 Total Number of persons (wearing it): 26 39 20 15 13 7 5 125
Find the modal shirt sizes, as observed from the survey.

The demand of different shirt sizes is given as

Here x denotes the size of the shirts and f denotes the number of peoples (frequency) wearing it.

The value 39 occur maximum number of times, that is, 39. Hence, the modal shirt size is 39.

#### Question 1:

Mark the correct alternative in each of the following:
Which one of the following is not a measure of central value?

(a) Mean

(b) Range

(c) Median

(d) Mode

We know that mean, median and mode are all measures of central tendency.

Hence, the correct choice is (b).

#### Question 2:

The mean of n observations is X. If k is added to each observation, then the new mean is

(a) X

(b) X + k

(c) X − k

(d) kX

Let us take n observations.

Ifbe the mean of the n observations, then we have

Add a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Hence, the correct choice is (b).

#### Question 3:

The mean of n observations is X. If each observation is multiplied by k, the mean of new observations is

(a) $k\overline{X}$

(b) $\frac{\overline{X}}{k}$

(c) $\overline{X}+k$

(d) $\overline{X}-k$

Let us take n observations.

Ifbe the mean of the n observations, then we have

Multiply a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Hence, the correct choice is (a).

#### Question 4:

The mean of a set of seven numbers is 81. If one of the numbers is discarded, the mean of the remaining numbers is 78. The value of discarded number is

(a) 98

(b) 99

(c) 100

(d) 101

Given that the mean of 7 numbers is 81. Let us denote the numbers by.

Ifbe the mean of the n observations, then we have

Hence the sum of 7 numbers is

If one number is discarded then the mean becomes 78 and the total numbers becomes 6.

Let the number discarded is x.

After discarding one number the sum becomesand then the mean is

But it is given that after discarding one number the mean becomes 78.

Hence we have

Thus the excluded number is. So, the correct choice is (b).

#### Question 5:

For which set of numbers do the mean, median and mode all have the same value?
(a) 2, 2, 2, 2, 4

(b) 1, 3, 3, 3, 5

(c) 1, 1, 2, 5, 6

(d) 1, 1, 1, 2, 5

For the data 2, 2, 2, 2, 4 of 5 numbers, we have

Since, 2 occurs maximum number of times,

For the data 1, 3, 3, 3, 5 of 5 numbers, we have

Since, 3 occurs maximum number of times,

Hence, the correct choice is (b).

Note that if it happens that the result is not true for the second data then we must follow the same procedure for the other data’s.

#### Question 6:

For the set of numbers 2, 2, 4, 5 and 12, which of the following statements is true?

(a) Mean = Median

(b) Mean > Mode

(c) Mean > Mode

(d) Mode = Median

For the data 2, 2, 4, 5, 12 of 5 numbers, we have

Since, 2 occurs maximum number of times,

So, the correct choice is (b).

#### Question 7:

If the arithmetic mean of 7, 5, 13, x and 9 is 10, then the value x is

(a) 10

(b) 12

(c) 14

(d) 16

The given data is 7, 5, 13, x and 9. They are 5 in numbers.

The mean is

But, it is given that the mean is 10. Hence, we have

Hence, the correct choice is (d).

#### Question 8:

If the mean of five observations x, x+2, x+4, x+6, x+8, is 11, then the mean of first three observations is

(a) 9

(b) 11

(c) 13

(d) none of these

The given data is x, x + 2,  x + 4, x + 6 and x + 8. They are 5 in numbers.

The mean is

But, it is given that the mean is 11. Hence, we have

Then the first three observations are 7, 7 + 2, 7+4, that is, 7, 9, 11. Their mean is

Hence, the correct choice is (a).

#### Question 9:

Mode is

(a) least frequent value

(b) middle most value

(c) most frequent value

(d) none of these

We know that, mode is the observation which occur maximum number of times.

Hence, the correct choice is (c).

#### Question 10:

The following is the data of wages per day : 5, 4, 7, 5, 8, 8, 8, 5, 7, 9, 5, 7, 9, 10, 8
The mode of the data is

(a) 7

(b) 5

(c) 8

(d) 10

The given data is 5, 4, 7, 5, 8, 8, 8, 5, 7, 9, 5, 7, 9, 10 and 8.

Make the following frequency table.

Since the values 5 and 8 occurs in the data maximum number of times, that is, 4. Hence, the modal value is 5 and 8. In this case the mode is not unique.

Hence, the correct options are (b) and (c).

#### Question 11:

The median of the following data : 0, 2, 2, 2, −3, 5, −1, 5, −3, 6, 6, 5, 6 is

(a) 0

(b) −1.5

(c) 2

(d) 3.5

The given data is 0, 2, 2, 2, -3, 5, -1, 5, 5, -3, 6, 6, 5 and 6.

Arranging the given data in ascending order, we have

-3, -3, -1, 0, 2, 2, 2, 5, 5, 5, 5, 6, 6, 6

Here, the number of observation, which is an even number.

Hence, the median is

So, the correct choice is (d).

#### Question 12:

The algebraic sum of the deviations of a set of n values from their mean is

(a) 0

(b) n − 1

(c) n

(d) n + 1

Ifbe the mean of the n observations, then we have

Letbe the mean of n values. So, we have

The sum of the deviations of n valuesfrom their meanis

Hence the correct choice is (a).

#### Question 13:

A, B, C are three sets of values of x:

(a) A: 2, 3, 7, 1, 3, 2, 3
(b) 7, 5, 9, 12, 5, 3, 8
(c) 4, 4, 11, 7, 2, 3, 4

Which one of the following statements is correct?

(a) Mean of A = Mode of C

(b) Mean of C = Median of B

(c) Median of B = Mode of A

(d)Mean, Median and Mode of A are equal.

For the data A: 2, 3, 7, 1, 3, 2, 3 of 7 numbers, we have

Arranging the data A in ascending order, we have

A: 1, 2, 2, 3, 3, 3, 7

Since, 3 occurs maximum number of times,

Hence, the correct choice is (d).

Note that if it happens that the result is not found in the first step then we must follow the same procedure for the other data’s.

#### Question 14:

The mean of five number is 30. If one number is excluded, their mean becomes 28. The excluded number is

(a) 28
(b) 30
(c) 35
(d) 38

We know

Mean =

⇒ Sum of observations = Mean × Number of observations

Mean of five numbers = 30     (Given)

∴ Sum of five numbers = 30 × 5 = 150        .....(1)

If one number is excluded, their mean is 28.

∴ Mean of four numbers = 28

⇒ Sum of four numbers = 28 × 4 = 112        .....(2)

Now,

Excluded number

= Sum of five numbers − Sum of four numbers

= 150 − 112                        [From (1) and (2)]

= 38

Thus, the excluded number is 38.

Hence, the correct answer is option (d).

#### Question 15:

If the mean of the observation : x, x + 3, x + 5, x + 7, x + 10 is 9, the mean of last three observation is

The given observations are x, x + 3, x + 5, x + 7, x + 10.

Mean of the given observations = 9

Number of observations = 5

Now,

$⇒9=\frac{x+\left(x+3\right)+\left(x+5\right)+\left(x+7\right)+\left(x+10\right)}{5}$

$⇒5x+25=45$

$⇒5x=45-25=20$

$⇒x=\frac{20}{5}=4$

x + 5 = 4 + 5 = 9

x + 7 = 4 + 7 = 11

x + 10 = 4 + 10 = 14

So, the last three observations are 9, 11 and 14.

∴ Mean of the last three observations = $\frac{9+11+14}{3}=\frac{34}{3}$

Thus, the mean of the last three observations is $\frac{34}{3}$.

Hence, the correct answer is option (c).

#### Question 16:

If $\overline{X}$ is the mean of x1,x2,.......,xn , then for $a\ne 0$, the mean of  ax1,ax2,......,axn,

It is given that, the mean of x1, x2,..., xn is $\overline{X}$.

$\therefore \overline{X}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}$

$⇒{x}_{1}+{x}_{2}+...+{x}_{n}=n\overline{X}$        .....(1)

Now, $a{x}_{1},a{x}_{2},...,a{x}_{n},\frac{{x}_{1}}{a},\frac{{x}_{2}}{a},...,\frac{{x}_{n}}{a}$ are 2n observations.

∴ Mean of these observations

$=\frac{a{x}_{1}+a{x}_{2}+...+a{x}_{n}+\frac{{x}_{1}}{a}+\frac{{x}_{2}}{a}+...+\frac{{x}_{n}}{a}}{2n}$

$=\frac{a\left({x}_{1}+{x}_{2}+...+{x}_{n}\right)+\frac{1}{a}\left({x}_{1}+{x}_{2}+...+{x}_{n}\right)}{2n}$

$=\frac{\left(a+\frac{1}{a}\right)\left({x}_{1}+{x}_{2}+...+{x}_{n}\right)}{2n}\phantom{\rule{0ex}{0ex}}$

$=\frac{\left(a+\frac{1}{a}\right)×n\overline{)X}}{2n}$                [Using (1)]

$=\frac{1}{2}\left(a+\frac{1}{a}\right)\overline{)X}$

Thus, the mean of $a{x}_{1},a{x}_{2},...,a{x}_{n},\frac{{x}_{1}}{a},\frac{{x}_{2}}{a},...,\frac{{x}_{n}}{a}$ is $\frac{1}{2}\left(a+\frac{1}{a}\right)\overline{)X}$.

Hence, the correct answer is option (b).

#### Question 17:

Let $\overline{\mathrm{X}}$ be the mean of x1,x2,...,xn, and $\overline{\mathrm{Y}}$ the mean of  y1,y2,...,yn, if  $\overline{\mathrm{Z}}$ is the mean of  x1,x2,...,xn, y1,y2,...,yn, then $\overline{\mathrm{Z}}$ is equal to

It is given that, the mean of x1, x2,..., xn is $\overline{X}$.

$\therefore \overline{X}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}$

$⇒{x}_{1}+{x}_{2}+...+{x}_{n}=n\overline{X}$        .....(1)

Also, the mean of y1, y2,..., yn is $\overline{Y}$.

$\therefore \overline{Y}=\frac{{y}_{1}+{y}_{2}+...+{y}_{n}}{n}$

$⇒{y}_{1}+{y}_{2}+...+{y}_{n}=n\overline{)Y}$        .....(2)

Now, ${x}_{1},{x}_{2},...,{x}_{n},{y}_{1},{y}_{2},...,{y}_{n}$ are 2n observations. The mean of these 2n observations is $\overline{Z}$.

$\therefore \overline{)Z}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}+{y}_{1}+{y}_{2}+...+{y}_{n}}{2n}$

$⇒\overline{)Z}=\frac{n\overline{)X}+n\overline{)Y}}{2n}$                [Using (1) and (2)]

$⇒\overline{)Z}=\frac{n\left(\overline{)X}+\overline{)Y}\right)}{2n}$

$⇒\overline{)Z}=\frac{\overline{)X}+\overline{)Y}}{2}$

Hence, the correct answer is option (b).

#### Question 18:

If ${\overline{\mathrm{X}}}_{1},{\overline{\mathrm{X}}}_{2},...,{\overline{\mathrm{X}}}_{k}$ are the means of  n group with n1,n2,...,nk number  of observations respectively, then the mean $\overline{\mathrm{X}}$ of all the groups taken together is govern by

We know

Mean =

⇒ Sum of observations = Mean × Number of observations

Mean of n1 observations of first group = $\overline{){X}_{1}}$

∴ Sum of n1 observations of first group = ${n}_{1}\overline{){X}_{1}}$

Mean of n2 observations of second group = $\overline{){X}_{2}}$

∴ Sum of n2 observations of second group = ${n}_{2}\overline{){X}_{2}}$

.            .            .            .            .            .            .
.            .            .            .            .            .            .

Mean of nk observations of kth group = $\overline{){X}_{k}}$

∴ Sum of nk observations of kth group = ${n}_{k}\overline{){X}_{k}}$

Now,

Sum of all observations in the k groups

= Sum of n1 observations of first group + Sum of n2 observations of second group + ... + Sum of nk observations of kth group

= ${n}_{1}\overline{){X}_{1}}+{n}_{2}\overline{){X}_{2}}+...+{n}_{k}\overline{){X}_{k}}$

= $\sum _{i=1}^{k}{n}_{i}\overline{){X}_{i}}$                      .....(1)

Total number of observations in the k groups  = ${n}_{1}+{n}_{2}+...+{n}_{k}=\sum _{i=1}^{k}{n}_{i}$           .....(2)

∴ Mean $\overline{)X}$ of all the groups =             [Using (1) and (2)]

Hence, the correct answer is option (c).

#### Question 19:

The median of the data : 4,4,5,7,6,7,7,12,3 is

(a) 4
(b) 5
(c) 6
(d) 7

Arranging the data in ascending order, we have

3, 4, 4, 5, 6, 7, 7, 7, 12

Here, the number of observations n = 9, which is odd

∴ Median = Value of $\left(\frac{9+1}{2}\right)\mathrm{th}$ observation = Value of $\left(\frac{10}{2}\right)\mathrm{th}$ observation = Value of 5th observation = 6

Thus, the median of given data is 6.

Hence, the correct answer is option (c).

#### Question 20:

The median of the data : 78,56,22,34,45,54,39,68,54,84 is

(a) 45
(b) 49.5
(c) 53.5
(d) 56

Disclaimer: Options (d) in the question changed to match the answer.

The median of the data : 78,56,22,34,45,54,39,68,54,84 is

(a) 45
(b) 49.5
(c) 53.5
(d) 54

Solution:

Arranging the data in ascending order, we have

22, 34, 39, 45, 54, 54, 56, 68, 78, 84

Here, the number of observations n = 10, which is even

∴ Median of the data

= $\frac{54+54}{2}$

= $\frac{108}{2}$

= 54

Thus, the median of the given data is 54.

Hence, the correct answer is option (d).

#### Question 21:

There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so obtained is found to be -3.5. The mean of the given numbers is

(a) 46.5
(b) 49.5
(c) 53.5
(d) 56.5

Let the 50 numbers be x1, x2, x3,..., x50.

When each number is subtracted from 53, the resulting numbers are 53 − x1, 53 − x2, 53 − x3,..., 53 − x50.

It is given that, the mean of resulting numbers 53 − x1, 53 − x2, 53 − x3,..., 53 − x50 is −3.5.

We know

Mean =

$\therefore -3.5=\frac{\left(53-{x}_{1}\right)+\left(53-{x}_{2}\right)+\left(53-{x}_{3}\right)+...+\left(53-{x}_{50}\right)}{50}$

$⇒\frac{53×50}{50}-\frac{{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{50}}{50}=-3.5$

$⇒\frac{{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{50}}{50}=53+3.5=56.5$        .....(1)

Now,

Mean of 50 given numbers = $\frac{{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{50}}{50}$

∴ Mean of 50 given numbers = 56.5               [From (1)]

Thus, the mean of the given numbers is 56.5.

Hence, the correct answer is option (d).

#### Question 22:

The mean of 100 observations is 50. If one of the observations which was 50 is replaced by 150, the resulting will be

(a) 50.5
(b) 51
(c) 51.5
(d) 52

We know

Mean =

⇒ Sum of observations = Mean × Number of observations

Mean of 100 observations = 50      (Given)

∴ Sum of 100 observations = 50 × 100 = 5000

If one of the observations which was 50 is replaced by 150, then

New sum of 100 observations = Sum of 100 observations − 50 + 150

⇒ New sum of 100 observations = 5000 − 50 + 150 = 5100

∴ New mean of 100 observations =

Thus, the new mean of 100 observations is 51.

Hence, the correct answer is option (b).

#### Question 23:

Mode of the data : 15,14,19,20,14,15,16,14,15,18,14,19,15,17,15 is

(a) 14
(b) 15
(c) 16
(d) 17

Arranging the given data in ascending order, we have

14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 17, 18, 19, 19, 20

Mode is the value of observation which occurs most frequently in a set of observations.

It can be seen that the value 15 occurs maximum number of times i.e. 5.

Thus, the mode of given data is 15.

Hence, the correct answer is option (b).

#### Question 24:

The empirical relation between mean, mode and median is

(a) Mode = 3 Median − 2 Mean

(b) Mode = 2 Median − 3 Mean

(c) Median = 3 Mode − 2 Mean

(d) Mean = 3 Median − 2 Mode

The relation between mean, median and mode is

Hence, the correct option is (a).

#### Question 25:

The mean of a, b, c, d and e is 28. If the mean of a, c, and e is 24, What is the mean of b and d?

(a) 31

(b) 32

(c) 33

(d) 34

Given that the mean of a, b, c, d and e is 28. They are 5 in numbers.

Hence, we have

But, it is given that the mean of a, c and e is 24. Hence, we have

Then, we have

Hence, the mean of b and d is.

Hence, the correct choice is (d).

#### Question 26:

The mean of 25 observations is 36. Out of these observations if the mean of first 13 observations is 32 and that the last 13 observations is 40, the 13th observations is

(a) 23
(b) 36
(c) 38
(d) 40

We know

Mean =

⇒ Sum of observations = Mean × Number of observations

Mean of 25 observations = 36                  (Given)

∴ Sum of 25 observations = 36 × 25 = 900

Mean of first 13 observations = 32           (Given)

∴ Sum of first 13 observations = 32 × 13 = 416

Mean of last 13 observations = 40           (Given)

∴ Sum of last 13 observations = 40 × 13 = 520

Now,

13th observation

= Sum of first 13 observations + Sum of last 13 observations − Sum of 25 observations

= 416 + 520 − 900

= 36

Thus, the 13th observation is 36.

Hence, the correct answer is option (b).

#### Question 1:

Fill In the Blanks

If  $\overline{\mathrm{X}}$ represents the mean of n observations x1,x2,...,xn, then the value of   is _____________.

It is given that, the mean of n observations x1, x2,..., xn is $\overline{X}$.

$\therefore \overline{X}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}$

$⇒{x}_{1}+{x}_{2}+...+{x}_{n}=n\overline{X}$        .....(1)

Now,

$\sum _{i=1}^{n}\left({x}_{i}-\overline{)X}\right)$

$=\left({x}_{1}-\overline{)X}\right)+\left({x}_{2}-\overline{)X}\right)+...+\left({x}_{n}-\overline{)X}\right)$

$=n\overline{)X}-n\overline{)X}$          [From (1)]

$=0$

Thus, the value of $\sum _{i=1}^{n}\left({x}_{i}-\overline{)X}\right)$ is 0.

If $\overline{\mathrm{X}}$ represents the mean of n observations x1,x2,...,xn, then the value of   is ____0____.

#### Question 2:

Fill In the Blanks

If each obsevation of the data is increased by 5, then their mean is _____________ by 5.

Let the mean of n observations x1, x2,..., xn be $\overline{X}$.

$\therefore \overline{X}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}$

$⇒{x}_{1}+{x}_{2}+...+{x}_{n}=n\overline{X}$        .....(1)

If each observation is increased by 5, then the new observations are x1 + 5, x2 + 5,..., xn + 5.

∴ Mean of new observations

$=\frac{\left({x}_{1}+5\right)+\left({x}_{2}+5\right)+...+\left({x}_{n}+5\right)}{n}$

$=\frac{n\overline{)X}+5n}{n}$

$=\frac{n\left(\overline{)X}+5\right)}{n}$

$=\overline{)X}+5$

Thus, the mean of new observations is increased by 5.

If each observation of the data is increased by 5, then their mean is ___increased___ by 5.

#### Question 3:

Fill In the Blanks

If the mean of the data x1,x2,...,xn is $\overline{\mathrm{X}}$, then the mean of ax1 + b,ax2+b,...,axn + b is ____________ .

It is given that, the mean of x1, x2,..., xn  is $\overline{X}$.

$\therefore \overline{X}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}$

$⇒{x}_{1}+{x}_{2}+...+{x}_{n}=n\overline{X}$        .....(1)

Let $\overline{)X\text{'}}$ be the mean of data ax1 + b, ax2 + b,..., axn + b.

$\therefore \overline{X\text{'}}=\frac{\left(a{x}_{1}+b\right)+\left(a{x}_{2}+b\right)+...+\left(a{x}_{n}+b\right)}{n}$

$⇒\overline{X\text{'}}=\frac{a\left({x}_{1}+{x}_{2}+...+{x}_{n}\right)+nb}{n}$

$⇒\overline{X\text{'}}=\frac{an\overline{)X}+nb}{n}$          [Using (1)]

$⇒\overline{X\text{'}}=\frac{n\left(a\overline{)X}+b\right)}{n}$

$⇒\overline{X\text{'}}=a\overline{)X}+b$

Thus, the mean of data ax1 + b, ax2 + b,..., axn + b is $a\overline{)X}+b$.

If the mean of the data x1, x2,..., xn is $\overline{X}$, then the mean of ax1 + b, ax2 + b,..., axn + b is .

#### Question 4:

Fill In the Blanks

If mode and median of the certain data are 3 and 3 respectively, then mean is ___________.

Mode of the data = 3

Median of the data = 3

Now,

Mode = 3Median − 2Mean

⇒ 3 = 3 × 3 − 2Mean

⇒ 2Mean = 9 − 3 = 6

⇒ Mean = $\frac{6}{2}$ = 3

Thus, the mean of the data is 3.

If mode and median of the certain data are 3 and 3 respectively, then mean is ___3___.

#### Question 5:

Fill In the Blanks

If Mode - Mean = 36, then Median - Mean = ___________.

It is given that,

Mode − Mean = 36

We know

Mode = 3Median − 2Mean

⇒ Mode − Mean = 3Median − 3Mean

⇒ Mode − Mean = 3(Median − Mean)

⇒ 3(Median − Mean) = 36           (Given)

⇒ Median − Mean = $\frac{36}{3}$ = 12

Thus, the value of Median − Mean is 12.

If Mode − Mean = 36, then Median − Mean = ____12____.

#### Question 6:

Fill In the Blanks

The mean and mode of a data are 24 and 12 respectively, then median of the data is ___________.

Mean of the data = 24

Mode of the data = 12

We know

Mode = 3Median − 2Mean

⇒ 12 = 3Median − 2 × 24

⇒ 3Median = 48 + 12 = 60

⇒ Median = $\frac{60}{3}$ = 20

Thus, the median of data is 20.

The mean and mode of a data are 24 and 12 respectively, then median of the data is ___20___.

#### Question 7:

Fill In the Blanks

If the mean of 12 observations is 15. If two observations 20 and 25 are removed, then the mean of remaining observations is _________.

We know

Mean =

⇒ Sum of observations = Mean × Number of observations

Mean of 12 observations = 15      (Given)

∴ Sum of 12 observations = 15 × 12 = 180

If two observations 20 and 25 are removed, then

Sum of remaining 10 observations = Sum of 12 observations − 20 − 25

⇒ Sum of remaining 10 observations = 180 − 20 − 25 = 135

∴ Mean of remaining 10 observations =

Thus, the mean of remaining observations is 13.5.

If the mean of 12 observations is 15. If two observations 20 and 25 are removed, then the mean of remaining observations is ___13.5___.

#### Question 8:

Fill In the Blanks

If Mean : Median = 2 : 3, then Mode : Mean = ________

Mean : Median = 2 : 3          (Given)

Let mean = 2x and median = 3x, where x is constant           .....(1)

We know

Mode = 3Median − 2Mean

⇒ Mode = 3 × 3x − 2 × 2x            [From (1)]

⇒ Mode = 9x − 4x = 5x

∴ Mode : Mean = 5x : 2x = 5 : 2

Thus, the ratio of mode to mean is 5 : 2.

If Mean : Median = 2 : 3, then Mode : Mean = ___5 : 2___.

#### Question 9:

Fill In the Blanks

The mean of a set of 12 observations is 10 and another set of 8 observations is 12. The mean of all 20 observations is _______.

We know

Mean =

⇒ Sum of observations = Mean × Number of observations

Mean of a set of 12 observations = 10                 (Given)

∴ Sum of a set of 12 observations = 10 × 12 = 120                   .....(1)

Mean of another set of 8 observations = 12         (Given)

∴ Sum of another set of 8 observations = 12 × 8 = 96              .....(2)

Now,

Sum of all 20 observations

= Sum of a set of 12 observations + Sum of another set of 8 observations

= 120 + 96                    [From (1) and (2)]

= 216

∴ Mean of all 20 observations = = 10.8

Thus, the mean of all 20 observations is 10.8.

The mean of a set of 12 observations is 10 and another set of 8 observations is 12. The mean of all 20 observations is ___10.8___.

#### Question 10:

Fill In the Blanks

The mean of the following distribution

x :    3    5    7    4
y :    2    a    5    b

is 5. Then the value of b is ____________.

We know

Mean =

The mean of given distribution is 5.

$\therefore 5=\frac{3×2+5×a+7×5+4×b}{2+a+5+b}$

$⇒5=\frac{6+5a+35+4b}{7+a+b}$

$⇒5\left(7+a+b\right)=41+5a+4b$

$⇒35+5a+5b=41+5a+4b$

$⇒5b-4b=41-35$

$⇒b=6$

Thus, the value of b is 6.

The mean of the following distribution

x :    3    5    7    4
y :    2    a    5    b

is 5. Then the value of b is ____6____.

#### Question 1:

If the ratio of mode and median of a certain data is 6 : 5, then find the ratio of its mean and median.

Given that the ratio of mode and median of a certain data is 6:5. That is,

We know that

#### Question 2:

If the mean of x + 2, 2x + 3, 3x + 4, 4x + 5 is x + 2, find x.

The given data is x+2, 2x+3, 3x+4, 4x+5. They are four in numbers.

The mean is

But, it is given that the mean is x+2. Hence, we have

#### Question 3:

If the median of scores $\frac{x}{2},\frac{x}{3},\frac{x}{4},\frac{x}{5}$ and $\frac{x}{6}$ (where x > 0) is 6, then find the value of $\frac{x}{6}$.

Given that the median of the scores, whereis 6. The number of scores n is 5, which is an odd number. We have to find

Note that the scores are in descending order. Hence the median is

But, it is given that the median is 6. Hence, we have

#### Question 4:

If the mean of 2, 4, 6, 8, x, y is 5, then find the value of x + y.

The given data is. They are 6 in numbers.

The mean is

But, it is given that the mean is 5. Hence, we have

#### Question 5:

If the mode of scores 3, 4, 3, 5, 4, 6, 6, x is 4, find the value of x.

The given data is.

The mode is the value which occur maximum number of times, that is, the mode has maximum frequency. If the maximum frequency occurs for more than 1 value, then the number of mode is more than 1 and is not unique.

Here it is given that the mode is 4. So, x must be 4, otherwise it contradicts that the mode is 4. Hence

#### Question 6:

If the median of 33, 28, 20, 25, 34, x is 29, find the maximum possible value of x.

The given data is. The total number of values is, is an even number. Hence the median depends on theobservation andobservation.

Since we have to find the maximum possible value of x.So we must put it in the 4th position when ordering in ascending order.

Arranging the data in ascending order, we have

Hence, the median is

Here it is given that the median is 29. So, we have

#### Question 7:

If the median of the scores 1, 2, x, 4, 5 (where 1 < 2 < x < 4 < 5) is 3, then find the mean of the scores.

The given data is 1, 2, x, 4 and 5. Since, the given data is already in ascending order.

Here, the number of observation, which is an odd number.

Hence, the median is

Here, it is given that the median is 3. Hence, we have.

The mean is

#### Question 8:

If the ratio of mean and median of a certain data is 2:3, then find the ratio of its mode and mean

Given that the ratio of mean and median of a certain data is 2:3. That is,

We know that

#### Question 9:

The arithmetic mean and mode of a data are 24 and 12 respectively, then find the median of the data.

Given that the arithmetic mean and mode of a data are 24 and 12 respectively. That is,

We have to find median

We know that

#### Question 10:

If the difference of mode and median of a data is 24, then find the difference of median and mean.

Given that the difference of mode and median of a data is 24. That is,

We have to find the difference between median and mean

We know that

#### Question 1:

If the height of 5 persons are 140 cm, 150 cm, 152 cm, 158 cm and 161 cm respectively, find the mean height.

Given that the heights of 5 persons are 140 cm, 150 cm, 152 cm, 158 cm and 161 cm, respectively.

We have to find the mean of their heights.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean height of the 5 persons is

#### Question 2:

Find the mean of 994, 996, 998, 1002, 1000.

The given 5 numbers are 994, 996, 998, 1002 and 1000, respectively.

We have to find their mean.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean of the given 5 numbers is

#### Question 3:

Find the mean of first five natural numbers.

The first 5 natural numbers are 1, 2, 3, 4 and 5.

We have to find their mean.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean of the first 5 natural numbers is

#### Question 4:

Find the mean of all factors of 10.

All the factors of 10 are 1, 2, 5, and 10. They are 4 in numbers.

We have to find the mean of the all factors of 10.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean of the all factors of 10 is

#### Question 5:

Find the mean of first 10 even natural numbers.

The first 10 even natural numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. They are 10 in numbers.

We have to find the mean of the first 10 even natural numbers.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean of the first 10 even natural numbers is

#### Question 6:

Find the mean of x, x+2, x+4, x+6, x+8.

We have to find the mean of, and. They are 5 in numbers.

Remember the definition of mean of n values x1, x2 xn is

Therefore the required mean is

#### Question 7:

Find the mean of first five multiples of 3.

The first five multiples of 3 are 3, 6, 9, 12, and 15. They are 5 in numbers.

We have to find the mean of first five multiples of 3.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean of first five multiples of 3 is

#### Question 8:

Following are the weights  (in kg)  of 10 new born babies in a hospital on a particular day:
3.4, 3.6, 4.2, 4.5, 3.9, 4.1, 3.8, 4.5, 4.4, 3.6. Find the mean X.

The weights (in kg) of 10 new born babies are 3.4, 3.6, 4.2, 4.5, 3.9, 4.1, 3.8, 4.5, 4.4, and 3.6. These are 10 in numbers.

We have to find the mean of their weights.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean of their weights is

#### Question 9:

The percentage of marks obtained by students of a class in mathematics are : 64, 36, 47, 23, 0, 19, 81, 93, 72, 35, 3, 1. Find their mean.

Given that the percentage of marks obtained by students of a class of mathematics are 64, 36, 47, 23, 0, 19, 81, 93, 72, 35, 3, and 1. These are 12 in numbers.

We have to find the mean of their percentage of marks.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean of their weights is

#### Question 10:

The number of children in 10 families of a locality are:
2, 4, 3, 4, 2, 0, 3, 5, 1, 1, 5. Find the mean number of children per family.

Given that the numbers of children in 10 families are 2, 4, 3, 4, 2, 3, 5, 1, 1, and 5.

We have to find the mean number of children per family.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean number of children per family is

#### Question 11:

Explain, by taking a suitable example, how the arithmetic mean alters by (i) adding a constant k to each term, (ii) subtracting a constant k from each them, (iii) multiplying each term by a constant k and (iv) dividing each term by a non-zero constant k.

Let us take n observations.

Ifbe the mean of the n observations, then we have

(i) Add a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 natural numbers are 1, 2, 3, 4, and 5. Their mean is

Add 2 to each of the numbers. Then the numbers becomes 3, 4, 5, 6, and 7. The new mean is

Therefore adding a constant number to each observation the mean increased by that constant

(ii) Subtract a constant k from each of the observations.

Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 even natural numbers are 2, 4, 6, 8 and 10. Their mean is

Subtract 1 from each of the numbers. Then the numbers becomes 1, 3, 5, 7, and 9. The new mean is

Therefore subtracting a constant number to each observation the mean decreased by that constant

(iii) Multiply a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 even natural numbers are 2, 4, 6, 8 and 10. Their mean is

Multiply 2 to each of the numbers. Then the numbers becomes 4, 8, 12, 16, and 20. The new mean is

Therefore multiplying by a constant number to each observation the mean becomes the constant multiplied by the old mean

(vi) Divide a nonzero constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Let us take an example to understand the above fact.

The first 5 even natural numbers are 2, 4, 6, 8 and 10. Their mean is

Divide 2 to each of the numbers. Then the numbers becomes 1, 2, 3, 4, and 5. The new mean is

Therefore dividing by a constant number to each observation the mean becomes the old mean divided by the constant

#### Question 12:

The mean of marks scored by 100 students was found to be 40. Later on it was discovered that a score of 53 was misread as 83. Find the correct mean.

Given that the mean of marks scored by 100 students is 40. Let us denote the marks of 100 students by.

Ifbe the mean of the n observations, then we have

Hence we have

This sum is incorrect. We have to find the correct sum

It was found that a score 53 was misread as 83. So, 83 must be replaced by 53.

To get the correct sum at first we have to remove the incorrect entry and then add the correct entry.

Therefore the correct sum is

Thus, the correct mean is

#### Question 13:

The  traffic police recorded the speed (in km/hr) of 10 motorists as 47, 53, 49, 60, 39, 42, 55, 57, 52, 48. Later on an error in recording instrument was found. Find the overage speed of the motorists if the instrument  recorded 5 km/hr less in each case.

Given that the recorded speed (in km/hr) of 10 motorists are 47, 53, 49, 60, 39, 42, 55, 57, 52, and 48. Let us denote the speeds of 10 motorists by.

Ifbe the mean of the n observations, then we have

Hence we have

This mean is incorrect. We have to find the correct mean.

It was found that the instrument recorded 5 km/hr less in each case.

To get the correct we have to add 5 with each entry. Then the correct entries are

Recall that ifbe the mean of the n observationsand if we add a constant k with each of the observations, then the new mean becomes

Therefore the correct mean speed of the motorists is

#### Question 14:

The mean of five numbers is 27. If one number is excluded, their mean is 25. Find the excluded number.

Given that the mean of 5 numbers is 27. Let us denote the numbers by.

Ifbe the mean of the n observations, then we have

Hence the sum of 5 numbers is

If one number is excluded then the mean becomes 25 and the total numbers becomes 4.

Let the number excluded is x.

After excluding one number the sum becomesand then the mean is

But it is given that after excluding one number the mean becomes 25.

Hence we have

Thus the excluded number is.

#### Question 15:

The mean weight per student in a group of 7 students is 55 kg. The individual weights of 6 of them (in kg) are 52, 54, 55, 53, 56 and 54. Find the weight of the seventh student.

Given that the mean weight (in kg) of 7 students is 55 kg. Let us denote their weights by.

Ifbe the mean of the n observations, then we have

Hence the sum of the weights of 7 students is

The individual weights (in kg) of 6 of them are 52, 54, 55, 53, 56 and 54.

Let the weight of the remaining student is x.

According to this information the sum of their weights is

Hence we have

Thus the weight of the seventh student is.

#### Question 16:

The mean weight of 8 numbers is 15. If each number is multiplied by 2, what will be the new mean?

Let us denote the 8 numbers by. Their mean is 15.

Ifbe the mean of the n numbers, then we have

Therefore the sum of the numbers is

If each numbers is multiplied by 2, then the numbers becomes.

The mean of the new numbers is

Note that the new mean is equal to the old mean multiplied by 2.

#### Question 17:

The mean of 5 numbers is 18. If one number is excluded, their mean is 16. Find the excluded number.

Given that the mean of 5 numbers is 18. Let us denote the numbers by.

Ifbe the mean of the n observations, then we have

Hence the sum of 5 numbers is

If one number is excluded then the mean becomes 16 and the total numbers becomes 4.

Let the number excluded be x.

After excluding one number the sum becomesand then the mean is

But it is given that after excluding one number the mean becomes 16.

Hence we have

Thus the excluded number is.

#### Question 18:

The mean of 200 items was 50, Later on, it was discovered that the two items were misread as 92 and 8 instead of 192 and 88, Find the correct mean.

Given that the mean of 200 items is 50. Let us denote the 200 items by.

Ifbe the mean of the n observations, then we have

Hence we have

This sum is incorrect. We have to find the correct sum

It was discovered that the two items were misread as 92 and 8 instead of 192 and 88. So, 92 and 8 are must be replaced by 192 and 8 respectively.

To get the correct sum at first we have to remove the incorrect items and then add the correct items.

Therefore the correct sum is

Thus, the correct mean is

#### Question 19:

If M is the mean of x1, x2, x3, x4, x5 and x6, prove that
(x1 − M) + (x2 − M) (x3 − M) (x4 − M) + (x5 − M) + (x6 − M) = 0.

Given that the mean of x1, x2, x3, x4, x5, and x6 is M.

We have to prove that

Remember the definition of mean of n values x1, x2 xn is

Therefore, we have

Now,

Hence the proof is complete.

#### Question 20:

Durations of sunshine (in hours) in Amritsar to first 10 days of August 1997 as reported by the Meteorological Department are given below:
9.6, 5.2, 3.5, 1.5, 1.6, 2.4, 2.6, 8.4, 10.3, 10.9

(i) Find the mean X.

(ii) Verify that

Given that the durations of sunshine (in hours) in Amritsar for 10 days are 9.6, 5.2, 3.5, 1.5, 1.6, 2.4, 2.6, 8.4, 10.3, and 10.9.

(i) We have to find their mean.

Remember the definition of mean of n values x1, x2 xn is

Therefore the mean duration of sunshine is

(ii) We have to prove that

We have

Hence the proof is complete.

#### Question 21:

Find the values of n and X in each of the following cases:

(i) $\sum _{t=1}^{n}\left({x}_{i}-12\right)=-10$ and $\sum _{t=1}^{n}\left({x}_{i}-3\right)=62$

(ii) $\sum _{t=1}^{n}\left({x}_{i}-10\right)=30$ and $\sum _{t=1}^{n}\left({x}_{i}-6\right)=150$

Ifbe the mean of the n observations, then we have

(i) The given two equations are

The equations can be rewritten as

We have to solve the above equations forand.

Subtracting the first equation from the second equation, we have

Substitute the value of n in the first equation, we have

(ii) The given two equations are

The equations can be rewritten as

We have to solve the above equations forand.

Subtracting the first equation from the second equation, we have

Substitute the value of n in the first equation, we have

#### Question 22:

The sum of the deviations of a set of n values x1, x2,...., xn, measured from 15 and −3 are −90 and 54 respectively. Find the value of n and mean.

Ifbe the mean of the n observations, then we have

Given that the sums of the deviations of a set of n valuesmeasured from 15 and -3 are -90 and 54 respectively. So that we get two equations

These equations can be rewritten as

We have to solve the above equations forandunknowns.

Subtracting the first equation from the second equation, we have

Substitute the value of n in the first equation, we have

#### Question 23:

Find the sum of the deviations of the variate values 3, 4, 6, 7, 8, 14 from their mean.

Ifbe the mean of the n observations, then we have

The given 6 variate values are 3, 4, 6, 7, 8 and 14. Let us denote the values by . Their mean value is given by

The sum of the deviations of the variate values from their mean is

$\sum _{i=1}^{6}\left({X}_{i}-\overline{)X}\right)\phantom{\rule{0ex}{0ex}}=\sum _{i=1}^{6}\left({X}_{i}-7\right)\phantom{\rule{0ex}{0ex}}=\sum _{i=1}^{6}{X}_{i}-\sum _{i=1}^{6}7\phantom{\rule{0ex}{0ex}}=6\overline{)X}-6×7\phantom{\rule{0ex}{0ex}}=6×7-6×7\phantom{\rule{0ex}{0ex}}=0$

Hence the proof is complete.

#### Question 24:

If $\overline{X}$ is the mean of the ten natural numbers x1, x2, x3,..., x10, show that
.

Ifbe the mean of the n observations, then we have

Given thatis the mean of the 10 natural numbers. So, we have

We have to show that

That is to show that the sum of the deviations of the 10 natural numbers from their mean is 0.

We have

Hence the proof is complete.

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