15

Q1. The mean deviation about the mean for the following data: Class interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 Frequency 2 5 6 4 3

• 9.5
• 12
• 11.3
• 8.2

Solution

Calculation of mean deviation: Class interval Frequency (f_i) Mid point (x_i) x_i.f_i 0 – 10 2 5 10 20 40 10 – 20 5 15 75 10 50 20 – 30 6 25 150 0 0 30 – 40 4 35 140 10 40 40 – 50 3 45 135 20 60   20   500   190  

Q2. The standard deviation of first 10 multiples of 4 is:

• 14
• 8
• 11.5
• 7

Solution

 First 10 multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 Calculation of standard deviation: x_i Deviations by mean 4 -18 324 8 -14 196 12 -10 100 16 -6 36 20 -2 4 24 2 4 28 6 36 32 10 100 36 14 196 40 18 324 Total = 220   1320  

Q3. For a given data, the standard deviation is 20.If 3 is added to each observation , what is the new variance of the resulting observations?

• 17
• 20
• 23
• 60

Solution

If  a constant is added to each observation, then the standard deviation of the resulting observations  does not change.

Q4. ……. about a central value ‘a’ is the mean of the absolute values of the deviations of the observations from ‘a’.

• Mean deviation
• Range
• Quartile deviation
• Standard deviation

Solution

Mean deviation about a central value ‘a’ is the mean of the absolute values of the deviations of the observations from ‘a’.

Q5.  Two offices P and Q show following results about the number of workers and the wages paid to them.   P Q No. of workers 50 75 Average monthly wages Rs 8000 Rs 8000 Variance of distribution of wages 64 81     In which office is there greater variability in individual wages?  

• Both P and Q
• Q
• P
• Neither P nor Q

Solution

Q6. The mean deviation about mean for the following data: x_i 4 8 12 16 f_i 5 6 4 3

• 4
• 3.58
• 5.6
• 6.2

Solution

Calculation of the mean deviation:   x f xf d=|x-mean| fd 4 5 20 5.1 25.5 8 6 48 1.1 6.6 12 4 48 2.9 11.6 16 3 48 6.9 20.7 Total 18 164   64.4  

Q7. The measure of variability which is independent of units is called:

• Coefficient of range
• Coefficient of quartile deviation
• Coefficient of variation
• Coefficient of median

Solution

The measure of variability which is independent of units is called coefficient of variation.

Q8. Mean of the squares of the deviations from mean is called the:

• Standard deviation
• Variance
• Mode
• Quartile deviation

Solution

Mean of the squares of the deviations from mean is called the variance.  

Q9. The arithmetic mean of the numerical values of the deviations of items from some average value is called the

• Range
• Quartile deviation
• Mean deviation
• Standard deviation

Solution

Q10. The difference of maximum and minimum values of each series is called ………….

• Mean
• Median
• Range
• Quartile deviation

Solution

The difference of maximum and minimum values of each series is called range of the data.

Q11. If the coefficient of variation and mean of sales of coffee and tea in one year is 75, 80 and 12, 10 respectively, and then the variance of coffee and tea is:

• 85, 75
• 102, 25
• 64, 100
• 81, 64

Solution

Coefficient of variation of sales of coffee = 75, Mean of sales of coffee = 12 Standard deviation of sales of coffee = 9 \  Variance of sales of coffee = (9)² = 81 Coefficient of variation of sales of tea = 80, Mean of sales of tea = 10 Standard deviation of sales of tea = 8 \  Variance of sales of tea = (8)² = 64

Q12. The mean of first n natural numbers =

• n(n-1)/2
• n/2
• (2n+1)/2
• (n+1)/2

Solution

The mean of any n observations= sum of all the observations/n So, in this case Mean ={n(n+1)/2}/n =(n+1)/2

Q13. The standard deviation for the following data: Class interval 0 – 5 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30 Frequency 6 7 4 3 7 3

• 14
• 12.35
• 8
• 7.11

Solution

Calculation of standard Deviation: Class interval Frequency (f_i) Mid-point (x_i) f_ix_i 0 – 5 6 2.5 15 124.7689 748.6134 5 – 10 7 7.5 52.5 38.0689 266.4823 10 – 15 4 12.5 50 1.3689 5.4756 15 – 20 3 17.5 52.5 14.6689 44.0067 20 – 25 7 22.5 157.5 77.9689 545.7823 25 – 30 3 27.5 82.5 191.2689 573.8067 Total 30   410   1514.167  

Q14. The following values are calculated in respect of marks of the students of sections A and B of Class X:   Section A Section B Mean 85 92 Variance 81 64 The marks of which section have more variability?

• Section B
• Section A
• Both section are equal variable
• Cannot be determined

Solution

Q15. If the coefficient of variation and standard deviation of the runs of team A is 80 and 15, and then the mean of runs of team A

• 18.75
• 16
• 20
• 21.23

Solution

Coefficient of variation = 80, Standard Deviation (σ) = 15

Q16. What is the range of the following data? 23, 45, 34, 21, 89, 45, 47, 91

• 69
• 70
• 56
• 71

Solution

Range of a data = Maximum value – Minimum Value = 91-21 = 70

Q17. For series with equal means, the series with _______ standard deviation is more consistent or less scattered.

• Lesser
• More
• Equal
• Different

Solution

For series with equal means, the series with lesser standard deviation is more consistent or less scattered.

Q18. The variance and standard deviation for the following data: x_i 2 4 6 8 10 12 14 f_i 3 2 4 6 5 3 2

• 4.21
• 5
• 3.12
• 5

Solution

 Calculation of variance and standard deviation: x_i f_i x_if_i 2 3 6 -6 36 108 4 2 8 -4 16 32 6 4 24 -2 4 16 8 6 48 0 0 0 10 5 50 2 4 20 12 3 36 4 16 48 14 2 28 6 36 72 Total 25 200     242  

Q19. The variance for the following data: Class interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

• 125
• 245
• 321
• 294.34

Solution

Calculation of variance: Class interval Frequency (f_i) Mid-point (x_i) f_ix_i 0 – 10 5 5 25 729 3645 10 – 20 4 15 60 289 1156 20 – 30 3 25 75 49 147 30 – 40 6 35 210 9 54 40 – 50 7 45 315 169 1183 50 – 60 5 55 275 529 2645 Total 30   960   8830  

Q20. If the standard deviation and mean of the data is 24 and 8, then the coefficient of variation of data is:

• 200
• 100
• 250
• 300

Solution

Standard Deviation (σ) = 24, Mean  = 8

Q21. The mean of a set of observations is expressed as positive square-root of the variance and is called:

• Variance
• Quartile deviation
• Range
• Standard deviation

Solution

The mean of a set of observations is expressed as positive square-root of the variance and is called standard deviation.  

Q22. From the prices of commodities X and Y below, find out which is more stable in value: X 42 54 62 48 60 56 58 Y 102 100 106 112 118 114 104

• Commodity X
• Commodity Y
• Variance X
• Mean Y

Solution

Calculation of mean, variance and standard deviation: X Y 42 -13 169 102 -6 36 54 -1 1 100 -8 64 62 7 49 106 -2 4 48 -7 49 112 4 16 60 5 25 118 10 100 61 6 36 114 6 36 58 3 9 104 -4 16 Total = 385   338 756   272   Since, coefficient of variation of commodity X is greater than Y, therefore commodity X is more stable in market than commodity Y.  

Q23. The variance of data: 0,10,20,30,40,50

• 290
• 291.67
• 230
• 12

Solution

Mean of data 0,10,20,30,40,50  is  = 25 Variance =

Q24. The series having greater C.V. is said to be …… variable than the other.

• Less
• More
• Equal
• Different

Solution

The series having greater C.V. is said to be more variable than the other.

Q25. The mean deviation of the following data 14, 15, 16, 17, 13 is:

• 2.3
• 4
• 3
• 1.2

Solution

Mean of the data = The deviations of the respective observations from the mean -1, 0, 1, 2, -2 The absolute values of the deviations i.e.,  are 1, 0, 1, 2, 2 Mean deviation about the mean is

Q26. The variance of following data 5, 6, 4, 2, 3, 6, 9 is:

• 3.5
• 4.58
• 2.36
• 5

Solution

 Calculation of variance: x_i Deviations from mean 5 0 0 6 1 1 4 -1 1 2 -3 9 3 -2 4 6 1 1 9 4 16 Total = 35   32  

Q27. The mean deviation about the mean for the following data:5, 6, 7, 8, 6, 9, 13, 12, 15 is:

• 3.2
• 5
• 1.5
• 2.89

Solution

Mean of the data = The deviations of the respective observations from the mean -4, -3, -2, -1, -3, 0, 4, 3, 6 The absolute values of the deviations i.e.,  are 4, 3, 2, 1, 3, 0, 4, 3, 6 Mean deviation about the mean is

Q28. For a given data, the variance is 15. If each observation is multiplied by 2, what is the new variance of the resulting observations?

• 7.5
• 15
• 30
• 60

Solution

If each observation is multiplied by a constant k, then the variance of the resulting observations becomes k² times the original variance. Here, k = 2. Hence new variance = 15 x 4 = 60

Q29. The mean deviation about the mean for the following data 3, 7, 8, 9, 4, 6, 8, 13, 12, 10 is:

• 5
• 2
• 2.4
• 3

Solution

Mean of the data = The deviations of the respective observations from the mean – 5, – 1, 0, 1, – 4, – 2, 0, 5, 4, 2 The absolute values of the deviations i.e.,  are 5, 1, 0, 1, 4, 2, 0, 5, 4,2 Mean deviation about the mean is

Q30. The mean deviation about the mean for the following data: x_i 1 5 6 8 9 f_i 2 5 3 4 1

• 5
• 1.47
• 6
• 3