Unit - II



Practice Set

1. AM is affected if every value of the variable is

(a) Decreased by same constant A

(b) Multiplied by same constant B

(c) Increased by same constant C

(d) Divide by same constant D

2. Find the AM and Variance of first n natural numbers

3. Effect of change of origin and scale on S.D.

4. Mean of 200 items is 48 and their S.D. is 3. Find sum of the squares of all the items.

5. Obtain a relationship to get central moments from non central moments

6. Mean square deviation cannot be less than variance. [Prove that S2 ( (2]

7. Obtain S.D & M.D about mean for the following series

a, a + d, a + 2d…... a + 2nd. Prove S.D ≥ M.D.

8. Prove β2 ≥ 1 & β2 ≥ β1

9. Prove that for any discrete distribution S.D can not be less than M.D from mean.

10. The mean & S.D of 100 observations were 40 & 5.1. Later on it was discovered an observation 40 was misread as 50. Find the correct S.D.

11. Why only 4 moments calculated. The first four moments of a frequency distribution about the value 5 are 2, 20, 50, 50.obtain as far as possible the various characteristic of this distribution on the basis of the information given.

12. Find variance of following frequency distribution

X : 0 1 2 3 -- n

Y : 1 nC1 nC2 - - nCn

13. Prove that mean square deviation cannot be less than the variance.

14. The first four moments of a distribution about 4 are 1, 3.5, 8.5 and 33.5 respectively. Calculate μ2, μ3, μ4 also measure kurtosis.

15. In calculating moments of a frequency distribution based on 100 observations the following results are obtained. Mean = 9, variance = 19, β1 = 0.7 μ3 = +ve, β2 = 4. But later on it was found that one observation 12 was read as 21. Obtain the correct values of first four central moments.

16. Prove that mean deviation is minimum when measured from median.

17. In a certain discrete distribution the first 4 moments about a point 4 are 1.5, 17, 30 and 108. Find the first 4 moments about mean.

18. Calculate the measures of Skewness

|Size |4-8 |8-12 |12-16 |16-20 |20-24 |

|frequency |4 |4 |9 |6 |2 |

20 Calculate first 4 moments about mean for the following:-

|Size |20 |21 |22 |23 |24 |25 |26 |

|Frequency | 5 |38 |65 |92 |70 |40 |10 |

22 The first 4 moment of a dist about the value 2 are 1, 16 and -40. Show that mean = 3; σ 2 = 15 and μ3 = -86. Also show that the first 3 moments about 0 are 3, 24 and 76 respectively.

23 The first 2 moment about the value X = 4 are -1.5 and 17 respectively. Find mean & variance.

24 The first 4 moments of a distribution about the value 5 of the variable X are -4, 22, -117 & 560. Find these moments about origin & about mean.

25 The first 4 moments of a distribution about the value 0 are -0.2, 1.7, -2.3 & 10.8. Find the moments about mean.

26 The first 4 moments of a distribution about the value 4 of the variable are -1.5, 17, -30 &108. Find the moments about mean, also β1, β2 .is there any doubt about the consistency of the given information.

< Hint: - Since β2 is less than 1 we can say the given information is not consistent.>

27 The first 4 moments about the value 5 of the variable are 7, 70,140 and 175. Is there any mistake in there calculation?

Note: - we verify the following relation. (a) β2 ≥ β1 (b) β2 ≥1 (c) μ2 ≥ 0

28 The first 3 moments about X = 22 are 0.43, 2.03 and 3.23 respectively. Find the moment about X=20.

29 The first 3 moments about zero for a distribution are 7.2, 71.2 and 738.8 respectively. Obtain the moments about point 7.

30 The first 4 moments of a frequency distribution are 0, 14.75, 39.75 and 142.3125 respectively. Find the first 4 moments about X = 4 given that the mean of the distribution is 2.5.

31 The first 4 moments of a distribution about 5 are 2, 20, 40 &150. Find the first 4 moments about origin.

32 The first 3 moments about 22 in a distribution are 0.43, 2.03, & 3.23 respectively. Find the central moments & the mean of the distribution.

33 For a distribution with the mean = 10, variance σ2 = 16, γ1 = 1, β2 = 4. Find the first 4 moments about origin.

34. The forth moment about mean of a frequency distribution is 48. What must be the value of its Standard deviation in order that the distribution is?

(i) Leptokurtic (ii) Measokurtic (iii) Platykurtic

36. Calculate Bowley’s coefficient of Skewness for the following distribution

|Class |0-10 |10-20 |20-30 |30-40 |40-50 |

|frequency |2 |7 |10 |5 |3 |

37. Effect of change of origin and scale on central moment

38. Effect of change of origin and scale on moment about origin

39. Effect of change of origin and scale on moment about any point

40. Obtain the variance for the combined variance for two sets of data.

41. Calculate the first four central moments, β1, β2, (1 and (2 for the following data.

|Class |0-10 |10-20 |20-30 |30-40 |40-50 |

|frequency |1 |3 |5 |7 |4 |

42. State the utility of Pearson’s constant β1, β2 & (1, (2

1. The first moment about the value of the variable 1.5 of a frequency of distribution is 4.5 Mean is equal to

(1) 5.5 (2)6 (3) 6.5 (4) 3

2. About which point the second moment is variance?

(1) Mean (2) zero (3) median (4) mode

3. If (’1=2, (’2 = 8 and (’3 =45 then (3 = is

(1) 45 (2)32 (3) 16 (4) 13

4. If (2 = 16 and (3 = -64 then (1 will be

(1) 1 (2) -1 (3) 4 (4) -4

5. (3 = for a frequency distribution. If each value of the variable is multiplied by 3, then value of (2 now will be

(1) 1 (2) 3 (3) 9 (4) 27

6. Sheppard’s Correction is applied to correct the error due to

(1) Wrong data (2) calculation mistakes

(3) grouping (4) All of these

7. Variance is equal to second central moments

(1) True (2) False (3) both (4) None

8. (2 is independent of the change of origin and scale

(1) True (2) False (3) both (4) None

9. (2 ( (1always

(1) True (2) False (3) both (4) None

10. In a symmetrical frequency distribution al odd ordered central moments are zero.

(1) True (2) False (3) both (4) None

11. Raw moments are moments about any point other than mean.

(1) True (2) False (3) both (4) None

12. A moderately asymmetrical frequency curve is said to be negatively skewed if

(1) mode < median < mean (2) mode > median > mean

(3) mode = median = mean (4) mode < median > mean

13. The curve of the frequency distribution

x: 1 2 3 4 5 6 7 8 9

f: 4 10 12 14 20 14 12 10 4

(1) Positively skewed (2) negatively skewed

(3) Asymmetrical about mode (4) symmetrical about mode

14. For symmetric frequency distribution

(1) Q2 – Q1 = Q3 - Q2 (2) Q2 – Q1 > Q3 - Q2

(3) Q2 – Q1 < Q3 - Q2 (4) Q2 – Q1 + Q3 = 0

15. If (1 = 9, we can say that frequency distribution is

(1) Positively skew (2) negatively skew (3) symmetric (4) none

16. The limits for the Bowley’s Skewness coefficient are

(1) -1, 1 (2) 0, 1 (3) -3, 3 (4) 0, 3

17. If (2 =16, (3 = -64 and (4 = 128, then will be (1

(1) 1 (2) -1 (3) 4 (4) -4

18. The measure of Skewness is

(1) (1 (2) (2 (3) (1 (4) none

19. The standard deviation of a frequency distribution is 5. The value of 4th central moment (4 in order that the distribution be measokurtic should be

(1) equal to 3 (2) greater than 1875

(3) equal to 1875 (4) less than 1875

20. For a platy kurtic curve, the measure of kurtosis is

(1) (2 = 0 (2) (2 0 (4) (1 = 0

21. (2 < 1, then we can say that

(1) The frequency distribution is platy kurtic

(2) The frequency distribution is leptokurtic

(3) The frequency distribution is measokurtic

(4) The information given is wrong

22. The forth moment about mean of a frequency distribution is 48. What must be the value of its variance in order that the distribution is leptokurtic?

(1) (2 < 4 (2) (2 > 4 (3) (2 = 4 (4) none

23. For a platy kurtic frequency curve, the measure of kurtosis is given by

(1) (1 =3 (2) (2 > 3 (3) (2 < 3 (4) (1 = 0

24. For a measokurtic distribution, the coefficient of kurtosis (2 is

(1) 0 (2) 3 (3) greater than three (4) less than three

25. If all odd order central moments are zero, then frequency distribution is

(1) Positively skewed (2) negatively skewed

(3) J-shaped (4) symmetric

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