# JSO SSC CGL Paper 3 : Part – 8

JSO SSC CGL Paper 3

Q106. In a distribution, the first four moments from the origin are 1, 4, 10 and 46 respectively. Then the nature of the distribution is

(a) symmetrical and mesokurtic

(b) negatively skewed and platykurtic

(c) negatively skewed and leptokurtic

(d) positively skewed and mesokurtic

Q107. In a distribution, the coefficient of skewness is 0.5. if the sum of lower and upper quartiles is 120 and median is 40, then upper and lower quartiles are respectively.

(a) 50, 70

(b) 60, 80

(c) 60, 60

(d) 100, 20

Q108. The Bowley’s coefficient of skewness lies between

(a) -1 and +1

(b) -3 and +3

(c) positive only

(d) negative only

Q109. If the mean, mode and standard deviation of a frequency distribution are 41, 45 and 8 respectively, then its Pearson’s coefficient of skewness is

(a)1/2

(b) -1/2

(c)1/4

(d) -1/4 Q110. Find the coefficient of variation, if it is given that the mean is 120, mode is 123 and Karl Pearson’s coefficient of skewness is – 0.3.

(a) 7.56%

(b) 8.33%

(c) 12.10%

(d) 6.54%

Q111. In a perfectly symmetric dist. 50% of the observations are above 60 and 75% are below 75. The coefficient of skewness is therefore

(a) 7.5

(b) 15

(c) 0

(d) Cannot be determined from the given data

Q112. Karl Pearson’s coefficient of skewness of a dist. Is 0.32, its sd. Is 6.5 and mean is 29.6. What is the mode of the dist.?

(a) 26.24

(b) 27.52

(c) 24.37

(d) 29.56

Q113. For leptokurtic curves β2 and v2 satisfies (β2 is measure of kurtosis and u2 is coefficient of kurtosis)

(a) β2 > 3, v2 < 0

(b) β2 = 3, v2 = 0

(c) β2 < 3, v2 < 0

(d) β2 > 3, v2 > 0

Q114. A symmetrical distribution is symmetric about the point h, then the mean is

(a) Less than h

(b) More than h

(c) Equal to h

(d) None of these

Q115. For any frequency distribution the measure of kurtosis (a) greater than one

(b) less than one

(c) equal to one

(d) greater than 3

Q116. If the odd ordered central moment of a distribution is 0, then distribution is

(a) Symmetric

(b) Mesokurtic

(c) Platykurtic

(d) Leptokurtic

Q117. By kurtosis of a frequency distribution, we mean

(a) the pattern of the spread of the total frequency over values of variables.

(b) the scatterness of the values of variable among themselves.

(c) the Peakedness of the corresponding frequency curve.

(d) the number of peaks of the corresponding frequency curve.

Q118. A central moment based measure of kurtosis is appropriate for

(a) bimodal distributions

(b) J – shaped distributions

(c) U – shaped distributions

(d) bell – shaped distributions Q119. If a test was generally very easy, except for a few students who had very low scores, then the distribution of scores would be

(a) Positively skewed

(b) Negatively skewed

(c) Not skewed at all

(d) Normal

Q120. The standard deviation of a distribution is 2. The distribution is leptokurtic fourth central moment is

(a) more than 12

(b) less than 12

(c) more than 48

(d) less than 48

SOLUTION BELOW:-

 Sr. No. Answers 106. A 107. D 108. B 109. B 110. B 111. C 112. B 113. D 114. C 115. A 116. A 117. C 118. D 119 A 120. C

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# Statistical Investigator CGL TIER 2 Paper 3 : Part – 7

Statistical Investigator CGL TIER 2 Paper 3 : Part – 7

Q91. The algebraic sum of deviations of a set of n values from their arithmetic mean is

(a) x

(b) 0

(c) 1

(d) None of these

Q92. The arithmetic mean of the numbers 1,2,3 ……….. n is Q93. Relation between range and S.D. is

(a) R = 3 S.D.

(b) R = 2 S.D.

(c) R = 6 S.D.

(d) R = 4 S.D.

Q94. Sum of squares of the deviations is minimum when the deviations are taken from

(a) Mean

(b) Median

(c) Mode

(d) None of these

Q95. Standard error of the sample correlation coefficient r based an n paired values is  Q96. The value of correlation ratios varies from

(a) -1 to 1

(b) -1 to 0

(c) 0 to 1

(d) 0 to ∞

Q97. The range of multiple correlation coefficient R is

(a) 0 to 1

(b) 0 to ∞

(c) -1 to 1

(d) -∞ to ∞

Q98. Measures of association usually deals with

(a) Attributes

(b) Quantitative factors

(c) Variables

(d) Numbers

Q99. Estimation of parameters in all scientific investigations is of

(a) Prime importance

(b) Secondary importance

(c) No use

(d) Deceptive nature

Q100. Estimate and estimator are

(a) Synonyms

(b) Different

(c) Related to population

(d) None of the above

Q101. If the observations recorded on five sampled items 3,4,5,6,7 the sample variance is

(a) 1

(b) 0

(c) 2

(d) 2.5

Q102. If the sample values are 1,3,5,7,9 the standard error of sample mean is

(a) S.E. = √2

(b) S.E. = 1/√2

(c) S.E. = 2.0

(d) S.E. = ½

Q103. How many types of optimum allocations are in common uses

(a) One

(b) Two

(c) Three

(d) Four

Q104. In an ordered series the data are

(a) In ascending order

(b) In descending order

(c) Either 1 or 2

(d) Neither 1 or 2 Q105. Classification is applicable in case of

(a) Quantitative characters

(b) Qualitative characters

(c) Both 1 and 2

(d) None of these

SOLUTION BELOW:-

 Sr. No. Answers 91. B 92. C 93. C 94. A 95. C 96. C 97. A 98. A 99. B 100. B 101. D 102. A 103. C 104. B 105. C

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# SSC CGL TIER 2 Statistical Investigator : Part – 6

SSC CGL TIER 2 Statistical Investigator  : Part – 6

Q76. The standard deviation of a series is 5.What should be the value of the fourth order central moment so that the distribution is mesokurtic?

(a) Less than 1875

(b) More than 1875

(c) Equal to 625

(d) Equal to 1875

Q77. A smoothed frequency polygon is called

(a) Frequency Curve

(b) Ogive

(c) Histogram

(d) None of the above

Q78. A relative frequency of a frequency distribution having more than one class

(a) must be a positive proper fraction

(b) must be a non-negative proper fraction

(c) may be greater than unity

(d) may be negative

Q79. In drawing the histogram of a given frequency distribution on each class interval are

(a) erect rectangles with heights proportional to the frequency of the corresponding class interval

(b) erect rectangles whose height is proportional to the ratio of the frequency to the width of the class interval

(c) erect rectangles whose height is proportional to the ratio of the frequencies to the sum of the frequencies

(d) None of the above

Q80. When both the upper and limits are included in the class intervals in case of a frequency distribution, the classes are called

(a) interval classes

(b) Exclusive classes

(c) inclusive classes

(d) Wide classes Q81. Relative frequency of a class is computed by

(a) dividing the midpoint of the class by the sample size

(b) dividing the frequency of the class by the midpoint

(c) dividing the sample size by the frequency of the class

(d) dividing the frequency of the class by the sample size

Q82. If the standard deviation of a distribution 15, the quartile deviation of the distrib is

(a) 15

(b) 12.5

(c) 10

(d) 15.5

Q83.The sum of deviations of the Ist 10 natural numbers from 5-5 is

(a) 11/2

(b) (102 – 1) / 12

(c) 0

(d) 102 – 1

Q84. If AM and coefficient of variation of y are 10 and 40 respectively what is the variance of (15-24) ?

(a) 8

(b) 4

(c) 64

(d) None of the above

Q85. The mean and SD of Y are p and q respectively , then SD of Y – P /q   is

(a) -1

(b) 1

(c) ab

(d) a/b

Q86. If the SD of Y is 3, what is the variance of (5-2Y)?

(a) 36

(b) 6

(c) 1

(d) 9

Q87. The mean and SD for p, q and 2 are 3 and 1 respectively the value of pq would be

(a) 5

(b) 6

(c) 12

(d) 3

Q88. ____ is useful in averaging ratios, rates and percentage.

(a) A.M.

(b) G.M.

(c) H.M.

(d) None

Q89. GM is useful in construction of index number

(a) True

(b) False

(c) Both

(d) None Q90. More laborious numerical calculations involves in G.M. than A.M.

(a) True

(b) False

(c) Both

(d) None

SOLUTION BELOW:-

 Sr. No. Answers 76. D 77. A 78. B 79. B 80. C 81. D 82. C 83. C 84. C 85. B 86. A 87. A 88. B 89. A 90. A

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# Statistical Investigator SSC CGL TIER 2 Paper 3 : Part – 5

Statistical Investigator SSC CGL TIER 2 Paper 3 : Part – 5

(a) 72%

(b) 68%

(c) 48%

(d) 15%

Q62. The coefficient of variance of a data is 25% and the mean is 20. Then standard deviation of the data set is

(a) 1.25

(b) 4.00

(c) 5.00

(d) 6.25

Q63. The variance of the data 5,11,13,19,20 is 37.5, the variance 7,13,15,21,22 is equal to

(a) 37.5

(b) 32.2

(c) 34.2

(d) 39.5

Q64. The variance of the data under consideration is 7. If each of the data points is multiplied by 3, then the variance of the new data is

(a) 7

(b) 10

(c) 21

(d) 63

Q65. The relation btw coefficient of variation C.V. and coefficient of dispersion based on standard deviations is  Q66. The mean of two samples of sizes 50 and 100 respectively are 54.1 and 50.3 and the standard deviations are 8 and 7. Then the mean and standard deviation of the sample of size 150 containing the two samples is

(a) 52.57 and 8.56

(b) 51.57 and 7.56

(c) 54.57 and 8.96

(d) 53.57 and 8.76

Q67. The type of error that will creep into the calculation of moments for continuous frequency distribution is

(a) Grouping error

(b) Standard error

(c) Non- sampling error

(d) Non- grouping error

Q68. If mean is 25 and standard deviation is 5 then C.V – coefficient of variation is

(a) 25%

(b) 20%

(c) None of these

(d) 100%

Q69. The mean of a distribution is 14 and the standard deviation is 5. What is the value of the coefficient of variation?

(a) 60.4%

(b) 48.3%

(c) 35.7%

(d) 27.8%

Q70. The mean of a distribution is 23, the median is 24, and the mode is 25.5. it is most likely that this distribution is

(a) Positively skewed

(b) Symmetrical

(c) Asymptotic

(d) Negatively skewed

Q71. If the standard deviation of a population is 9, the population variance is

(a) 9

(b) 3

(c) 2

(d) 81

Q72. Given X1 = 12, X2 = 19, X3 = 10, X4 = 7, then ∑4i = 1  equals ?

(a) 36

(b) 48

(c) 41

(d) 29

Q73. If a distribution is abnormally tall and peaked, then is can be said that the distribution is:

(a) Leptokurtic

(b) Pyrokurtic

(c) Platykartic

(d) Mesokurtic

Q74. Kurtosis of a set observations on temperature of Kolkata measured in degree Celsius is 3, 2. If the temperature are measured in degree Fahrenheit, the value of the kurtosis

(a) Will be changed

(b) Remains unchanged

(c) Will increase

(d) Will decrease Q75. The magnitude of correlation coefficient is the …….. of two regression coefficients.

(a) HM

(b) Angle

(c) AM

(d) GM

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# Statistical Investigator. SSC CGL TIER 2 Paper 3, SSC CGL JSO

Statistical Investigator SSC CGL TIER 2 Paper 3 : Part – 4

Q46. Which of the following diagrams would be appropriate for cumulative frequency of a discrete variable?

(a) Step diagram

(b) Histogram

(c) Frequency curve

(d) Frequency polygon

Q47. Indirect interview method for collecting primary data demands.

(a) Creating influence of interviewee

(b) Frequently chasing the interviewee

(c) Friendly and to – the – point discussion

(d) Gossiping mood

Q48. Which of the following diagrams would be appropriate corresponding to frequency table for a continuous variable?

(a) Pie diagramme

(b) Histogram

(c) Step diagramme

(d) Ogive

Q49. Pie chart is very much useful for

(a) Assessing the contribution of different groups in a comparative way

(b) Ordering different groups

(c) Comparing different groups

(d) Measuring and evaluating different groups

Q50. If  m1 and m2 and  are average heights of two classes of 50 and 60 students, the overall average is a

(a) Geometric mean of  m1 and m2

(b) Median of  m1 and m2

(c) Simple mean of  m1 and m2

(d) Weighted mean of  m1 and m2 Q51. If the two ogives of n observations  x1,x2 ,x3, ……….,xn  meet at the point (x , y), then consider the following statements :

(a) x is the arithmetic mean of x1,x2 ,x3, ……….,xn

(b) x is the median of  x1,x2 ,x3, ……….,xn

(c) y = n/2

(d) y = 1/2

Q52. Suppose x has 10 values 1,2 ,…….10 with corresponding more than type cumulative frequencies F1, F2 , ………. F10. Then the AM of this data set will be equal to Q53. The number of seeds in 10 fruits of a variety are counted and the first 9 are found to be 10,9,6,11,8,9,12,10 and 7. If the number of seeds in the tenth fruit is x, then for the mean number of seeds to be at least 9,it is necessary and sufficient that

(a) x ≥ 9

(b) x > 9

(c) x > 6

(d) x ≥ 8

Q54. A computer supplier sells computers of some well – known brands. To improve his profits reasonably he must study the average sales of computers where th appropriate measure of average will be

(a) Median

(b) Mode

(c) Geometric mean

(d) Arithmetic mean

Q55. The function (x-3)+ (x+3)2 + (x-2)2 + (x+2)2 Will take its minimum value at x=

(a) -3

(b) 0

(c) 2

(d) 3

Q56. Which of the following is true for a symmetric distribution?

ƍ3 = Third Quartile

ƍ1= First Quartile

ƍ2 = Second Quartile or median

(a) (ƍ3 – ƍ2) = (ƍ2 – ƍ1)

(b) (ƍ3 – ƍ1) = 2ƍ2

(c) (ƍ3 – ƍ2) < (ƍ2 – ƍ1)

(d) (ƍ3 – ƍ2) > (ƍ2 – ƍ1)

Q57. Harmonic mean is

(a) The mean of the reciprocal of the observations

(b) The reciprocal of the mean of the observations

(c) The reciprocals of the arithmetic mean of the reciprocal of the observations

(d) None of the above

Q58. If the median of a first set of observations is M1 and the median of a second set of observations is M2 , then the combined median M

(a) Lies to the right of M2

(b) Lies in between M1 and M2

(c) Lies to the left of M1

(d) None of the above

Q59. Coefficient of variation is not applicable if

i. S.D. is very large

ii. Mean is zero or negative

iii. The grouped distribution has open-end classes

iv. Extreme values are present in the data

Which situations do you think to be justified?

(a) i, ii and iv

(b) iii and iv

(c) ii and iii

(d) i and ii

Q60. In a symmetric distribution the upper and the lower Quartile are not equidistant from

(a) Harmonic mean

(b) Median

(c) Mode

(d) Arithmetic mean Sr. No. Ans 1 A 2 C 3 B 4 A 5 D 6 D 7 D 8 D 9 B 10 B 11 A 12 C 13 B 14 C 15 A

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# JSO SSC CGL TIER 2  Paper 3 : Part – 3

JSO SSC CGL TIER 2  Paper 3 : Part – 3

Q31. The mean of a variable x having 50 observations is 45. If a new variable is defined by u=x +45, the mean of the new variable is

(a) 45

(b) 0

(c) 90

(d) 95

Q32. White dividing each entry in a data by a non zero number, a the arithmetic mean of the new data .

(a) Is multiplied by a

(b) Does not change

(c) Is divided by a

(d) Is diminished by a

Q33. The sum of deviations taken from the actual arithmetic mean is

(a) Zero

(b) Two

(c) Negative

(d) Infinite

Q34. The mean age of 40 students is 16 years and the mean age of another group of 60 students is 20 years. The mean age of all the 100 students is (in years)

(a) 16.8

(b) 18

(c) 18.4

(d) 18.8

Q35. If mean = K (3 median – mode) for asymmetrical distribution, the value of K is

(a) 1/2

(b) -1/2

(c) 1

(d) -1 Q36. The point of intersection of two ogives, if a frequency distribution with median is 30 and the total frequency is 100, is

(a) (35, 50)

(b) (30, 50)

(c) (40, 50)

(d) (50, 30)

Q37. Which is the correct relation for a moderately asymmetrical distribution?

(a) A.M. < G.M. < H.M.

(b) A.M. > G.M. > H.M.

(c) G.M > A.M. > H.M.

(d) G.M < A.M. < H.M

Q38. If  – 50) = 0 then the mean of the distribution is

(a) 0

(b) 45

(c) 40

(d) 50

Q39. In a distribution, the values of observations are 1,2……, 10 with frequencies 1,2,…..,10, then the mean of the distribution is

(a) 7

(b) 5

(c) 6

(d) 4

Q40. Mean of 25 observations was found to be 78.4. but later it was found that 96 was misread as 69. The correct mean is

(a) 78.48

(b) 76.98

(c) 79.48

(d) 77.58

Q41. The most obvious measure to describe the characteristic of data is

(a) Mean

(b) Mode

(c) Median

(d) Standard deviation

Q42. The number 4 and 9 have frequencies x and (x-1) respectively. If their A.M. is 6, then x is equal to

(a) 2

(b) 3

(c) 4

(d) 5

Q43. A variable takes 11 values which are arranged in ascending order of magnitude. It was found that the 4th, 6th and 8th observations are 8,6 and 4 respectively. The median of the distribution is

(a) 4

(b) 6

(c) 8

(d) 10

Q44. For the number 3,1,2,4,5,3,3, which one of the following is correct?

(a) Mean = Mode = 3 and Median = 4

(b) Mean= Median = Mode = 3

(c) Mean = Median = 3 and Mode = 5

(d) Mean = 3, Median = 4, Mode = 5

Q45. The mean grade of a section of 20 students is 66% and that of another section of 15 students is 70%. The combined mean grade is

(a) 66.7%

(b) 67.7%

(c) 68.7%

(d) 69.7% SOLUTIONS:

 Sr. No. Ans 1 C 2 C 3 A 4 C 5 A 6 B 7 B 8 D 9 A 10 C 11 A 12 B 13 B 14 B 15 B

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# Statistical Investigator SSC CGL TIER 2  Paper 3 : Part – 2

Statistical Investigator SSC CGL TIER 2  Paper 3 : Part – 2

Q16. What is the purpose of a summary table?

(a) This is the only way to present categorical data in numerical form.

(b) To see differences between or among categories.

(c) To sum the values of responses to a survey.

(d) To list data to create a bar or pie chart.

Q17. A graphical representation of a frequency distribution is called a__________________ .

(a) scatter diagram

(b) stem-and-leaf plot

(c) time-series plot

(d) histogram

Q18. You have a summary table and a simple bar chart (like the ones at the beginning of the           chapter) indicating where customers prefer to do their banking. How could you                         enhance the bar chart to provide both visual and actual information?

(a) Use vertical lines on the bar chart to show the values more precisely.

(b) Add values to the bar chart like what is commonly done on a pie chart.

(c) Only the summary table can show the actual values for the data.

(d) The bar chart and summary table must be presented together in order to represent this data.

Q19. It might be said that the stem-and-leaf display is really a quick and easy way of                      creating a rudimentary chart or diagram for numerical data. If so, which chart is used to describe categorical data does it most closely resemble?

(a) The stem-and-leaf display most closely resembles a rudimentary bar chart.

(b) The stem-and-leaf display most closely resembles a rudimentary pie chart.

(c) The stem-and-leaf display most closely resembles a rudimentary Pareto chart.

(d) The stem-and-leaf display does not resemble any of the above charts or diagrams.

Q20. The width of a class interval in a frequency distribution (or bar chart) will be                              approximately equal to the range of the data divided by the ______________ .

(a) Average of the data set

(b) Number of class intervals

(c) Highest value in the data set

(d) Lowest value in the data set Q21. Which of the above histograms represents the graph of city restaurant meal prices with an interval of \$8?

(a) Neither histogram

(b) Both histograms

(c) Only the histogram on the right

(d) Only the histogram on the left

Q22. When constructing a frequency distribution, which of the following rules must be followed?

(a) The width of each class is equal to the lowest value in the data set.

(b) The midpoint of each class must be an integer.

(c) The number of classes must be an even number.

Q23. The rule of thumb for creating a frequency distribution is to divide the data into 5-15 classes. While larger numbers of classes allow for larger data sets, how do you know exactly how many classes to use?

(a) If in doubt about the number of classes, select 10 since it is the midpoint between 5 and 15 classes.

(b) Any number of classes between 5 and 15 is sufficient.

(c) Determine the width of the class interval, then calculate the number of classes.

(d) Select the number of classes that provides definition to the shape of the data.

Q24. The following numbers represent exam scores in an accounting class:

78, 93, 85, 81, 73, 96, 72, 86, 90, 85

If a stem-and-leaf diagram is developed from this data, how many stems will be used?

(a) 10

(b) 4

(c) 3

(d) 5

Q25. In the above figure of wine imports into the United States, what three guidelines for developing good graphs have not been followed?

(a) Distorted data, unnecessary adornment, no scales for either axis.

(b) Distorted data, X axis not labeled, no title.

(c) Distorted data, unnecessary adornments, vertical axis not properly labeled.

(d) Unnecessary adornment, vertical scale does not begin at zero, no title.

Q26. Considering the various types of tables and charts introduced in this chapter, which table, chart, diagram or plot would you use to depict categorical data for two variables in a visual format?

(a) scatter plot

(b) contingency table

(c) pie chart

(d) side-by-side bar chart

Q27. The cumulative frequency for a particular class is equal to 35. The cumulative frequency for the next class will be _________________ .

(a) 35 minus the next class frequency.

(b) equal to 65.

(c) 35 plus the next class frequency.

(d) less than 35.

Q28. Which of the following would be most helpful in the construction of a pie chart?

(a) Ogive

(b) Frequency distribution

(c) Relative frequencies

(d) Cumulative percentages

Q29. The table above shows the frequency and relative frequencies for 7 groups of restaurant meal prices. How was the value of 0.36 obtained for the relative frequency of meals costing \$32 but less than \$40?

(a) The number of data points is 50, so divide 18 by 50.

(b) (18 x 2)/100 = 0.36.

(c) The midpoint of the class is \$36, so divide 36 by 100.

(d) (18/100) x 2 = 0.36. Q30. The highest bar in a histogram represents?

(a) The class with the lowest relative frequency.

(b) The class with the highest cumulative frequency.

(c) The class with the lowest frequency.

(d) The class with the highest frequency.

SOLUTIONS:

 Sr. No. Answers 1. B 2. D 3. B 4. A 5. B 6. B 7. D 8. D 9. C 10. C 11. D 12. C 13. C 14. A 15. D

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# Statistical Investigator SSC CGL TIER 2  Paper 3 : Part – 1

Statistical Investigator SSC CGL TIER 2 Paper 3 : Part – 1

Q1. The science of collecting, organizing, presenting, analyzing and interpreting data to                 assist in making more effective decisions is called:

(a) Statistic

(b) Parameter

(c) Population

(d) Statistics

Q2. Methods of organizing, summarizing, and presenting data in an informative way are               called:

(a) Descriptive statistics

(b) Inferential statistics

(c) Theoretical statistics

(d) Applied statistics

Q3. The methods used to determine something about a population on the basis of a sample         is called:

(a) Inferential statistics

(b) Descriptive statistics

(c) Applied statistics

(d) Theoretical statistics

Q4. When the characteristic being studied is non – numeric, it is called a:

(a) Quantitative variable

(b) Qualitative variable

(c) Discrete variable

(d) Continuous variable

Q5. When the variable studied can be reported numerically, the variable is called a:

(a) Quantitative variable

(b) Qualitative variable

(c) Independent variable

(d) Dependent variable Q6. A specific characteristic of a population is called:

(a) Statistic

(b) Parameter

(c) Variable

(d) Sample

Q7. A specific characteristic of a sample is called:

(a) Variable

(b) Constant

(c) Parameter

(d) Statistic

Q8. A set of all units of interest in a study is called:

(a) Sample

(b) Population

(c) Parameter

(d) Statistic

Q9. A part of the population selected for study is called a:

(a) Variable

(b) Data

(c) Sample

(d) Parameter

Q10. Listing of the data in order of numerical magnitude is called:

(a) Raw data

(b) Arrayed data

(c) Discrete data

(d) Continuous data

Q11. Listings of the data in the form in which these are collected are known as:

(a) Secondary data

(b) Raw data

(c) Arrayed data

(d) Qualitative data

Q12. Data that are collected by anybody for some specific purpose and use are called:

(a) Qualitative data

(b) Primary data

(c) Secondary data

(d) Continuous data

Q13. The data which have under gone any treatment previously is called:

(a) Primary data

(b) Secondary data

(c) Symmetric data

(d) Skewed data

Q14. The data obtained by conducting a survey is called:

(a) Primary data

(b) Secondary data

(c) Continuous data

(d) Qualitative data

Q15. A survey in which information is collected from each and every individual of the                      population is known as:

(a) Sample survey

(b) Pilot survey

(c) Biased survey

(d) Census survey SOLUTIONS :

 Sr. No. Answers 1. D 2. A 3. A 4. B 5. A 6. B 7. D 8. B 9. C 10. B 11. B 12. B 13. B 14. A 15. D

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# SSC CGL Tier 2 Paper 3 Study Material Day 7 [statistics]

SSC CGL Tier 2 Paper 3  Study Material Day 7 [statistics]:-

#### Introduction

19th century under the framework of the group theory and of the theory of algebraic invariants. The theory of algebraic invariants was thoroughly studied by famous German mathematicians P.A. Gordan and D. Hilbert  and was further developed in the 20th century in and among others.

Moment invariants were first introduced to the pattern recognition and image processing community in 1962 when Hu employed the results of the theory of algebraic invariants and derived his seven famous invariants to rotation of 2-D objects. Since that time, hundreds of papers have been devoted to various improvements, extensions and generalizations of moment invariants and also to their use in many areas of application. Moment invariants have become one of the most important and most frequently used shape descriptors. Even though they suffer from certain intrinsic limitations (the worst of which is their globalness, which prevents direct utilization for occluded object recognition), they frequently serve as ”first-choice descriptors” and as a reference method for evaluating the performance of other shape descriptors. Despite a tremendous effort and huge number of published papers, many open problems remain to be resolved.

Moments in mathematical statistics involve a basic calculation.  These calculations can be used to find a probability distribution’s mean, variance and skewness.

Suppose that we have a set of data with a total of n discrete points. One important calculation, which is actually several numbers, is called the sth moment. The sth moment of the data set with values x1, x2, x3, . . . , xn is given by the formula:

(x1s + x2s + x3s + . . . + xns)/n

Using this formula requires us to be careful with our order of operations. We need to do the exponents first, add, then divide this sum by n the total number of data values.

The term moment has been taken from physics. In physics the moment of a system of point masses is calculated with a formula identical to that above, and this formula is used in finding the center of mass of the points. In statistics the values are no longer masses, but as we will see, moments in statistics still measure something relative to the center of the values.

#### Definition

Moments are scalar quantities used for hundreds of years to characterize function and to capture its significant features. They have been widely used in statistics for description of the shape of a probability density function and in classic rigid-body mechanics to measure the mass distribution of a body. From the mathematical point of view, moments are ”projections” of a function onto a polynomial basis (similarly, Fourier transform is a projection onto a basis of harmonic functions). For the sake of clarity, we introduce some basic terms and propositions, which we will use throughout the book.

Definition 1: By an image function (or image) we understand any piecewise continuous real function f(x, y) of two variables defined on a compact support D ⊂ R × R and having a finite nonzero integral.

Definition 2: General moment M(f) pq of an image f(x, y), where p, q are non-negative integers and r = p + q is called the order of the moment, is defined as

####  where p00(x, y), p10(x, y), . . . , pkj(x, y), . . . are polynomial basis functions defined on D. (We omit the superscript (f) if there is no danger of confusion.)

Depending on the polynomial basis used, we recognize various systems of moments.

The nth raw moment µn  (i.e., moment about zero) of a distribution P(x) is defined by  µn = (xn)

Where

µthe mean, is usually simply denoted u=u1 If the moment is instead taken about a point a,

µn(a) = <(x-a)n> = ∑(x-a)n P(x).

A statistical distribution is not uniquely specified by its moments, although it is by its characteristic function.

The moments are most commonly taken about the mean. These so-called central moments are denoted un and are defined by

µn = <(x-µ)n>

∫(x-µ)n P(x)dx,

with µ1 = 0 The second moment about the mean is equal to the variance

µ22,

where σ = √µ2 is called the standard deviation

The related characteristic function is defined by

(n) (0) = [dø/ d tn]t = 0

in µn (0).

The moments may be simply computed using the moment-generating function,

#### Different types of Moments-

The types of moments are-

µn = M(n) (0).

#### 1. First Moment

For the first moment we set s = 1. The formula for the first moment is thus:

(x1 x2 + x3 + . . . + xn)/n

This is identical to the formula for the sample mean.

The first moment of the values 1, 3, 6, 10 is (1 + 3 + 6 + 10) / 4 = 20/4 = 5.

#### 2. Second Moment

For the second moment we set s = 2. The formula for the second moment is:

(x21 + x22 + x23 + . . . + x2n)/n

The second moment of the values 1, 3, 6, 10 is (12 + 32+ 62 + 102) / 4 = (1 + 9 + 36 + 100)/4 = 146/4 = 36.5.

#### 3.Third Moment

For the third moment we set s = 3. The formula for the third moment is:

(x31 + x32 + x33 + . . . + x3n)/n

(The third moment of the values 1, 3, 6, 10 is (13 + 33 + 63 + 103) / 4 = (1 + 27 + 216 + 1000)/4 = 1244/4 = 311.

Higher moments can be calculated in a similar way. Just replace s in the above formula with the number denoting the desired moment #### 4. Fourth (s=4).

The 4th moment = (x14 + x24 + x34 + . . . + xn4)/n

####  A related idea is that of the sth moment about the mean. In this calculation we perform the following steps:

1. First calculate the mean of the values.
2. Next, subtract this mean from each value.
3. Then raise each of these differences to the sth power.
4. Now add the numbers from step #3 together.
5. Finally, divide this sum by the number of values we started with.

The formula for the sth moment about the mean m of the values x1, x2, x3, . . . , xn is given by:

MS = (x1 – m)S + (x2– m)S + (x3 – m)S + . . . + (xn – m)s)/n

#### First Moment about the Mean

The first moment about the mean is always equal to zero, no matter what the data set is that we are working with. This can be seen in the following:

M1 = ((x1 – m) + (x2 – m) + (x3 – m) + . . . + (xn – m))/n = ((x1+ x2 + x3 + . . . + xn) – nm)/n = m – m = 0.

#### Second Moment about the Mean

The second moment about the mean is obtained from the above formula by settings = 2:

m2 = ((x1 – m)2 + (x2 – m)2 + (x3 – m)2 + . . . + (xn – m)2)/n

This formula is equivalent to that for the sample variance.

For example, consider the set 1, 3, 6, 10. We have already calculated the mean of this set to be 5. Subtract this from each of the data values to obtain differences of:

1 – 5 = -4

3 – 5 = -2

6 – 5 = 1

10 – 5 = 5

We square each of these values and add them together: (-4)2 + (-2)2 + 12 + 52 = 16 + 4 + 1 + 25 = 46. Finally divide this number by the number of data points: 46/4 = 11.5

#### Applications of Moments

As mentioned above, the first moment is the mean and the second moment about the mean is the sample variance. Pearson introduced the use of the third moment about the mean in calculating skewness and the fourth moment about the mean in the calculation of kurtosis. Uses of Moments In Statistics-

The central question in statistics is that given a set of data, we would like to recover the random process that produced the data (that is, the probability law of the population). This question is extremely difficult in general and in the absence of strong assumptions on the underlying random process you really can’t get very far (those who work in nonparametric statistics may disagree with me on this). A natural way to approach this problem would be to look for simple objects that do identify the population distribution if we do make some reasonable assumptions.

The question then becomes what type of objects should we search for. The best arguments I know about why we should look at the Laplace (or Fourier; I’ll show you what this is in a second if you don’t know) transform of the probability measure are a bit complicated, but naively we can draw a good heuristic from elementary calculus: given all the derivatives of an analytic function evaluated at zero we know everything there is to know about the function through its Taylor series.

Suppose for a moment that the function f(t)=E[etX] exists and is well behaved in a neighborhood of zero. It is a theorem that this function (when it exists and behaves nicely) uniquely identifies the probability law of the random variable XX. If we do a Taylor expansion of what is inside the expectation, this becomes a power series in the moments of XX: X: and so to completely identify the law of XX we just need to know the population moments. In effect we reduce the question above “identify the population law of XX” to the question “identify the population moments of XX”.

It turns out that (from other statistics) population moments are extremely well estimated by sample moments when they exist, and you can even get a good feel on how far off from the true moments it is possible to be under some often realistic assumptions. Of course we can never get infinitely many moments with any degree of accuracy from a sample, so now we would really want to do another round of approximations, but that is the general idea. For nice random variables, moments are sufficient to estimate the sample law.

I should mention that what I have said above is all heuristic and doesn’t work in most interesting modern examples. In truth, I think the right answer to your question is that we don’t need moments because for many relevant applications (particularly in economics) it seems unlikely that all moments even exist. The thing is that when you get rid of moment assumptions you lose an enormous amount of information and power: without at least two, the Central Limit Theorem fails and with it go most of the elementary statistical tests. If you do not want to work with moments, there is a whole theory of nonparametric statistics that make no assumptions at all on the random process.

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# SSC CGL Tier 2 JSO / Paper 3 Statistics Study Material Day 6

SSC CGL Tier 2 JSO / Paper 3 Statistics Study Material Day 6

#### Standard Deviation

The standard deviation (often SD) is a measure of variability. When we calculate the standard deviation of a sample, we are using it as an estimate of the variability of the population from which the sample was drawn. For data with a normal distribution, about 95% of individuals will have values within 2 standard deviations of the mean, the other 5% being equally scattered above and below these limits. Contrary to popular misconception, the standard deviation is a valid measure of variability regardless of the distribution. About 95% of observations of any distribution usually fall within the 2 standard deviation limits, though those outside may all be at one end. We may choose a different summary statistic, however, when data have a skewed distribution.

The standard deviation is the most important and widely used measure of dispersion. This is also known as root mean square deviation, since it is the square root of the mean of the squared deviation of the sizes from arithmetic mean.

The standard deviation measures the absolute dispersion or variability of the distribution. The greater the amount of dispersion, the greater is the standard deviation.

Thus, standard deviation is defined as the positive square root of the squares of the deviation of all the sizes from their arithmetic mean. Standard Deviation for Ungrouped Data #### Standard Deviation for Frequency Distribution

Formula for computing standard deviation for a frequency distribution is #### Merits and Demerits of Standard Deviation

1. Its most important beauty is that it is free from the of Its most important beauty is that it is free from the compulsion of taking only absolute value in estimating mean deviation. So it is frequently applicable in different algebraic operations.
2. It took into account all individual observations and so any slight variation in any observations automatically got representation in standard deviation.
3. Through variance it easily reflects the aberration in data series.
4. It is the basis of relative measure of dispersion coefficient of variation (CV).
5. It is also an absolute measure of dispersion and so comparisons of data series in different units of comparisons of data series in different units of measurement are not tenable.
6. Its value changes if unit of measurement changed.
7. In a normal distribution, data are symmetrically distributed around mean(mean, median or mode all become identical) and mean σ covers 68.27 per cent of observations; mean 2σ covers 95 45 per cent of observations; mean 2σ covers 95.45 per cent of observations and mean 3σ covers 99.73 per cent of observations. This property is useful in dividing a data series into suitable groups or class.

#### Measures of Relative Dispersion

The relative dispersion of a data set, more commonly referred to as its coefficient of variation, is the ratio of its standard deviation to its arithmetic mean. In effect, it is a measurement of the degree by which an observed variable deviates from its average value. It is a useful measurement in applications such as comparing stocks and other investment vehicles because it is a way to determine the risk involved with the holdings in your portfolio.

Determine the arithmetic mean of your data set by adding all of the individual values of the set together and dividing by the total number of values.

Square the difference between each individual value in the data set and the arithmetic mean.

Add all of the squares calculated in Step 2 together.

Divide your result from Step 3 by the total number of values in your data set. You now have the variance of your data set.

Calculate the square root of the variance calculated in Step 4. You now have the standard deviation of your data set.

Divide the standard deviation calculated in Step 5 by the absolute value of the arithmetic mean calculated in Step 1.

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