STATISTICS.pdf

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STATISTICS

By:- Nishant Gupta For any help contact: 9953168795, 9268789880

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880

1. The data obtained in a statistical investigation is called raw data and when it is arranged in ascending or

descending order of magnitude, it is called an array.

2. A variable which can assume any value between two given values is called a continuous variable, otherwise it is called a discrete variable.

MEASURES OF CENTRAL TENDENCY (OR AVERAGES)

An average of a distribution is that value of the variable which is representative of the entire distribution. Following are the five measures of central tendency.

1. Arithmetic Mean or just Mean x

2. Geometric Mean

3. Harmonic Mean

4. Median

5. Mode.

AIRTHMETIC MEAN

(i) If a variable x takes values x1, x2, …, xn, then the A.M. is denoted by x and is given by

n

1ii

n21 xn

1

n

x........xxx

(ii) For a ungrouped frequency distribution

x = x1 x2 …. xn f = f1 f2 …… fn

n21

nn2211

f........ff

xf........xfxfx

.fwhereNxf

N

1 n

1ii

n

1i1ii

(iii) For a grouped frequency, formula listed in (ii) is applicable where xi denotes the mid point of ith class.

(iv) Weighted Arithmetic Mean. If x takes values x1, x2, .......x:n with their respective weights w1, w2, ……..wn, then weighted A.M. is given by

n

1ii

n

1iii

n21

nn2211

w

xw

w........ww

xw........xwxwx

SHORT-CUT METHOD IN COMPUTING

Arithmetic Mean We take a number 'a' (generally in the middle of the greatest and the least values of the variable) called the assume mean.

(i) For simple distribution

STATISTICS


n

1iidaA where di = xi - a, n is the number of terms.

(ii) For ungrouped frequency distribution

n

1ii

n

1iii

f

df

aA where di = xi – a.

(iii) Step deviation or Shift of origin and change of scale for grouped frequency distribution :

uhaufN

1hax

n

1iii

where .fN;h

axu

n

1ii

ii

(iv) Mean of the composite of the k groups. If k21 x....,,.........x,x are means of k groups having n1, n2,.............,

nk members, then mean of the k groups, combined is give

k21

kk2211

n..............nn

xn...............xnxnx

.

Some Algebraic Properties of A.M.

(i) Algebraic sum of deviations of all values of variable from their A.M. is always zero.

Thus, for simple distribution. ,0xxn

1ii

And for a frequency distribution. ,0xxfn

1iii

(ii) The mean of the sum of two (or more) variables is equal to sum of their means.

(iii) If u, v are two variables and w = au + bv for some constants a, b then vbuaw .

(iv) Sum of squares of deviations of variable is minimum when taken about A.M.

GEOMETRIC MEAN

(i) If x takes positive values x1, x2,……...,xn then G.M. of x is G = (x, x2 ... xn)1/N. Using logarithm, we see that

G = antilog

n

1iixlog

x

1

(ii) For a frequency distribution :

x = x1, x2, …..., xn f = f1, f2, ……., fn

G.M. is given by N1fn

f2

f1

n21 x..........x.xG

In terms of log, G = antilog

n

1iii xlogflog

x

1

For a grouped frequency distribution, xi is the mid-point of the ith class interval.

(iii) If G1 and G2 are the geometric means of the two series of sizes n1 and n2 respectively, then the G.M. G of the combined series is given by

log G21

2211

nn

GlognGlogn

(iv) It is useful in the construction of index numbers, averaging ratios, percentages etc.


HARMONIC MEAN

If x assumes non-zero values x1, x2,...., xn, then H.M. is denoted by H and is given by

n

1i i

i

x

f

n

1

1H

For a frequency distribution : (xi, fi), i = 1, 2, …....., n,

n

1i i

i

x

f

N

1

1H

It is useful in problems related with rates, ratios, times, etc. Note. A G H.

MEDIAN AND OTHER PARTITION VALUES

Median is that value of the variable which divides the total observations into two equal halves.

(i) If x takes values x1, x2, ..., xn (n odd), then the median is

2

1nth value after the values have been

arranged in ascending or descending order of magnitude.

If n is even, then the A.M. of

2

nth and

1

2

nth values is the median.

(ii) For a frequency distribution (xi, fi), i = 1,2,….., n, median is calculated as follows :

First, find the cumulative frequencies. Then, see the cumulative frequency just greater than 2

N. The

corresponding value of x is the median.

(iii) For a grouped frequency distribution. Median is calculated by the formula

f

hC

2

NlMe

Where l = lower limit of median class

f = frequency of median class

h = width of median class

c = c.f. of the class preceding the median class.

The class corresponding to cumulative frequency just greater than 2

N is the median class.

Graphical Method: Here we draw 'less than' and 'more than' ogive. The abscissa of point of intersection of these ogives is the median.

Like median, the other partition values — quar-tiles, deciles, percentiles, etc. can be determined- The ith

quartile Qi is given by etc3,2,1i,f

hC4

iN

lQ

MODE

The mode or modal value of a distribution is that value of the variable which has the maximum frequency.

For a grouped frequency distribution, mode is given by

hfff2

fflMode

21m

1m

Where l = lower limit of modal class (i.e., the class in which frequency is maximum)


h = width of modal class

f1 = frequency of the class preceding the modal class.

f2 = frequency of the class following the modal class

fm = maximum frequency.

Note: (i) The length of intervals should be equal (ii) If 2fm – f1 – f2 = 0 then use :

hffff

fflMode

2m1m

1m

MEASURES OF DISPERSION

Averages are not sufficient to give a complete picture of the distribution as they do not tell us how the values vary about some central value. There can be more than one distributions having the same average but have wide disparities in the formation of the distribution. Dispersion measures the scatteredness of various observation about some central value. Following are the measures of dispersion :

(i) Range

(ii) Quartile Deviation

(iii) Mean Deviation and

(iv) Standard Deviation

(i) Range of a distribution is the difference of the largest and the smallest values.

Coefficient of range = SL

SL

(ii) Quartile Deviation = Q3 – Q1 Coefficient of quartile deviation = 13

13

QQ

QQ

(iii) Mean Deviation. For a frequency distribution (xi, fi),i = 1,2, ...,n

Mean Deviation (M.D.) from 'a' .axfN

1i

n

1ii

where 'a' can be mean, mode or median

Coefficient of dispersion =a

a'' fromdeviation Mean

(iv) Standard Deviation (S.D.) For a frequency distribution (xi, fi),i = 1,2,…..,n,

S.D. is denoted by and is given by

n

1i

2

1i xxfN

1

n

1i

2

ii2

ii xfN

1xf

N

1

(for calculation)

n

1i

2

ii2

ii ufN

1uf

N

1h Where

h

axu i

i

ux h

Thus S.D. is independent of shift of origin but depends upon change of scale,

Coefficient of Dispersion (C.D.) = x

Coefficient of Variation (C.V.) = 100

x

If s denotes the root mean square deviation from some number a, i.e.,

n

1i

2ii axf

N

1s and is the S.D. s2 = 2 + d2 where d = ax

clearly, s is least when d = 0 i.e., ax


Thus, root mean square deviation is least when deviation are taken from x .

Square of S.D. is called variance. S.D. ( ) of the combined mp of two groups having means, 21 x,x ;

standard deviation 21, and number of elements n1, n2 is given by

22

222

21

211

21

2 dndnnn

1

Where .xxd,xxd 2211

And 21

2211

nn

xnxnx

Also, note that 2 (Range)2.

SYMETRIC AND SKEW-SYMMETRIC

In a symmetrical distribution, Mean, Median, Mode coincide. Here, frequencies are symmetrically distributed both sides of some central value.

A distribution which is not symmetrical, is called skew- symmetrical. In a moderately skew-symmetric distribution,

Mean - Mode = 3 (Mean - Median)

In a positively skew-symmetric distribution, the value of mean is maximum and that of mode is least, and the median lies between the two.

In a negatively skew-symmetric distribution, the value of mode is maximum and that of mean is least, and the median lies between the two.

Absolute measures of skewness are

(i) ,Mx e (ii) ,Mx 0 (iii) Q3 + Q1- 2Q2.


1. A.M. of squares of first n natural numbers is

(a) 6

1n (b)

6

1n 2

(c) 6

)1n2)(1n( (d)None of these.

2. The A.M. of nC0, nC1, nC2, …….. , nCn is

(a) 1n

2n

(b)

n

2 n

(c) 1n

2 1n

(d) None of these

3. The mean wage of 1000 workers in a factory running in two shifts of 700 and 300 workers is Rs, 500. The mean wage of 700 workers working in day shift is Rs. 450. The mean wage of workers working in the night shift is

(a)Rs.570 (b) Rs.616.67

(c) Rs.543.67 (d) None of these.

4. The average weight of 25 boys was calculated to be 78.4 kg. If was later discovered that one weight was misread as 69 kg instead of 96 kg. The correct average is

(a) 79 kg (b) 79.48 kg

(c) 81.32 kg (d) N/T

5. Which of the following is not a measure of central tendency?

(a) Mean (b) Median

(c) Mode (d) Range.

6. The weighted mean of first n natural numbers whose weights are equal to the squares of the corresponding numbers is

(a) 2

1n (b)

1n22

1nn3

(c)

6

1n21n (d)

2

1nn

7. The relationship between mean, median and mode for a moderately skewed distribution is

(a) Mode = Median - 2 Mean

(b) Mode = 2 Median – Mean

(c) Mode = 3 Median - 2 Mean

(d) Mode = 2 Median - 3 Mean.

8. Median of 16, 10, 14, 11, 9, 8, 12, 6, 5 is

(a) 10 (b) 12

(c) 11 (d) 14.

9. In an arranged series of an even number n of the median is

(a) th2

n term

(b) th12

n

term

(c) the mean of th2

n

and th1

2

n

term

(d) None of these

10. Which of the following is not a measure of dispersion?

(a) Variance (b) Mode

(c) Mean deviation (d)Standard deviation

11. If each observation of a raw data whose

variance is 2 , is increased by then the

variance of the new set is

(a) 2 (b) 22

(c) 22 (d) None of these.

12. If each observation of a raw data, whose

variance is 2 , is multiplied by , then the

variance of the new set is

ASSIGNMENT STATISTICS


(a) 2 (b) 22

(c) 2 (d) 22

13. If x is the mean of a distribution, then

xxf 11

(a) 0 (b) M.D.

(c) S.D. (d) None of these.

14. The variance of the first n natural number is

(a) 12

1n 2 (b)

12

1n 2

(c) 6

1n 2 (d)

12

1n 2

15. The sum of squares of deviations of a set of values is minimum when taken about

(a) A.M. (b) Median

(c) Mode (d) H.M.

16. Median can be graphically determined from

(a) Ogive (b) Histogram

(c) Frequency curve (d) None of these.

17. A person purchased one kg of potatoes from each of 4 places at the rate of 1 kg, 2 kg, 3 kg and 4 kg per rupee respectively. If he has purchased x kg of potatoes per rupee, then x

(a) 1.92 (b) 2

(c)2.10 (d)None of these.

18. A market with 3900 operating firms has the following distribution:

Income group of workers No. of firms

150 – 300

300 – 500

500 – 800

800 – 1200

1200 – 1800

300

500

900

1000

1200

If the histogram is constructed with the above data, the highest bar in the histogram would correspond to the class

(a) 500 - 800 (b) 1200 - 1800

(c) 800 - 1200 (d) 150 – 300.

19. The mean of a set of observation is x. If each observation is divided by a, a 0 and then is increased by 10, then mean of the new set is

(a) a

x (b)

a

10x

(c) a

a10x (d) bxa

20. The mean age of a combined group of men and women is 30 years. If the means of the age of men and women are respectively 32 and 27, then the percentage of women in the group is

(a) 30 (b) 40

(c) 50 (d) 60.

21. Which one of the following measures is the most suitable one of central location for computing intelligence of students ?

(a) Mode (b) A.M.

(c) G.M. (d) Median.

22. Variance of the data 2, 4, 6, 8,10 is

(a) 6 (b) 7

(c)8 (d) None of these.

23. The mean deviation from the median is

(a) greater than that measured from any other value

(b) less than that measured from any other value

(c) equal to that measured from any other value

(d) maximum if all observation are positive.

24. If a variable x takes values a:; such that bxa i for i = 1,2, ...,n, then

(a) bxvara (b) 22 bxvara

(c) xvar4

a 2

(d) xvarab2

25. If variance of x1, x2, …….. , xn is 2 , then

variance of ax1, ax2, ……….. ,axn 0a , is

(a) 2 (b) a 2

(c) a2 2 (d) 2

2

a

26. If in an examination different weights are assigned to different subjects. Physics (2), Chemistry (1), English (1). Mathematics (2). If a student scored 60 in Physics, 70 in


Chemistry, 70 in English and 80 in Mathematics, then his weighted A.M. is

(a) 60 (b) 70

(c) 80 (d) None of these.

27. Workers work in three shifts I, II, III in a factory. Their wages are in the ratio 4:5:6 depending upon the shift. Number of workers in the shifts are in the ratio 3 : 2 : l. If total number of workers working is 1500 and wages per worker in shift I is Rs.400. Then mean wage of a worker is

(a) Rs.467 (b) Rs.500

(c) Rs.600 (d) Rs.400.

28. A group of 10 items has A.M. 6 and A.M. of four items in 7.5, then A.M. of remaining items is

(a) 6.5 (b) 5.5

(c) 4.5 (d) None of these.

29. If 25% of the items are less than 15 and 25% are more than 45, then coefficient of quartile deviation is

(a) l (b) 1/2

(c) 1/4 (d) 1/8

30. The A.M. of 9 items is 15. If one more item is added to this series, the A.M. becomes 16. The value of 10th item is

(a) 23 (b) 25

(c) 27 (d) 30.

31. A car owner buys petrol at Rs.7.50, Rs.8.00 and Rs.8.50 per litre for the 3 successive years. If he spends Rs.4000 each year, then the average cost per litre of petrol is

(a) Rs.8 (b) Rs.8.25

(c) Rs.7.98 (d) None of these.

32. The mean of following frequency table is 50.

Class Frequency

0 – 20

20 – 40

40 – 60

60 – 80

80 – 100

17

f1

32

f2

19

Total 120

The missing frequencies are

(a) 28, 24 (b) 24, 36

(c) 36, 28 (d) None of these.

33. Geometric mean of 1, 2, 22, 23, .....,, 2n is

(a) n

2

2 (b) 2

n

2

(c) 2

1n

2

(d) 2

1n

2

34. The mean square deviation of n observations x1 , x2, ... xn about - 2 and 2 are 18 and 10 respectively. Then, S.D. of the given set is

(a) 1 (b) 2

(c) 3 (d) 4.

35. If G is the G.M. of the product of K sets of observations, with G.M.'s G1, G2, ..., GK respectively, then G is equal to

(a) log G1 + log G2 + ... + log GK

(b) log G1 log G2 ... log GK

(c)G1 G2 ...GK

(d) None of these.

36. Mean of n times is x . If these x items are successively increased by 2, 22, 23, ..., 2n, then the new mean is

(a)n

2x

1n

(b) n

2

n

2x

1n

(c) n

2x

n

(d) None of these.

37. If 1X and 2X are means to two distributions

such that 1X < 2X and X is the mean of the

combined distribution, then

(a) 1XX (b) 2XX

(c) 2

XXX

21 (d) 21 XXX

38. The A.M. of n observation is x . If the sum n – 5 observations is a, then the mean of remaining 5 observations is

(a) 5

axn (b)

5

axn

(c) axn (d) None of these.


39. Karl-Pearson's coefficient of skewness of a distribution is 0.4. If S.D. is 6 and mean 40, then median of the distribution is

(a) 39.5 (b) 39

(c) 39.2 (d) None of these.

40. The mean of the values 0, 1, 2, ..., n with the corresponding weights nC0, nC1,..., nCn, respectively is

(a) 1n

2n

(b)

1nn

2 1n

(c) 2

1n (d)

2

n

41. A car completes the first half of its journey with a velocity v1 and the rest half with velocity v2. Then the average velocity of the car for the whole journey.

(a) 2

vv 21 (b) 21vv

(c) 21

21

vv

vv2

(d) None of these.

42. The quartile deviation of daily wages (in Rs.) of 7 persons is given below :

12, 7.15,10, 17,17, 25 is

(a) 14.6 (b) 5

(c) 9 (d) 4.5.

43. Mean deviation of numbers 3, 4, 5,6, 7 is

(a) 0 (b) 1.2

(c) 5 (d) 25.

44. In a class of 100 students there are 70 boys whose average marks in a subject are 75, If the average marks of the complete class is 72, then what is the average marks of the girls ?

(a) 73 (b) 65

(c) 68 (d) 74.

45. In an experiment with 15 observations on x, the following results were available Sx2 = 2830, Ix =a 170. One observation 20 found to be wrong and was replaced by the correct value 30- Then, the corrected variance is

(a) 188, 66 (b)177,33

(c) 8.33 (d) 78.00.

46. Consider the following statements :

(i) Mode can be computed from histogram

(ii) Median is not independent of change of scale

(iii) Variance is independent of change of origin and scale.

Which of these is/are correct

(a) only (i) (b)only (ii)

(c) only (i) and (ii) (d) (i), (ii) and (iii).

47. In a series of2n observations, half of them equal a and the remaining equal - a. If the S.D. is 2 then |a| equals

(a) n

1 (b) 2

(c) 2 (d) n

2

48. If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

(a) 25.5 (b)24.0

(c) 22.0 (d) 20.5.

49. A random variable X has Poisson distribution with mean 2. Then P(x > 1,5) equals

(a) 2e

31 (b)

2e

3

(c) 2e

2 (d) 0

50. Suppose a population A has 100 observations 101, 102, ......., 200, and another population B has 100 observations 151,152, .„..., 250. If VA

and VB represent the variances of the two

populations respectively, then , B

A

V

Vis

(a) 4/9 (b) 2/3

(c) 1 (d) 9/4


ANSWER (STATISTICS)

1 2 3 4 5 6 7 8 9 10

c b b b d b c a c b

11 12 13 14 15 16 17 18 19 20

d b a a a a a b c b

21 22 23 24 25 26 27 28 29 30

d c b d c b a d b B

31 32 33 34 35 36 37 38 39 40

c a b c c b d b b d

41 42 43 44 45 46 47 48 49 50

c b b − − − − − − −

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