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STATISTICS

By:- Nishant Gupta

For any help contact:

9953168795, 9268789880


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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 ordescending 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 thefive measures of central tendency.

1.

Arithmetic Mean or just Meanx

2. Geometric Mean3. Harmonic Mean4. Median5. 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 distributionx = 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 ofith 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


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n

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

(ii) For ungrouped frequency distribution

n

1i i

n

1iii

f

df

aA where di = xi a.

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

N

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,.............,nkmembers, 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 N1fnf2f1 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 ofthe combined series is given by

log G21

2211

nn

GlognGlogn

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


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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 than2

Nis 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 maximumfrequency.

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)


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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

fflMode2m1m

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, ...,nMean Deviation (M.D.) from 'a' .axf

N

1i

n

1ii

where 'a' can be mean, mode or median

Coefficient of dispersion =a

a''fromdeviationMean

(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 ii

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


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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

222222121121

2 dndnnn

1

Where .xxd,xxd 2211

And21

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.


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1. A.M. of squares of first n natural numbers is(a)

6

1n (b)

6

1n2

(c)6

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

2. The A.M. ofnC0, nC1, nC2, .. , nCn is(a)

1n

2n

(b) n

2n

(c)1n

21n

(d) None of these

3. The mean wage of 1000 workers in a factoryrunning 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 calculatedto 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 ofcentral tendency?

(a) Mean (b) Median

(c) Mode (d) Range.

6. The weighted mean of first n natural numberswhose 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 andmode 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 ofthe median is

(a) th2

nterm

(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 ofdispersion?

(a) Variance (b) Mode

(c) Mean deviation (d)Standard deviation

11. If each observation of a raw data whosevariance 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, whosevariance is 2 , is multiplied by , then thevariance of the new set is

ASSIGNMENT

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(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

1n2

(b)12

1n2

(c)6

1n2

(d)12

1n2

15. The sum of squares of deviations of a set ofvalues 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 fromeach 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 thefollow- ing 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 eachobservation is divided by a, a 0 and then isincreased 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 menand women is 30 years. If the means of theage 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 themost 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) xvarab 2

25. If variance of x1, x2, .. , xn is 2 , thenvariance 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 areassigned to different subjects. Physics (2),

Chemistry (1), English (1). Mathematics (2). If

a student scored 60 in Physics, 70 in


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39. Karl-Pearson's coefficient of skewness of adistribution 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 thecorresponding weights

n

C0,n

C1,...,n

Cn,respectively is

(a)1n

2n

(b)

1nn2 1n

(c)2

1n (d)

2

n

41. A car completes the first half of its journeywith a velocity v1 and the rest half with

velocity v2. Then the average velocity of the

car for the whole journey.

(a)2

vv21

(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 boyswhose 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 correctvalue 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 themequal 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 andmedian 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 distributionwith mean 2. Then P(x > 1,5) equals

(a)2

e

31 (b)

2

e

3

(c)2

e

2(d) 0

50. Suppose a population A has 100 observations101, 102, ......., 200, and another population B

has 100 observations 151,152, ...., 250. If VAand 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


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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 20d 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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