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Subject: Introduction to Statistics BS Mathematics Morning/Evening program spring semester 2020
Chapter # 04 Measures of Dispersion, Moments, Skewness and Kurtosis
Measures of Dispersion: Sometimes when two or more different data sets are to be compared using measure of central tendency or averages, we get the same result. Consider the runs scored by two batsmen in their last ten matches as follows: Batsman A : 30, 91, 0, 64, 42, 80, 30, 5, 117, 71 Batsman B : 53, 46, 48, 50, 53, 53, 58, 60, 57, 52 Clearly, mean of the runs scored by both the batsmen A and B is same i.e. 53 Can we say that the performance of two players is same? Clearly No, because the variability in the scores of batsman A is from 0 to 117, whereas, the variability of the runs scored by batsman B is from 46 to 60. Let us now plot the above scores as dots on a number line. We find the following diagrams:
Batsman A (^) 0 10 20 30 40 50 60 7080 90 100 110 (^120)
Batsman B (^) 0 10 20 30 40 50 60 7080 90 100 110 12
We can see that the dots corresponding to batsman B are close to each other and is clustering around the measure of central tendency (mean), while those corresponding to batsman A are scattered or more spread out. Thus, the measures of central tendency are not sufficient to give complete information about a given data. In such a situation the comparison becomes very difficult. We therefore, need some additional information for comparison, concerning with, how the data is dispersed about (more spread out) the average. This can be done by measuring the dispersion. Like „ mea sures of central tendency ‟ we want to have a single number to describe variability. This single number is called a ‘measure of dispersion’.
Disp ersion: “T he v a r i a b i l i t y (s p r e a d ) th a t e x i s t s be t w e e n th e v a l u e of a da t a i s c a l l e d di s p e r s i o n ” OR “The extent to which the observations are spread around an average is called dispersion or scatter ”. As we know that, there are quite a few ways of measuring the central tendency of a data set i.e. A.M, G.M, H.M, Mode and Median. Similarly, we have different ways of measuring and comparing the dispersion of the distribution(s). There are two important types of measures of dispersion.
Types of Measures of Dispersion:
There are two types of measure of dispersion
I) Absolute Measure of Dispersion II) Relative Measure of Dispersion
Absolute Measu re of Disp ersion: “An absolute measure of dispersion measures the variability in terms of the same units of the data” e.g. if the units of the data are R s, mete rs, kg, etc. The units of the mea sures of dispe rsio n will a lso be Rs , meters, kg, etc.
The common absolute measures of dispersion are: Range Quartile Deviation or Semi Inter-Quartile Range Average Deviation or Mean Deviation Standard Deviation
Relative Measu re of Disp ersion: “A relative measure of dispersion compares the variability of two or more data that are independent of the units of measurement”
In other word “A relative measure of dispersion, expresses the absolute measure of dispersion relative to the relevant average and multiplied by 100 many times” i.e.
R X (^) m X 0 X (^) m =48, X (^) 0 = R 48 32 R 16 marks
Coefficient of Range 0 0
.. m m
C o f X^ X X X
C o f (^) 48 32
C o f 80 C o f.. 0.
Qu artile Deviat ion or S emi-inter-qu artile Ran ge: “half of the difference between the upper quartile and lower quartile is called the semi-inter quartile range or quartile deviation”i.e. 3 1 2 Quartile deivation ^ Q^ Q
Coefficient of Quartile Deviation: The coefficient of quartile deviation is a relative measure of dispersion and is given by: 3 1 3 1
Coefficient Q D. QQ^^ QQ
Numerical example of quartile deviation and coefficient of quartile deviation
Ex # calculate quartile deviation and coefficient of quartile deviation for ungrouped data. The marks obtained by 9 students are given below
3 1 2 Quartile deviation ^ Q^ Q
“ n =9” is odd then we use odd case formulae
n^ th Q Marks obtained by^ ^ ^ student 1
Q =36 marks, Q 3 =
45 36 Quartile deviation (^) 2 4.5marks (Answer).
Coefficient of Quartile deviation:
x i^45 32 37 46 39 36 41 48
3 1 3 1
Cofficient of Q D. QQ^^ QQ
. 45 32 Cofficient of Q D (^) 45 32 ^ 0.11 (Answer).
Ex # calculate quartile deviation and coefficient of quartile deviation for continuous grouped data.
Class boundaries Midpoints xi Frequency fi Cumulative frequency c f.
1
n
i ^ fi =
Q l h^ n C f
^
Q 56.40marks
3
Q l h^ n C f
^
Q 73.76marks
Calculate Quartile deviation and coefficient of quartile deviation from above results?
Mean Absolute Deviation or Mean Deviation (Average Deviation): “ T h e a r i t h m e t i c m e a n o f t h e a b s o l u t e d e v i a t i o n f r o m a n a v e r a g e ( m e a n , m e d i a n e t c. ) i s c a l l e d m e a n d e v i a t i o n o r a v e r a g e d e v i a t i o n ”
Coefficient of Mean Deviation: The coefficient of mean deviation is a relative measure of dispersion and is given by:
Calculate coefficient of mean deviation for both mean and median.
.... ( ) C o M D M D (^) mean x ... 4. C o M D 40 C o M D... =0.11 (Answer).
C o M D... M D^. (^) median
... 4. C o M D 39 C o M D... =0.11 (Answer).
Calculate mean deviation and coefficient of mean deviation from mean in continuous grouped case, showing the weights of 60 apples.
Weight (grams)
Midpoints ( xi )
Frequency ( fi ) f xi i i
x x fi xi x
65---- 85---- 105---- 125---- 145---- 165---- 185----
1
n
i ^ fi^ 1
n
i ^ f xi i^
f i^ xi^ x
i i i
x f x (^) f
x 60 x =122.5 grams
. i^ i i
M D f^ x^ x f
^
M D 60 M D. =28.27 grams (Answer).
Weights (grams)
Frequency 09 10 17 10 05 04 05
Stand ard Deviation : “ T he p o sit iv e square roo t o f v a r i a n c e i s c a l l e d a s s ta nda rd de v i a t i o n ”. OR “T he p o sit ive square r oo t o f th e a rithmet ic mea n o f th e sq uared devia ti ons fro m th e m ea n is ca lled the s t a nda rd d evia ti o n”
Methods of Calculating Variance and Standard Deviation.
Coefficient of Standard Deviation OR Coefficient of Variation: The coefficient of standard deviation is a relative measure of dispersion and is given by:
Coefficient of S.D Standard Deviation
Mean
The coefficient of standard deviation is also called the coefficient of variation, denoted by C.V and is given by:
Coefficient of Variation was introduced by Karl Pearson. It is used to compare the variation or to compare the performance of two sets of data. A large value of C.V indicates that there is greater variability and vice versa. Similarly, the smaller the C.V the more consistent is the performance and vice versa.
Numerical example of Standard deviation (S.D),Variance and Coefficient of Variation (C.V) in case of ungrouped data, using direct method:
Ex # Calculate the variance, S.D and C.V from the following marks obtained by 9 students.
x i xi x x i x ^2 x i^2
x i =360^ x i^ x ^2 =232^ x i^2 =
x x^ i (^) n
x 9 x =40 marks
Variance or ^
2 S^2 xi^ x n
^ ^2
S 9
S^2 =25.78 marks ^2
Standard deviation or S.D or ^
x i x^2
S ^ n^ S 25.78 S =5.08 marks
(Answer). Using the alternative method. Calculate Variance:
x i^45 32 37 4846 3639 36
2 2 S^2 xi^ xi n n
^ ^
^
S 9 9
^
S^2 =1625.78 1600
S^2 =25.78 marks ^2 (Answer).
Calculate Standard deviation: 2 2 S x^ i^ xi n n
^ ^
^
S 9 9
^
S 25.78 S =5.08 marks
(Answer).
Calculate Coefficient of Variation (C.V): C V. S D^. 100 (^) x
. 5.08 100 C V 40 C V. =12.70 (Answer).
Calculate the Variance, Standard deviation and Coefficient of Variation from the following weight of 60 apples in Continuous grouped data:
Weight (grams)
Midpoints ( xi ) Frequency ( fi ) f xi i (^) f x i i^2 65---- 85---- 105---- 125---- 145---- 165---- 185----
1
n
i ^ fi^ 1
n
i ^ f xi i^ ^735
f xi i^^2 973 335.
Calculate Variance: 2 2 2 i i i i i i
S f x^ f x f f
^ ^
^
2 973335.00^ 7350.0^2
S 60 60
^
S^2 16222.25 15006.
S^2 =1216 grams ^2 (Answer).
Calculate Standard deviation:
Moments: “T he arit hmet ic mean o f th e rth^ po we r o f devia ti ons taken eit her fro m mean , z ero o r fro m a n y a r b i t r a r y ori g in ( p rovisiona l , m e a n) a r e c a lled moment s ”. When the deviations are computed from the arithmetic mean, then such moments are called moments about mean (mean moments) or
sometimes called central moments , denoted by mr and given as
follows:
x i xi x x i x ^2 x i x ^3 x i x ^4
x i =360^ ^ xi^ x
^ x i^ x ^2
^ ^ x i^ x ^3
^ ^ x i^ x ^4
m 1 = 0 , m 2 = 25.78 (marks)^2 , m 3 = 20.67 (marks)^3 , m 4 = 1189.78 (marks)^4 (Answer).
Question # Calculate first four moments about mean for grouped data (using a continuous grouped case formula). The following distribution relates to the number of assistants in 50 retail establishments, the data are given below:
No. of assistants
f 3 4 6 7 10 6 5 5 3 1 Using these formulae
1^ i^ ii m f^ x^ x f
^
, ^
2 2^ i^ ii m f^ x^ x f
^
, ^
3 3^ i^ ii m f^ x^ x f
^
and
^4
4^ i^ ii m f^ x^ x f
^
Numerical example of Moment in continuous grouped data: Compute the first four moments and measure of Skewness and Kurtosis for the following distribution of wages using a short cut method:
Weekly earnings (Rupees)
No. of men 1 2 5 10 20 51 22 11 5 3 1
1^ i^ ii m f D (^) f
m 131 m 1 =0.06,
2 2^ i^ ii m f D (^) f
m 131 m 2 =
3 3^ i^ ii m f D (^) f
m (^131) m 3 = 0.56,
4 4^ i^ ii m f D (^) f
m 131 m 4 =28.
m 1 m 1 m 1 0 (always zero), m 2 m 2 m 1 ^2 m 2 2.64 0.06^2
m 2 = 2.64;
m 3 m 3 3 m m 2 1 2 m 1 ^3 m 3 0.56 3 2.64 0.06 2 0.06 ^3 m 3 = 0.08;
m 4 m 4 4 m m 3 1 6 m 2 m 1 ^ 2 3 m ^4
m 4 28.38 4 0.56 0.06 6 2.64 0.06 2 3 0.06 ^4
m 4 28.30 (Answer).
Earnings in Rs. ( xi^ )
Men fi
Di xi A
A 10
f D i i (^) f D i i^2 f D i i^3 f D i i^4
f i =131^ f Di i = 8^ f D i i^2 =
f D i i^3 =
f D i i^4
