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Measures Of Central Tendency And Dispersion

1 Arithmetic mean:
(i) For ungrouped data (individual series) ` x =

(ii) For grouped data (continuous series)
(a) Direct method `x = where xi , I = 1 .. n be n observations and fi be their corresponding frequencies
(b) short cut method : `x = A + åfidi / åfi) where A = assumed mean, di = xi - A = deviation for each term
2 Properties of A.M.
(i) In a statistical data the sum of the deviation of items form A.M. is alwalys zero.
(ii) If each of the n given observation be doubled, then their mean is doubled
(iii) If `x is the mean of x1, x2, , xn . the mean of ax1, x2, .. , axn is a `x where a is any number different form zero
(iv) Arithmetic mean is independent of origin i.e. it is x effected by any change in origin.
3 Geometric mean:
(i) For ungrouped data G.M. = (x1 x2 x3 .. xn)1/n or
G.M. = antilog
(ii) For grouped data G.M. = (xf11 xf22 .. xfnn)1/N , where N = i=1 ån fi

4 Harmonic mean Harmonic mean is reciprocal of arithmetic mean of reciprocals.
(i) For ungrouped data H.M. =
(ii) For grouped data H.M. =
5 Relation between A.M., G.M and H.M.
A.M. ³ G.M. ³ H.M.
Equation holds only when all the observations in the series are same
6 Msdian:
(a) Individual series (ungrouped data) : If data is raw, arrange in ascending or descending order and n be the no. of observations . If n is odd, Median = value of ((n+1) / 2))th observation If n is even, median = (1/2) [value of (n/2)th + value of ((n/2) + 1)th] observation.
(b) Discrete series: First find cumulative frequencies of the variables arranged in ascending or descending order and Mediain = {(n+1) / 2}th observation, where n is cumulative frequency.
(c) Continuous distribution (grouped data)
(i) For series in ascending order
Median = e +
Where
e = Lower limit of the median class.
f = Frequency of the median class.
N = Sum of the all frequencies.
i = The width of the median class.
C = Cumulative frequency of the class preceding to median class.
(ii) For series in descending order Median = u -
Where u = upper limit of median class.
7 Mode:
(i) For individual series: In the case of individual series, the value which is repeated maximum number of times is the mode of the series.
(ii) For discrete frequency distribution series: In the case of discrete frequency distribution, mode is the value of the variate corresponding to the maximum frequency.
(iii) For continuous frequency distribution : first find the model class i.e. the class which has maximum frequency for continuous series

Where
e1 = Lower limit of the model class.
f1 = Frequency of the model class.
f0 = Frequency of the class preceding mode class.
f2 = Frequency of the class succeeding model class.
i= Size of the model class.
8 Relation between mean, mode & median:
(i) In symmetrical distribution : mean = mode = median
(ii) In Moderately symmetrical distribution : mode = 3 median 2 mean

Measure of Dispersion:
The degree to which numerical data tend to spread about an average value is called variation or dispersion popular methods of measure of dispersion.
(a) Individual series (ungrouped data)
1 Mean deviationMean deviation = (å|x-S| / n)
Where n = number of terms, S = deviation of variety form mean mode , median
(b) Continuous series (grouped data)

Note:
Mean deviation is the least when measured from the median.
2 Standerd Deviation :
S.D. (s) is the squere root of the arithmetic mean of the squares of the deviations of the terms from their A.M.
(a) for individual series (ungrouped data)
where `x = Arithmetic mean of the series N = Total frequency
(b) For continuous series (grouped data)
(i) Direct method s =
Where
`x = Arithmetic mean of series
X1 = mid value of the class
f1 = Frequency of the corresponding x1
N = åf = Total frequency
(ii) Short cut method

Where
d = x - A = Derivation from assumed mean A
f = Frequency of item (term)
N = åf = Total frequency.
Variance Square of standard direction
i.e. variance = (S.D.)2 = (s)2
Coefficient of variance = Coefficient of S.D. x 100 = (s / x) x 100


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