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Page 1 Page # 47 STATISTICS 1. Arithmetic Mean / or Mean If x 1 , x 2 , x 3 ,.......x n are n values of variate x i then their A.M. x is defined as x = n x ....... x x x n 3 2 1 n x n 1 i i ? ? If x 1 , x 2 , x 3 , .... x n are values of veriate with frequencies f 1 , f 2 , f 3 ,.........f n then their A.M. is given by x = n 3 2 1 n n 3 3 2 2 1 1 f ...... f f f f f ...... x f x f x f ? ? ? ? ? ? ? = N x f n 1 i i i ? ? , where N = ? ? n 1 i i f 2. Properties of Arithmetic Mean : (i) Sum of deviation of variate from their A.M. is always zero that is ? ? ? x x i ? = 0. (ii) Sum of square of deviation of variate from their A.M. is minimum that is ? ? ? 2 i x x ? is minimum (iii) If x is mean of variate x i then A.M. of (x i + ?) = x + ? A.M. of ? i . x i = ?. x A.M. of (ax i + b) = a x + b 3. Median The median of a series is values of middle term of series when the values are written is ascending order or descending order. Therefore median, divide on arranged series in two equal parts For ungrouped distribution : If n be number of variates in a series then Median = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? even is n when term 2 2 n and 2 n of Mean odd is n when , term 2 1 n th th th ? ? ? ? = Page 2 Page # 47 STATISTICS 1. Arithmetic Mean / or Mean If x 1 , x 2 , x 3 ,.......x n are n values of variate x i then their A.M. x is defined as x = n x ....... x x x n 3 2 1 n x n 1 i i ? ? If x 1 , x 2 , x 3 , .... x n are values of veriate with frequencies f 1 , f 2 , f 3 ,.........f n then their A.M. is given by x = n 3 2 1 n n 3 3 2 2 1 1 f ...... f f f f f ...... x f x f x f ? ? ? ? ? ? ? = N x f n 1 i i i ? ? , where N = ? ? n 1 i i f 2. Properties of Arithmetic Mean : (i) Sum of deviation of variate from their A.M. is always zero that is ? ? ? x x i ? = 0. (ii) Sum of square of deviation of variate from their A.M. is minimum that is ? ? ? 2 i x x ? is minimum (iii) If x is mean of variate x i then A.M. of (x i + ?) = x + ? A.M. of ? i . x i = ?. x A.M. of (ax i + b) = a x + b 3. Median The median of a series is values of middle term of series when the values are written is ascending order or descending order. Therefore median, divide on arranged series in two equal parts For ungrouped distribution : If n be number of variates in a series then Median = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? even is n when term 2 2 n and 2 n of Mean odd is n when , term 2 1 n th th th ? ? ? ? = Page # 48 4. Mode If a frequency distribution the mode is the value of that variate which have the maximum frequency. Mode for For ungrouped distribution : The value of variate which has maximum frequency. For ungrouped frequency distribution : The value of that variate which have maximum frequency. Relationship between mean, median and mode. (i) In symmetric distribution, mean = mode = median (ii) In skew (moderately asymmetrical) distribution, median divides mean and mode internally in 1 : 2 ratio. ? median = ? ? ? ? 3 Mode Mean 2 ? 5. Range values extreme of sum values extreme of difference = S L S L ? ? where L = largest value and S = smallest value 6. Mean deviation : Mean deviation = n  A x  n 1 i i ? ? ? Mean deviation = N  A x  f n 1 i i i ? ? ? (for frequency distribution) 7. Variance : Standard deviation = + iance var formula ? x 2 = ? ? n x x 2 i ? ? Page 3 Page # 47 STATISTICS 1. Arithmetic Mean / or Mean If x 1 , x 2 , x 3 ,.......x n are n values of variate x i then their A.M. x is defined as x = n x ....... x x x n 3 2 1 n x n 1 i i ? ? If x 1 , x 2 , x 3 , .... x n are values of veriate with frequencies f 1 , f 2 , f 3 ,.........f n then their A.M. is given by x = n 3 2 1 n n 3 3 2 2 1 1 f ...... f f f f f ...... x f x f x f ? ? ? ? ? ? ? = N x f n 1 i i i ? ? , where N = ? ? n 1 i i f 2. Properties of Arithmetic Mean : (i) Sum of deviation of variate from their A.M. is always zero that is ? ? ? x x i ? = 0. (ii) Sum of square of deviation of variate from their A.M. is minimum that is ? ? ? 2 i x x ? is minimum (iii) If x is mean of variate x i then A.M. of (x i + ?) = x + ? A.M. of ? i . x i = ?. x A.M. of (ax i + b) = a x + b 3. Median The median of a series is values of middle term of series when the values are written is ascending order or descending order. Therefore median, divide on arranged series in two equal parts For ungrouped distribution : If n be number of variates in a series then Median = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? even is n when term 2 2 n and 2 n of Mean odd is n when , term 2 1 n th th th ? ? ? ? = Page # 48 4. Mode If a frequency distribution the mode is the value of that variate which have the maximum frequency. Mode for For ungrouped distribution : The value of variate which has maximum frequency. For ungrouped frequency distribution : The value of that variate which have maximum frequency. Relationship between mean, median and mode. (i) In symmetric distribution, mean = mode = median (ii) In skew (moderately asymmetrical) distribution, median divides mean and mode internally in 1 : 2 ratio. ? median = ? ? ? ? 3 Mode Mean 2 ? 5. Range values extreme of sum values extreme of difference = S L S L ? ? where L = largest value and S = smallest value 6. Mean deviation : Mean deviation = n  A x  n 1 i i ? ? ? Mean deviation = N  A x  f n 1 i i i ? ? ? (for frequency distribution) 7. Variance : Standard deviation = + iance var formula ? x 2 = ? ? n x x 2 i ? ? Page # 49 ? x 2 = n x n 1 i 2 i ? ? – 2 n 1 i i n x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = n x n 1 i 2 i ? ? – ? ? 2 x ? d 2 = n d 2 i ? – 2 n di ? ? ? ? ? ? ? , where d i = x i – a , where a = assumed mean (ii) coefficient of S.D. = ? ? ? ? ? ? ? x coefficient of variation = ? ? ? ? ? ? ? x × 100 (in percentage) Properties of variance : (i) var(x i + ?) = var(x i ) (ii) var( ?.x i ) = ? 2 (var x i ) (iii) var(a x i + b) = a 2 (var x i ) where ?, a, b are constant.Read More
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