(6) Consider a statistical population consisting of N values Vi, ,UN, and let m\
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Question
(6) Consider a statistical population consisting of N values Vi, ,UN, and let m', denote the population mean value, variance and standard deviation. (a) Suppose that the vi values are coded to wi,... , wn, where w cvi, where ci is a known constant. Show that the mean value, variance and standard deviation of the statistical population wi,... ,wv are Hint: Use the Definition 12 from the lecture note Chapter 1 part 1 to calculate the variance of wi,..., wv, instead of computational formula. The same is true for parts (b) and (c).) (b) Suppose that the Vi values are coded to wi, , wy, where wi = au, where c2 is a known constant. Show that the mean value, variance and standard deviation of the statistical population wi,... , wv are (c) Suppose that the vi values are coded to wi,...,wv, where w,C C2vi, where ci and c2 are known constants. Show that the mean value, variance and standard deviation of the statistical population wi,.. . , wN areExplanation / Answer
a). any coded value is equal to value of the variable +C.
Now try to visualize the coded values . all the variables are C+ vi variable.
The mean of coded variables will hence be equal to C+mean of vi variables . Thisi s so because all the variables of coded variables is increased by C amount .Hence the mean will also increase by C.
The standerd deviation and variance are measures of difference between the variable and mean. Now in the coded variable series since all the variables are increased by C amount hence there difference of variable to mean will be equal to the difference between variable and mean of initial sample (vi).
Thus the standard deviation and variance of the coded variable sereis will be equal to the std deviation and variance of the actual initial variable series(vi). thisi s so because both are measures of difference which are same in both cases.
b). any coded value is equal to C2 times the value of the initial variable Vi.
Now try to visualize the coded values . all the variables are C*Vi variable.
The mean of coded variables will hence be equal to C2 times mean of Vi variables . This is so because all the variables of coded variables are multiplied by C2 amount .Hence the mean will also multiply by C2.
The standard deviation and variance are measures of difference between the variable and mean. Now in the coded variable series since all the variables are multiplied by C2 amount hence the difference of variable to mean in this case will be equal to the C2 times the difference between variable and mean of initial sample (Vi).
Thus the standard deviation and variance of the coded variable series will be equal to C2 times the std deviation and variance of the actual initial variable series(Vi). this is so because the measure of spread in coded series is C2 times the measures of difference of V1 series.
c).This is a situation in which each variable is added with a constant and multiplied by a given constant. This is what we have analysed in part a and b already.
This case is the addition of part a and part b cases. Hence it will show the results obtained by adding the effects noticed in part a nad part b of this question.
The mean will be equal to mean of initial mean multiplied by the constant added to another given constant.
The standard deviation will be equal to the saure of the constant multiplied to thew SD of initial series
and the variance will be equal to the product of variance of initial series and the constan t C2
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