I have come across the following question, which requires me to compute different Z-score value of 5 numbers, ie
110, 120, 130, 160, 170
and their respective Z-score are:
-1.08, -0.69, -0.31, 0.85, 1.24
I would like to know what are the meaning of
1) The mean of the above 5 z-scores and
2) SD of the above 5 z-scores
Also, my professor taught me that the mean and the SD of of the above 5 z-scores can be obtained without calculation. Can it be? and Why can it be?
Thanks
The mean and the SD of the Z-scores?
2010-06-10 11:31 am
回答 (2)
2010-06-10 2:36 pm
Also, my professor taught me that the mean and the SD of of the above 5 z-scores can be obtained without calculation. Can it be? and Why can it be?
Because Z ∼ n(0, 1)
x1 = 110, x2 = 120, x3 = 130, x4 = 160, x5 = 170
Xmean = 138
S^2 = (110^2 + 120^2 + 130^2 + 160^2 + 170^2 - 5*138^2)/4 = 670
z1 = (110 - 138)/sqrt(670.) = -1.081734373
z2 = (120 - 138)/sqrt(670.) = - 0.6954006685
z3 = (130 - 138)/sqrt(670.) = - 0.3090669638
z4 = (160 - 138)/sqrt(670.) = 0.8499341503
z3 = (170 - 138)/sqrt(670.) = 1.236267855
E(Z) = ( - 1.081734373 - 0.6954006685 - 0.3090669638 + 0.8499341503 + 1.236267855)/5
E(Z) = - 2.000000000*10^(-10) = 0
V(Z) = (( - 1.081734373)^2 + ( - 0.6954006685)^2 + (- 0.3090669638)^2 + (0.8499341503)^2 +(1.236267855)^2 - 5*(- 2.000000000*10^(-10))^2)/4 = 1
Because Z ∼ n(0, 1)
x1 = 110, x2 = 120, x3 = 130, x4 = 160, x5 = 170
Xmean = 138
S^2 = (110^2 + 120^2 + 130^2 + 160^2 + 170^2 - 5*138^2)/4 = 670
z1 = (110 - 138)/sqrt(670.) = -1.081734373
z2 = (120 - 138)/sqrt(670.) = - 0.6954006685
z3 = (130 - 138)/sqrt(670.) = - 0.3090669638
z4 = (160 - 138)/sqrt(670.) = 0.8499341503
z3 = (170 - 138)/sqrt(670.) = 1.236267855
E(Z) = ( - 1.081734373 - 0.6954006685 - 0.3090669638 + 0.8499341503 + 1.236267855)/5
E(Z) = - 2.000000000*10^(-10) = 0
V(Z) = (( - 1.081734373)^2 + ( - 0.6954006685)^2 + (- 0.3090669638)^2 + (0.8499341503)^2 +(1.236267855)^2 - 5*(- 2.000000000*10^(-10))^2)/4 = 1
2010-06-10 11:39 am
he probably meant that it shouldn't need a calculation as the arithmetic works out as quite neat: the sum of all the means is 0.01.
Also the standard deviation is a simple arithmetic minus and subtract of the mean from the extremes.
Hope this helps.
Also the standard deviation is a simple arithmetic minus and subtract of the mean from the extremes.
Hope this helps.
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