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Question
Find the variance of the following data: 6,8,10,12,14,16,18,20,22,24
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Solution
Step 1: Finding Step-Deviation
Given data: 6,8,10,12,14,16,18,20,22,24
Let,
a = assumed mean be = 14
h= common factor = 2
di = step deviation
n = number of observations = 10
di=xi−ah=6−142=−82=−4
xidi=xi−ah66−142=−488−142=−31010−142=−21212−142=−11414−142=01616−142=11818−142=22020−142=32222−142=42424−142=5∑101di=5
Step 2 : Finding mean
Mean (¯¯¯x) = assumed mean +∑din×h
Mean (¯¯¯x)=14+510×2=15
Step 3: Finding variance and standard deviation
For xi=6,¯¯¯x=15
xi−¯¯¯x=6−15=−9
(xi−¯¯¯x)2=(−9)2=81
xixi−¯¯¯x(xi−¯¯¯x)266−15=−9(−9)2=8188−15=−7(−7)2=491010−15=−5(−5)2=251212−15=−3(−3)2=91414−15=−1(−1)2=11616−15=1(1)2=11818−15=3(3)2=92020−15=5(5)2=252222−15=7(7)2=492424−15=9(9)2=81∑101(xi−¯¯¯x)2=330
Variance (σ2)=1n∑(xi−¯¯¯x)2
=110×330=33
Given data: 6,8,10,12,14,16,18,20,22,24
Let,
a = assumed mean be = 14
h= common factor = 2
di = step deviation
n = number of observations = 10
di=xi−ah=6−142=−82=−4
xidi=xi−ah66−142=−488−142=−31010−142=−21212−142=−11414−142=01616−142=11818−142=22020−142=32222−142=42424−142=5∑101di=5
Step 2 : Finding mean
Mean (¯¯¯x) = assumed mean +∑din×h
Mean (¯¯¯x)=14+510×2=15
Step 3: Finding variance and standard deviation
For xi=6,¯¯¯x=15
xi−¯¯¯x=6−15=−9
(xi−¯¯¯x)2=(−9)2=81
xixi−¯¯¯x(xi−¯¯¯x)266−15=−9(−9)2=8188−15=−7(−7)2=491010−15=−5(−5)2=251212−15=−3(−3)2=91414−15=−1(−1)2=11616−15=1(1)2=11818−15=3(3)2=92020−15=5(5)2=252222−15=7(7)2=492424−15=9(9)2=81∑101(xi−¯¯¯x)2=330
Variance (σ2)=1n∑(xi−¯¯¯x)2
=110×330=33
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