Question 477232: Find the range, standard deviation, and variance for the following sample data:
33, 55, 29, 40, 43, 8, 90, 61, 41, 17, 80, 56, 17, 59, 21, 78
Thank you
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Range = Largest Value - Smallest Value
Range = 90 - 8
Range = 82
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Finding the sample variance and sample standard deviation:
First, we need to find the sample mean xbar
xbar = (33+55+29+40+43+8+90+61+41+17+80+56+17+59+21+78)/16 = 728/16 = 45.5
So the sample mean is xbar = 45.5
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Now subtract the sample mean from EVERY data value:
33-45.5 = -12.5
55-45.5 = 9.5
29-45.5 = -16.5
40-45.5 = -5.5
43-45.5 = -2.5
8-45.5 = -37.5
90-45.5 = 44.5
61-45.5 = 15.5
41-45.5 = -4.5
17-45.5 = -28.5
80-45.5 = 34.5
56-45.5 = 10.5
17-45.5 = -28.5
59-45.5 = 13.5
21-45.5 = -24.5
78-45.5 = 32.5
So the differences are: -12.5, 9.5, -16.5, -5.5, -2.5, -37.5, 44.5, 15.5, -4.5, -28.5, 34.5, 10.5, -28.5, 13.5, -24.5, 32.5
Now square each difference:
(-12.5)^2 = 156.25
(9.5)^2 = 90.25
(-16.5)^2 = 272.25
(-5.5)^2 = 30.25
(-2.5)^2 = 6.25
(-37.5)^2 = 1406.25
(44.5)^2 = 1980.25
(15.5)^2 = 240.25
(-4.5)^2 = 20.25
(-28.5)^2 = 812.25
(34.5)^2 = 1190.25
(10.5)^2 = 110.25
(-28.5)^2 = 812.25
(13.5)^2 = 182.25
(-24.5)^2 = 600.25
(32.5)^2 = 1056.25
Now add up each square:
156.25+90.25+272.25+30.25+6.25+1406.25+1980.25+240.25+20.25+812.25+1190.25+110.25+812.25+182.25+600.25+1056.25 = 8966
Now divide that sum by n-1 = 16-1 = 15 to get 8966/15 = 597.733333333333
So the sample variance is 597.733333333333
Finally, take the square root of 597.733333333333 to get 24.448585507823
So the sample standard deviation is 24.448585507823
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