Question 1068455: Hello, Given the class boundaries and frequencies I must find the variance and standard deviation.
Class Boundaries
52.5 – 63.5
63.5 – 74.5
74.5 – 85.5
85.5 – 96.5
96.5 – 107.5
I found the class limits as follows:
Class Limits
53 – 63
64 – 74
75 – 85
86 – 96
97 – 107
Frequencies given (f):
10
13
28
15
14
N = 80
Found the midpoints (x):
(63 + 53)/2 = 58
(74 + 64)/2 = 69
(85 + 75)/2 = 80
(96 + 86)/2 = 91
(107 + 97)/2 = 102
n= 400
Found the (f)(x):
(10)(58) = 580
(13)(69) = 897
(28)(80) = 2240
(15)(91) = 1365
(14)(102) = 1428
n = 6510
I need to find the variance s^2 and once I find this I can take its square root and get the standard deviation.
I believe I need to find the mean which would be:
6510 is the sum of your (f)(x) or (frequencies) (midpoints)
80 is the sum of your (x) or frequencies
The MEAN = 6510/80 = 81.38
I know that the variance is s^2 but I am confused on how to find this from the information I found above. I am not sure I got the above table filled in correctly.
Your help is greatly appreciated
Ann
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! there are two ways to get the variance.
the formulas they give you are:
s^2 = (x-m)^2/n
s^2 = sum(x^2)/n - m^2
s^2 = variance
x = data
m = mean
n = number of occurrences
here's a reference.
http://www.sciencebuddies.org/science-fair-projects/project_data_analysis_variance_std_deviation.shtml
the second formula is actually derived from the first and is a simpler way of calculating the variance.
the variance will be the same either way.
if you're dealing with just data (not frequency * data), it's fairly straight forward.
when you're dealing with frequency * data, it becomes a little bit more complicated and definitely more confusing.
when you're dealing with just data (not frequency * data), the formulas are:
m = sum(x)/n
s^2 = sum(x-m)^2/n
the alternate formula is:
s^2 = sum(x^2)/n - m^2
when you're dealing with frequency * data, as in your data set, the formulas become:
m = sum(f*x) / sum(f)
s^2 = sum (f*(x-m)^2)/sum(f)
the alternate formula becomes:
s^2 = sum (f*x^2)/sum(f) - m^2 ***** see note 1 immediately following.
***** note 1.
the subtraction of m^2 was missing in the formula presented earlier.
the formula is correct now.
the picture below has also been corrected.
the actual formula used was correct.
it was, unfortunately, not displayed correctly before.
it is now.
the following picture of the excel spreadsheet i used to do the calculations is shown below.
your variance, based on these calculations, should be 185.6394 rounded to the nearest decimal place.
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