Question 820:

Asked on October 17, 2009

Tags: Standard Deviation , F-test , Standard Deviation on Frequency Data , Standard Deviation of Grouped Data

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Answer:

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1. Q820_1_q820_sd_frequency.xls

To find the standard deviation and variance of frequency data we first need to find the mid-point for each category. To do so we just find the median (mid-point) so for example the class of 2.3-2.9 becomes 2.6.  I've attached a spreadsheet which does all the calculations.

1. Multiply the category-mid point times the frequency, we'll call this one x*f.
2. Square the category-mid point and multiply this times the frequency. We'll cal this one x^2*f
3. We do this for all values for both sets of data.
4. Next add up all the frequency data to get the sample size, we'll call this Sf
5. Then add-up all the x*f and x^2*f values, we'll call this Sx^2*f and Sx*f.
   
6. Find the sum of squares values which is [ (Sx^2*f) -(Sx*f)^2 ] /Sf. You should get 173.95 and 115.39 for the two sets of data.
   
7. Finally divide the sum of squares values for each data-set by (Sf-1) and you should get 1.2518 and 1.2162, these are the standard deviations.
   
8. The variance is just the square of the standard deviation, so squaring those values gets us variances of 1.567 and 1.479 respectively.

Now that we have the variances, we need to see if they are statistically different by comparing their ratios using a F-test.The ratio of the variances are 1.567/1.479 = . To determine the significance of this ratio we look it up in an F-table or using an excel function (see the excel file). We use the degrees of freedom in the numerator and denominator as the sample size -1. This gets us the DF of 111 and 78 and a two tailed p-value of .77 which is well above .05.

We conclude there is not enough evidence to conclude there is a different in the variances between the groups.