Khan Public Lessons-Statistics study notes: (11) Sum of squares, F statistics

The sum of squares and the sum of degrees of freedom

This part is actually deduced through the χ2, but the concrete mathematical proof is not speaking, can be explained by the visual case. There is a 3 (m) x3 (n) array.

A total of 9 samples, the sample mean value is 4, is also the mean value of each group, that is, mean of means. For sum of Square there is: total sum of square = group Sum of square + inter-group square, or overall fluctuation = intra-group fluctuation + component fluctuation . We perform visual verification

SST = (3-4) A (2-4) a (1-4) a (5-4) a (3-4) a (4-4) (5-4) (6-4) (7-4) 2 = 30, 9 number, know mean, from which 8 can be introduced another, so freedom is mn-1=3*3-1=8.

Ssw= (3-2), (2-2), (1-2), (5-4) (3-4) (4-4) (5-6) 6-6) (7-6) (2) (+)-= 6, in each group: Know the group's mean, each group of degrees of freedom is n-1, total m (n-1), in this case 3 (3-1) =6

Ssb= (2-4), (2-4), (2-4), (4-4), (4-4), (4-4) (6-4) and (6-4) (6-4) 2 = 24, knowing the total mean, the most group of mean can be pushed to the processing, so the freedom is m-1=3-1=2

There are SST=SSW+SSB, and total degrees of freedom = Group Freedom + inter-group degrees of freedom.

example: F-Statistic

3 groups ate different foods, with 3 measurements sampled, asking whether there were any effects on different foods. Or the sampled values are normal fluctuations.

H0:food does not make a difference,→μ1=μ2=μ3;

H1:it does

Assume H0 is true, will be used to f-statistic, f statistics, subject to F distribution,f to denote Fisher, biologist and statistician, F statistics is actually a comparison of two χ2 distributions

A higher value indicates that the fluctuations between groups are greater than the fluctuations within the group, and there is a high likelihood of differences between groups.

In this example f-statistics= (24/2)/(6/6) =12

For α=10%, you can http://www.socr.ucla.edu/applets.dir/f_table.html check, ("Note" to check the different α F distribution table, do not go to Baidu, do not find out, Google will be able to check the F Table forα= 0.10, check df1=2,df2=6, find critical f value=3.46,fc=3.46, and this example of the F value is much larger than this, all the probability of such sampling is far below 10%, can be rejected H0.

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