## ← Grand Mean - Intro to Inferential Statistics

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Showing Revision 5 created 05/25/2016 by Udacity Robot.

1. The answer is sometimes. When sample sizes are equal, in other words, there
2. could be five values in each sample, or n values in each sample. The grand mean
3. is the same as the mean of sample means. Since there's an equal weight to each
4. sample in calculating the grand mean. However, when sample sizes are unequal.
5. We have to find the grand mean by adding all the values from each sample and
6. dividing by the total number of values, the sum of the sample sizes. When
7. sample sizes are equal. The grand mean is the same as the mean of sample means.
8. Since there is an equal weight to each sample. And this is the same as the
9. means of all values. Where capital N is the total number of values in all the
10. samples. But when the sample sizes are unequal. We can't use the mean of sample
11. means. We have to add all the values in all the samples and divide by the total
12. number, the sum of the sample sizes. But in this lesson we'll only work with
13. samples of the same size, so it's fine to use the mean of means. Here's a quick
14. explanation in symbols, if you're interested. Let's say we have three samples.
15. X, Y, and Z. The mean of X is X bar. The sum of each value, divided by the
16. number in that sample. The mean of Y is the sum of all the values in Y, divided
17. by the number in Y. And likewise for sample Z. We want to know if the mean of
18. means. Equals the total mean, the sum of all the values in x, y, and z, divided
19. by the total number in each sample. Well, we know that because the mean of x is
20. the sum of the x values divided by the total number. That the sum of x is just
21. x bar times the number. So we can replace that in each of these. The sum of the
22. x i's equals the average times the number in x. The sum of the y values, is the
23. average of y times the number, et cetera. Well these are not the same if the
24. number in each sample is different. But if there is the same number in each
25. sample, let's just call it n. Then we can rewrite this as n times the sum of
26. the sample means divided by 3n. The ends cancel out and then you get the same
27. thing. So, that's why if the sample sizes are the same, which they will be
28. throughout this lesson, we can just use the mean of means. But, just remember,
29. for later in lesson 13, we'll work with different sample sizes. So to calculate
30. our grand mean we'll have to add all the values and then divide by the total
31. number of values.