
タイトル：
Grand Mean  Intro to Inferential Statistics

概説：

The answer is sometimes. When sample sizes are equal, in other words, there

could be five values in each sample, or n values in each sample. The grand mean

is the same as the mean of sample means. Since there's an equal weight to each

sample in calculating the grand mean. However, when sample sizes are unequal.

We have to find the grand mean by adding all the values from each sample and

dividing by the total number of values, the sum of the sample sizes. When

sample sizes are equal. The grand mean is the same as the mean of sample means.

Since there is an equal weight to each sample. And this is the same as the

means of all values. Where capital N is the total number of values in all the

samples. But when the sample sizes are unequal. We can't use the mean of sample

means. We have to add all the values in all the samples and divide by the total

number, the sum of the sample sizes. But in this lesson we'll only work with

samples of the same size, so it's fine to use the mean of means. Here's a quick

explanation in symbols, if you're interested. Let's say we have three samples.

X, Y, and Z. The mean of X is X bar. The sum of each value, divided by the

number in that sample. The mean of Y is the sum of all the values in Y, divided

by the number in Y. And likewise for sample Z. We want to know if the mean of

means. Equals the total mean, the sum of all the values in x, y, and z, divided

by the total number in each sample. Well, we know that because the mean of x is

the sum of the x values divided by the total number. That the sum of x is just

x bar times the number. So we can replace that in each of these. The sum of the

x i's equals the average times the number in x. The sum of the y values, is the

average of y times the number, et cetera. Well these are not the same if the

number in each sample is different. But if there is the same number in each

sample, let's just call it n. Then we can rewrite this as n times the sum of

the sample means divided by 3n. The ends cancel out and then you get the same

thing. So, that's why if the sample sizes are the same, which they will be

throughout this lesson, we can just use the mean of means. But, just remember,

for later in lesson 13, we'll work with different sample sizes. So to calculate

our grand mean we'll have to add all the values and then divide by the total

number of values.