Is the mean of equal length subsets always equal to the mean of the set?

Question

I've been fooling around with some numbers while learning R and would like to know if the following is generalizable.

When I calculated the mean of numbers 1 through 100, I got this:

> mean(1:100)
[1] 50.5

Then when I calculated the mean of equal length subsets, I got the same:

> mean25 <- mean(1:25)
> mean50 <- mean(26:50)
> mean75 <- mean(51:75)
> mean100 <- mean(76:100)
> mean(c(mean25, mean50, mean75, mean100))
[1] 50.5

As an aside, when calculating the mean of different length subsets, the mean is expectedly not the same:

> mean5 <- mean(1:5)
> mean37 <- mean(6:37)
> mean91 <- mean(38:91)
> mean100 <- mean(92:100)
> mean(c(mean5, mean37, mean91, mean100))
[1] 46.25

My question:

Would it be accurate to say the mean of equal length subsets is always equal to the mean of the set? Are there any counterexamples to this?

Thanks in advance.

Answer 1

Would it be accurate to say the mean of equal length subsets is always equal to the mean of the set?

It's always true!

Consider $n=mk$ observations, where you take $k$ mutually-exclusive groups of size $m$.

Label the observations in group $i$ as $x_{ij},\: j=1,2,...,m$.

The individual means are $\bar{x}_i = \frac{1}{m}\sum_{j=1}^m x_{ij}$.

The mean-of-means is

$\overline{\bar{x}_i} = \frac{1}{k}\sum_{i=1}^k \bar{x}_{i}$

$\hspace{.5cm}=\frac{1}{k}\sum_{i=1}^k (\frac{1}{m}\sum_{j=1}^m x_{ij})$

$\hspace{.5cm}=\frac{1}{km}\sum_{i=1}^k \sum_{j=1}^m x_{ij}$

$\hspace{.5cm}=\frac{1}{n}\sum_{i,j} x_{ij}$

which is just the overall mean of the data.

For the unequal-length case, they're also equal (that is, the mean of means equals the overall mean) if you do an appropriately weighted average when taking the mean of means.