1. Sampling Distribution of Means: Basic Theorems
is an unbiased estimate of
Consider N samples each consisting of 2 observations sampled at random
from a single population with mean and variance is the i-th
observation in the j-th sample. T.j is the sum (total) of the 2 observations in
the j-th sample.
Observation T
X X T.1
X X T.2
j X1j X2j T.j
N X1N X2N T.N
Sample
Or, more generally, for samples of n observations:
As N approaches infinity the distribution of the i-th observation over
repeated random samples approaches the distribution of the population
from which the i-th observation was drawn. If all Xiare sampled from the
same population, then is a constant for all i,
Q.E.D.

2. 2. The variance of the sampling distribution of means for random and
independent samples of size n is given by the variance of the population
from which the samples were divided by the sample size:
Note that
For samples of size 2:
If X1j is independent of , then and
Or more generally, if all Xij are independent, for random samples of n
observations:
As N approaches infinity the distribution of the i-th observation over repeated
random samples approaches the distribution of the population from which the
i-th observations was drawn. If all Xij are sampled from the same population,
then is a constant for all i,

3. Observation 1 2 ….. i …… n ……….M 1 X11 X21 Xi1 Xn1 M.1
is an unbiased estimate of
Consider the following data matrix.
Note that
To obtain the mean squared deviation of each observation in the data matrix
about the grand mean of all observations, we may proceed as follows:
As N approaches infinity the distribution of all N observations in the data
matrix approaches the distribution of the population from which each
was sampled.
Observation
….. i …… n ……….M
X X Xi Xn M.1
X X Xi Xn2 M.2
j X1j X2j Xij Xnj M.j
N X1N X2N XiN XnN M.N
Sample

4. 3. continued 4. From Theorems 2 and 3it follows that:
Is an unbiased estimate of
(with n-1 degrees of freedom).
Note that: