Question:

IQ-scores are normally distributed with a mean of 100 points and a standard deviation of 15 points. A teacher is in charge of collecting 12 samples of size 20 in her Grade One schools.

If the IQ-scores are from a general population what do we expect of the 12 means?

Samples of size n are drawn from a population.

1) If samples of size n where n > 30 are drawn from any population with a mean of µ and a standard deviation of , then the sampling distribution of sample means approximates a normal distribution.

2) If the population itself is normally distributed the sampling distribution of sample means is normally distributed for any sample size n.

Mean of sample means

Stand. deviation of sample means

Try to place some real numbers into the CLT above :

Next we rather look at the 300 trials as a series of 10 samples (each sample has 30 trials)

With the 10 samples (n=30) we get the following means:

This is called a sampling distribution of means. (Notice that there is less variation than in individual trials.)

Mean of sampling distribution of means is still 2.580 (As expected.)
Standard deviation of this distribution is a small 0.298

For Means of samples of size n=30 we expect means to be between 2.000 and 3.000 but you can see it would be difficult to get a "smaller" mean or a "larger" mean.

Moral of the Story

The mean of a sample is a straight-forward calculation as we saw in Chapter 2. The symbols we used were X₁, X₂, X₃, X₄, ... X_n for the n different data points.

Mean =

With different means we must use repeatedly

For 10 means we have :

There is a mean of these means.
There is a standard deviation of these means.

 

This Chapter features sampling distributions.
We need "many samples" to visualize our needs:

In a previous assignment we have used the Random Number Generator function in Microsoft Excel to produce near-instant experiments. We applied the spreadsheet to simulate the flipping of a coin five times per trial when we used 300 trials.
Typical trial outcomes are : 3 Heads out of 5 flips ; or 4 heads out of 5 flips.

Display of our 300 trials :

The mean was 2.58 Heads per 5 flips. (Makes common sense)
The standard deviation was 1.11 Heads.

Obviously there is a good deal of variation in the 300 trials and we do expect to get 1 Heads out of 5 flips some time as well as 4 Heads out of 5 flips a few times.