Sampling Distribution of the Sample Mean

Example

• Take random sample of 1 hour periods in an ER.

• Ask “how many patients arrived in that one hour period ?”

• Calculate statistic, say, the sample mean.Sample 1: 2 3 1 Mean = 2.0

Sample 2: 3 4 2 Mean = 3.0

Situation• Different samples produce different results.

• Value of a statistic, like mean, SD, or proportion, depends on the particular sample obtained.

• But some values may be more likely than others.

• The probability distribution of a statistic (“sampling distribution”) indicates the likelihood of getting certain values.

Let’s investigate how sample means and standard deviations vary….

(click here for Live Demo)

Web link to try it yourself:http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/

Sampling Distribution of Sample Mean

IF:

• data are normally distributed with mean and standard deviation , and

• random samples of size n are taken, THEN:

The sampling distribution of the sample means is also normally distributed.

The mean of all of the possible sample means is .

The standard deviation of the sample means (“standard error of the mean”) is

nXSE )(

Example

• Adult nose length is normally distributed with mean 45 mm and standard deviation 6 mm.

• Take random samples of n = 4 adults.• Then, sample means are normally distributed with

mean 45 mm and standard error 3 mm [from ].mm 34/6/)( nXSE

Using empirical rule...

• 68% of samples of n=4 adults will have a sample mean nose length between 42 and 48 mm.

• 95% of samples of n=4 adults will have a sample mean nose length between 39 and 51 mm.

• 99% of samples of n=4 adults will have a sample mean nose length between 36 and 54 mm.

What happens if we take larger samples?

• Adult nose length is normally distributed with mean 45 mm and standard deviation 6 mm.

• Take random samples of n = 36 adults.

• Then, sample means are normally distributed with mean 45 mm and standard error 1 mm [from 6/sqrt(36) = 6/6].

Again, using empirical rule...• 68% of samples of n=36 adults will have a sample

mean nose length between 44 and 46 mm.

• 95% of samples of n=36 adults will have a sample mean nose length between 43 and 47 mm.

• 99% of samples of n=36 adults will have a sample mean nose length between 42 and 48 mm.

• So … the larger the sample, the less the sample means vary.

What happens if data are not normally distributed?

Let’s investigate that, too …

Sampling Distribution Demo: (Live Demo)

http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/

Central Limit Theorem• Even if data are not normally distributed, as long as

you take “large enough” samples, the sample averages will at least be approximately normally distributed.

• Mean of sample averages is still • Standard error of sample averages is still

• In general, “large enough” means more than 30 measurements, but it depends on how non-normal population is to begin with.

nXSE /)(

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

n = 2

0.0 0.2 0.4 0.6 0.8 1.00

1

2

Population: Triangular

0 .0 0 .2 0 .4 0.6 0 .8 1 .0

0

1

2

3

4

n = 4

The Shape of the Sampling Distribution

X

0.0 0.2 0.4 0.6 0.8 1.00

1

2

n = 1

0 .0 0 .2 0 .4 0 .6 0 .8 1 .00

1

2

3

4

5

n = 10

sX ' ofon Distributi

0.0 0.2 0.4 0.6 0.8 1.00

1

2

n = 100.0 0.2 0.4 0.6 0.8 1.0

1

2

3

n = 2

The Shape of the Sampling Distribution

Population: Uniform

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

n = 4

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

n = 10

0.0 0.2 0.4 0.6 0.8 1.00

1

2 XsX ' ofon Distributi

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

n = 4

0 1 20.0

0.4

0.8

1.2

n = 10

The Shape of the Sampling Distribution

Population: Exponential

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

n = 1

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

X

0 1 2 3 40.0

0.2

0.4

0.6

0.8

n = 2

sX ' ofon Distributi

The Shape of the Sampling Distribution

Population: U-shaped

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

n = 1

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

n = 2

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

n = 4

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

n = 10

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3 X

sX ' ofon Distributi

__ (Central Limit Theorem)

X is approximately Normally distributed for large samples.

(i.e. when the sample size n is large)

The Shape of the Sampling Distribution

Big Deal?

Let’s look at some useful applications...