Standard Error of the Mean

Central Limit Theorem

Simple Random Sample

Sampling distribution of meanIF:• 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.

What is the mean of all of the sample means ?

What is the standard deviation of the sample means ?

Mean and Standard Deviation of

mean =

standard deviation =

X

xn

x

If the population is finite of size N then

nx

N nN 1

is also called standard error of the mean

x

finite population correction factor

used if

n

N0.05

Averages are less variable than individual observations

Example: mean wagesThe mean wage per hour for all 5000 employees is $ 17.50

and the standard deviation is $ 2.90.

Let be the mean wage per hour for a random sample of 30

employees.

What is the mean and standard deviation of ?

x

N = 5000 , = $ 17.50, = $ 2.90

xThe mean of the sampling distribution is = $ 17.50

x

n = 30, N = 5000, so n/N = 30/5000 = 0.006 < 5% so we use

nx

2.90

30 $ 0.529

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 /√ n.

• In general, “large enough” means more than 30 measurements.

Distribution of when sampling from a normal distribution

has a normal distribution with

mean =

and

standard deviation (standard error) =

X

x

nx

X

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 6/ = 6/2].

4

Using empirical rule...

• 68% of samples of n=4 adults will have an average nose length between 42 and 48 mm.

• 95% of samples of n=4 adults will have an average nose length between 39 and 51 mm.

• 99% of samples of n=4 adults will have an average 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 / = 6/6].

36

Again, using empirical rule...

• 68% of samples of n=36 adults will have an average nose length between 44 and 46 mm.

• 95% of samples of n=36 adults will have an average nose length between 43 and 47 mm.

• 99% of samples of n=36 adults will have an average nose length between 42 and 48 mm.

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

What happens if data are not normally distributed ?

Let’s investigate that, too …

distribution of original population

distribution of x for 2 observationsn=2

distribution of x for 10 observations n=10

distribution of x for 25 observations n=25

Central Limit Theorem

If the sample size (n) is large enough, has a normal distribution with

mean =

and

standard deviation =

regardless of the population distribution

(normal or not ! )

X x

nx

30n

Sampling Distribution of a Sample Statistic

• Sampling Distribution of a Sample Statistic: The distribution of values for a sample statistic obtained from repeated samples, all of the same size and all drawn from the same population

1) Make a list of all samples of size 2 that can be drawn from this set (Sample with replacement)

2) Construct the sampling distribution for the sample mean for samples of size 2

3) Construct the sampling distribution for the minimum for samples of size 2

Example: Consider the set {1, 2, 3, 4}:

{1, 1} 1.0 1 1/16{1, 2} 1.5 1 1/16{1, 3} 2.0 1 1/16{1, 4} 2.5 1 1/16{2, 1} 1.5 1 1/16{2, 2} 2.0 2 1/16{2, 3} 2.5 2 1/16{2, 4} 3.0 2 1/16{3, 1} 2.0 1 1/16{3, 2} 2.5 2 1/16{3, 3} 3.0 3 1/16{3, 4} 3.5 3 1/16{4, 1} 2.5 1 1/16{4, 2} 3.0 2 1/16{4, 3} 3.5 3 1/16{4, 4} 4.0 4 1/16

Sample x Minimum Probability

This table lists all possible samples of size 2, the mean for each sample, the minimum for each sample, and the probability of each sample occurring (all equally likely)

Table of All Possible Samples

1.0 1/161.5 2/162.0 3/162.5 4/163.0 3/163.5 2/164.0 1/16

Sampling Distributionof the Sample Mean

x P x( )

1.0 1.5 2.0 2.5 3.0 3.5 4.00.00

0.05

0.10

0.15

0.20

0.25

x

P x( )

Histogram: Sampling Distributionof the Sample Mean

Sampling Distribution

• Summarize the information in the previous table to obtain the sampling distribution of the sample mean and the sample minimum: