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# Standard Error of the Mean

## Central Limit Theorem

Simple Random Sample
Sampling distribution of mean
IF:
• data are normally distributed with mean µ and standard
deviation σ, 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 X
mean = µ x = µ
σ
standard deviation = σ x =
n finite population
correction factor
If the population is finite of size N then
used if
σ N −n n
σx = ≥0.05
n N −1 N

## is also called standard error of the mean

Averages are less variable
than individual observations
Example: mean wages
The mean wage per hour for all 5000 employees is \$ 17.50
and the standard deviation is \$ 2.90.
Let x be the mean wage per hour for a random sample of 30
employees.
What is the mean and standard deviation of x ?
N = 5000 , µ = \$ 17.50, σ = \$ 2.90
The mean of the sampling distribution is µ x = µ = \$ 17.50
n = 30, N = 5000, so n/N = 30/5000 = 0.006 < 5% so we use
σ 2.90
σx = = = \$ 0.529
n 30
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 µ
• Standard error of sample averages is still
∀ σ /√ n.
• In general, “large enough” means more than 30
measurements.
Distribution of X when sampling
from a normal distribution

## X has a normal distribution with

mean = µx = µ
and
σ
standard deviation (standard error) = σ x =
n
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 / 36 = 6/6].
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 distribution of x
original for 2 observations
population n=2

distribution of x distribution of x
for 10 observations for 25 observations
n=10 n=25
Central Limit Theorem

## If the sample size (n) is large enough,

X has a normal distribution with
mean = µ x = µ
and
σ
standard deviation = σ x =
n

## regardless of the population distribution

(normal or not ! )
n ≥ 30
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
Example: Consider the set {1, 2, 3, 4}:
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
Table of All Possible Samples

## Sample x Minimum Probability

{1, 1} 1.0 1 1/16
This table lists all {1, 2} 1.5 1 1/16
possible samples of {1, 3} 2.0 1 1/16
{1, 4} 2.5 1 1/16
size 2, the mean for {2, 1} 1.5 1 1/16
each sample, the {2, 2} 2.0 2 1/16
minimum for each {2, 3} 2.5 2 1/16
{2, 4} 3.0 2 1/16
sample, and the {3, 1} 2.0 1 1/16
probability of each {3, 2} 2.5 2 1/16
sample occurring {3, 3} 3.0 3 1/16
{3, 4} 3.5 3 1/16
(all equally likely) {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
Sampling Distribution
• Summarize the information in the previous table to obtain the sampling
distribution of the sample mean and the sample minimum:

## Sampling Distribution Histogram: Sampling Distribution

of the Sample Mean of the Sample Mean

x P( x ) P( x )
1.0 1/16 0.25

2.5 4/16
3.0 3/16 0.10

## 4.0 1/16 0.00

1.0 1.5 2.0 2.5 3.0 3.5 4.0 x
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