Download Sampling distributions

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 SE ( 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 SE ( X )   / n  6 / 4  3 mm ].
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
SE ( X )   / n
• In general, “large enough” means more than 30
measurements, but it depends on how non-normal
population is to begin with.
The Shape of the Sampling Distribution
Population: Triangular5
2
X
4
1
3
0
0 .0
0.2
0.4
0.6
0 .8
1 .0
Distributi on of X ' s
3
2
2
2
4
1
nn == 10
11
1
0
0
0
0.0
.0
00.0
0.2
.2
00.2
.4
00
.4
0.4
.6
000.6
.6
.8
000.8
.8
.0
111.0
.0
The Shape of the Sampling
Distribution
Population: Uniform 32
2
4
X
3
1
3
2
2
0
1
2
0.0 0.2 0.4 0.6 0.8 1.0
1
1
1
nn == 4
2
110
0
0.0
Distributi on of X ' s
0.2
0.4
0.6
0.8
1.0
The Shape of the Sampling
Distribution
Population: Exponential
0.8
1.0
0.8
0.6
X
1.2
0.4
0.8
0.2
0.6
0.0
0
1
2
3
4
5
6
0.6
0.8
0.4
0.4
10
4 0.4
1
n=2
0.2
0.2
1.0
0.0
00
Distributi on of X ' s
1 1 12 12 3
23
4
25 4 3 6
The Shape of the Sampling
Distribution
Distributi on of X ' s
Population: U-shaped
3
X
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
n=2
10
4
1
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
The Shape of the Sampling
Distribution
__
(Central Limit Theorem)
X is approximately
Normally distributed for large samples.
(i.e. when the sample size n is large)
Big Deal?
Let’s look at some useful applications...