Chapter 8 & (part) Chapter 12: Distribution of Sample
Means
We’ve been examining Z-scores & the probability of obtaining
individual scores within a normal distribution
But inferential statistics involve samples of more than 1
To transition into inferential statistics, it is important that we understand
how probability relates to sample means, not just individual scores
Inferential statistics: sample  infer something about population
Often not possible to measure everyone in a population
Samples are convenient representations of them
Chapters 8 & 12: Page 1
If you take multiple samples of the same size from a population, they are
likely to give different results
Samples vary!
Quite likely that a particular sample won’t reflect the population exactly
Discrepancy b/n sample & population = sampling error
The term “sampling error” does not mean a sampling mistake – rather it
indicates that means drawn from multiple samples taken from a
population will vary from each other due to random chance and
therefore may deviate from the population mean
Chapters 8 & 12: Page 2
What is a distribution of Sample Means (Sampling Distribution of
the Mean)?
A distribution of sample means ( X ); a “distribution of a statistic [in this case
a sample mean] over repeated sampling from a specified population”
Based on all possible random samples of size n, from a population
Can inform us of the degree of sample-to-sample variability we should expect
due to chance
Chapters 8 & 12: Page 3
Suppose we have a population:
6
7
8
9
 = 7.5
Let’s take all possible samples of size n = 2 from this population:
1.
2.
3.
4.
5.
6.
6
6
6
6
7
7
6
7
8
9
6
7
X=
X=
X=
X=
X=
X=
6.0
6.5
7.0
7.5
6.5
7.0
7.
8.
9.
10.
11.
12.
7
7
8
8
8
8
8
9
6
7
8
9
X=
X=
X=
X=
X=
X=
7.5
8.0
7.0
7.5
8.0
8.5
13. 9
14. 9
6
7
15. 9
8
16. 9
9
X=
X=
X=
X=
7.5
8.0
8.5
9.0
What do you notice?
X is rarely exactly 
But, most X are close to (or cluster around) 
Extreme values of X are rare
You can determine the exact probability of obtaining a particular X
p( X < 7)? = 3 / 16
Chapters 8 & 12: Page 4
Important properties of the sampling distribution of means:
1. Mean
2. Standard Deviation
3. Shape
1. The Mean
The mean of the distribution of sample means is the mean of the population
The mean of the distribution of sample means is called expected value of X
X is an unbiased estimate of : on average, the sample mean produces a
value that exactly matches the population mean
Chapters 8 & 12: Page 5
2. The Standard Deviation of the Distribution of Sample Means
 X = Standard Error of the Mean
X =

n
Variability of X around 
Special type of standard deviation, type of “error”
Average amount by which X deviates from 
Less error = better, more reliable estimate of population parameter
Chapters 8 & 12: Page 6
 X influenced by two things:
(1) Sample size (n)
Larger n = smaller standard errors
Note: when n = 1   X = 
 as “starting point” for  X ,
 X gets smaller as n increases
(2) Variability in population ()
Larger  = larger standard errors
Chapters 8 & 12: Page 7
3. The Shape
Central Limit Theorem = Distribution of sample means will approach a
normal distribution as n approaches infinity
Very important!
True even when raw scores NOT normal!
What about sample size?
(1) If raw scores ARE normal, any n will do
(2) If raw scores are not normal but are symmetrically distributed, a small
n will usually suffice
(3) If the raw scores are severely skewed, n must be “sufficiently large”
For most distributions  n  30
Chapters 8 & 12: Page 8
Why are Sampling Distributions Important?
 Tell us the probability of getting a particular X , given  & 
 Critical for inferential statistics!
 Allow us to estimate population parameters
 Allow us to determine if a sample mean differs from a known population
mean just because of chance
 Allow us to compare differences between sample means – due to chance or
to experimental treatment?
Sampling distribution is the most fundamental concept underlying all statistical
tests
Chapters 8 & 12: Page 9
Working with the Distribution of Sample Means
If we assume DSM is normal (again, we can do this if raw scores are normally
distributed or n is at least 30)
AND
If we know  & 
We can use the Normal Curve & Table E.10!
z
x
x
where:  X =

n
Chapters 8 & 12: Page 10
Example: Suppose you take a sample of 25 high-school students, and measure
their IQ. Assuming that IQ is normally distributed with  = 100 and  = 15,
what is the probability that your sample’s IQ will be 105 or greater?
Step 1: Convert to Z-score:
z
x
x
 X = 15  15  3
25
5
105  100
z
 1.67
3
Step 2: See Table E. 10
The probability of the sample having a mean of 105 or greater is: 0.0475
Chapters 8 & 12: Page 11
Example: Repeat the same problem in the previous example, but assume your
sample size is 64
Step 1: Convert to Z-score:
z
x
x
 X = 15  15 1.875
64
8
105  100
z
 2.67
1.875
Step 2: See Table E. 10
The probability of the sample having a mean of 105 or greater is: 0.0038
Chapters 8 & 12: Page 12
Example: What X marks the point above which sample means are likely to
occur only 15% of the time, if n = 36?
Step 1: Find  X =

n
 X = 15  15  2.5
36
6
Step 2: See Table E. 10 : Z = ?
Step 3: Solve for X : X =  + Z X
X = 100 + (?)(2.5) = 102.6
Chapters 8 & 12: Page 13