Download A Distribution of Sample Means

THE DISTRIBUTION OF SAMPLE MEANS
Inferential statistics:
Generalize from a sample to a population
Statistics vs. Parameters
Why?
Population not often possible
Limitation:
Sample won’t precisely reflect population
Samples from same population vary
“sampling variability”
Sampling error = discrepancy between sample
statistic and population parameter
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 Extend z-scores and normal curve to SAMPLE
MEANS rather than individual scores
 How well will a sample describe a population?
 What is probability of selecting a sample that has a
certain mean?
 Sample size will be critical
 Larger samples are more representative
 Larger samples = smaller error
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THE DISTRIBUTION OF SAMPLE MEANS
Population of 4 scores:
2
4
6
8

=5
4 random samples (n = 2):
X 1= 4
X3 = 5
X2 = 6
X4 = 3

X is rarely exactly 

Most X a little bigger or smaller than 

Most X will cluster around 

Extreme low or high values of X are relatively rare

With larger n, X s will cluster closer to µ (the DSM
will have smaller error, smaller variance)
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A Distribution of Sample Means
X= 4
X= 5
X= 6
Figure 7-3 (p. 205)
The distribution of sample means for n = 2. This distribution shows the 16
sample means obtained by taking all possible random samples of size n=2 that
can be drawn from the population of 4 scores (see Table 7.1 in text). The
known population mean from which these samples were drawn is µ = 5.
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THE DISTRIBUTION OF SAMPLE MEANS

A distribution of sample means ( X )

All possible random samples of size n

A distribution of a statistic (not raw scores)
“Sampling Distribution” of X

Probability of getting an X , given known  and 

Important properties
(1) Mean
(2) Standard Deviation
(3) Shape
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PROPERTIES OF THE DSM

Mean?
X = 
Called expected value of X
X is an unbiased estimate of 

Standard Deviation?
Any X can be viewed as a deviation from 
 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 
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Less error = better, more reliable, estimate of
population parameter
 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
Note:  X = M
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Figure 7-7 (p.215)
The distribution of sample means for random samples of size (a) n = 1, (b) n = 4, and (c) n = 100
obtained from a normal population with µ = 80 and σ = 20. Notice that the size of the standard error
decreases as the sample size increases.
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
Shape of the DSM?
Central Limit = DSM will approach a normal dist’n
Theorem
as n approaches infinity
Very important!
True even when raw scores NOT normal!
True regardless of  or 
What about sample size?
(1) If raw scores ARE normal, any n will do
(2) If raw scores NOT normal, n must be
“sufficiently large”
For most distributions  n  30
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Why are Sampling Distributions important?
 Tells us probability of getting X , given  & 
 Distribution of a STATISTIC rather than raw scores
 Theoretical probability distribution
 Critical for inferential statistics!
 Allows us to estimate likelihood of making an error
when generalizing from sample to popl’n
 Standard error = variability due to chance
 Allows us to estimate population parameters
 Allows 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
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WORKING WITH THE
DISTRIBUTION OF SAMPLE MEANS

If we assume DSM is normal

If we know  & 

We can use Normal Curve & Unit Normal Table!
z = X 
x
Example #1:
 = 80  = 12
What is probability of getting X  86 if n = 9?
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Example #1b:
 = 80  = 12
What if we change n =36
What is probability of getting X  86
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Example #2:
 = 80  = 12
What X marks the point beyond which sample means are
likely to occur only 5% of the time? (n = 9)
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Homework problems:
Chapter 7: 3, 10, 11, 17
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