The Sampling Distribution of the Mean

THE DISTRIBUTION OF SAMPLE MEANS

Inferential statistics:

Generalize from a sample to a population

Statistics vs. Parameters

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

1  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

2 THE DISTRIBUTION OF SAMPLE MEANS

Population of 4 scores: 2 4 6 8   = 5

4 random samples (n = 2):

X 1= 4 X 3 = 5

X 2 = 6 X 4 = 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)

3 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.

4 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

5 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 

6 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

7 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.

8  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

9 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

10 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?

11 Example #1b:  = 80  = 12

What if we change n =36

What is probability of getting X  86

12 Example #2:

 = 80  = 12

What X marks the point beyond which sample means are likely to occur only 5% of the time? (n = 9)

13 Homework problems: Chapter 7: 3, 10, 11, 17