THE SAMPLING DISTRIBUTION OF THE MEAN
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
• 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
The Distribution of Sample Means
Population of 4 scores: 2 4 6 8 ( ( = 5
4 random samples (n = 2):
[pic]1= 4 [pic]3 = 5
[pic]2 = 6 [pic]4 = 3
[pic] is rarely exactly (
Most [pic] a little bigger or smaller than (
Most [pic] will cluster around (
Extreme low or high values of [pic] are relatively rare
With larger n, [pic]s will cluster closer to µ (the DSM will have smaller error, smaller variance)
A Distribution of Sample Means
The Distribution of Sample Means
1. A distribution of sample means ([pic])
2. All possible random samples of size n
3. A distribution of a statistic (not raw scores)
“Sampling Distribution” of [pic]
4. Probability of getting an [pic], given known ( and (
5. Important properties
(1) Mean
(2) Standard Deviation
(3) Shape
Properties of the DSM
Mean?
([pic] = (
Called expected value of [pic]
[pic] is an unbiased estimate of (
Standard Deviation?
Any [pic] can be viewed as a deviation from (
([pic] = Standard Error of the Mean
([pic] = [pic]
Variability of [pic] around (
Special type of standard deviation, type of “error”
Average amount by which [pic] deviates from (
Less error = better, more reliable, estimate of
population parameter
([pic] influenced by two things:
(1) Sample size (n)
Larger n = smaller standard errors
Note: when n = 1 ( ([pic] = (
( as “starting point” for ([pic],
([pic] gets smaller as n increases
(2) Variability in population (()
Larger ( = larger standard errors
Note: ([pic] = (M
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
Why are Sampling Distributions important?
• Tells us probability of getting [pic], 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
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 =
Example #1:
( = 80 ( = 12
What is probability of getting [pic] ( 86 if n = 9?
Example #1b:
( = 80 ( = 12
What if we change n =36
What is probability of getting [pic] ( 86
Example #2:
( = 80 ( = 12
What [pic] marks the point beyond which sample means are likely to occur only 5% of the time? (n = 9)
Homework problems:
Chapter 7: 3, 10, 11, 17
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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 à = 20. Notice that the si = 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.
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.
[pic]= 4
[pic]= 5
[pic]= 6
[pic]
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