1. PROBABILITY AND
DISTRIBUTIONS
STA 250

2. So Far
• Learned to describe a distribution through its shape,
central tendency, and variability
• Used z-scores to locate and compare individual
scores
• Applied the rules of probability to determine the
likely hood of obtaining that score in a sample
• However, we have only dealt with samples consisting
of a single score.
• Most research uses far larger samples to represent a
population.

3. More About Populations and
Samples
• A POPULATION is a universe of individuals who share at least one
characteristic the study is interested in.
• A SAMPLE is a subgroup from within the population.
• The natural discrepancy or difference between a SAMPLE and the
POPULATION it was drawn from is SAMPLING ERROR
• Multiple SAMPLES can be drawn from the same population
• Statistics can be calculated for each of these SAMPLES
• Each SAMPLE will be different from the POPULATION and other SAMPLES

4. Distribution of Sample Means
• So far we have seen two types of distributions:
1. Distribution of scores for a population of
individuals
2. Distribution of scores for a particular sample
drawn from a population
• Now we add a third
3. Distribution of means of all possible samples of a
particular size taken from a distribution

5. Distribution of Sample Means
• The Distribution of Sample Means is the
collection of sample for all the possible
random samples of a particular size (n) that
can be obtained from a population
– Contains all possible combination for a specific n
– Comprised of the statistics (means) for each of the
samples
– Also referred to as a sampling distribution, or
sampling distribution of M.

6. SAMPLING DISTRIBUTION
• Sampling distribution is a distribution of samples
from the same population distribution

7. Distribution of Sample Means
• We would expect that if you repeatedly drew samples and
recorded the means the following would be true
– The sample means would pile up around the population mean
– The pile of sample means would tend to form a normal-shaped
distribution
• The most often occurring in the middle close to population mean
• The least often occurring on the outside away from population mean
– The larger the sample size the closer the sample means will be to the
population mean

8. Think of Each Square as a
Individual Sample Mean

9. Consider
• If the population consisted of only 4 scores: 2,
4, 6, 8, and we wanted to construct a
distribution of sample means for the sample
size n=2
• When we listed every possible sample that
could be drawn from this population (16)
• Calculated the mean for each sample
• Then graphed the means using a histogram

10. We Would Find

11. We Would Find

12. Central Limit Theorem
For any population with mean μ and standard
deviation σ, the distribution of sample means
for a sample size n will have a mean of μ and
a standard deviation of
and will
approach a normal distribution as n
approaches infinity
• Includes central tendency, variability, and
shape of distribution

13. What This Means
• Describes the distribution of sample means for any
population no matter what shape, mean, or standard
deviation.
• The mean of all the sample means will be the same
as the population mean
• The normality of the distribution increases as the
sample size increases. When n=30 the distribution is
almost perfectly normal.

14. Mean of the Distribution of Sample
Means
• The mean of a distribution of sample means is
is called the expected value of M
• Signified by M
• The mean expected value of M will always be
equal to the population mean μ
M=μ

15. Standard Deviation of the
Distribution of Sample Means
• The standard deviation for the distribution of sample
means is called the standard error of the mean or M.
• Just like the standard deviation the standard error of
the mean represents the average distance between
each sample mean and the mean of the distribution
of means.
• Signified by

16. Standard Error of M
• Tells us
• How much difference is expected from one sample
to another.
– The larger the standard error the more spread out the
distribution
– The smaller the standard the more clustered the
distribution
• How well an individual sample mean represents
entire distribution.
– Because M=μ it also tells us how much difference there is
between the M and μ. Check of sampling error

17. Standard Error of M
• Magnitude of the standard error determined by :
– Sample size (Law of Large Numbers)
• The larger the sample size the more probable it is that the sample mean
will be close to the population mean
– Standard deviation of the population
• The starting point for standard error. When n=1 standard error and
standard deviation are the same
• Inverse relationship between sample size and standard error
• Formula

18. As Sample Size Increases Standard
Error Decreases

19. Example
• The GRE has mean of 500 and standard deviation of
100. If many samples of n=50 students are taken:
– Mean of distribution of means is 500
– What is the SE of Mean?
• Formula:
– Shape of distribution will be normal.

20. Probability and the Distribution of
Sample Means
• Because the distribution of sample means is a
normal distribution, z-scores and the unit
normal table can be used to find probability
• The z-score formula does change in notation
but not concept