PROBABILITY AND DISTRIBUTIONS

STA 250

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.

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

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 third3. Distribution of means of all possible samples of a

particular size taken from a distribution

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.

SAMPLING DISTRIBUTION• Sampling distribution is a distribution of samples

from the same population distribution

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

Think of Each Square as a Individual Sample Mean

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

We Would Find

We Would Find

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

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.

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

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

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

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

As Sample Size Increases Standard Error Decreases

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.

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