Sampling distribution and central limit theorem

Inferential statistics

infer from samples to population

English letters

measure statistics of samples

Greek letters

Infer paramètres of population

Know:

Estimate:

• Know: sample statistics (mean, standard deviation)

• Estimate: population parameters (mean, standard deviation

What is the way we do this estimation?

sampling theory

A population in statistics =

all the scores for a particular variable

Sample mean

unbiased estimator of the population mean, but rarely will it be the same because of error

What is equal to the population mean

Mean of all possible sample means

Sampling distribution of the mean =

the distribution of all possible sample means

Central limit theorem part 1

The sampling distribution of the mean will have a mean equal to μ (population)

Estimating population parameters

• To estimate the mean of a population: take a sample and find its mean

•  the mean for our sample is an unbiassed estimator of the population mean

Sampling error:

Sampling mean usually ≠ population mean (μ)

What is the standard error of the mean?

The standard deviation of the sampling distribution of the mean

The standard error of the mean is a measure of

• How spread out the sample means are from the population mean

• How much we might be wrong when we estimate the population mean from the sample

Standard error

 measure of the amount that a sample mean could be different from the population mean

Central limit theorem part 2

The sampling distribution of the mean will have a standard deviation of σ / √N

As N is denominator

If have bigger samples then variability will be smaller

Standard error formula

σ / √N

Estimated standard error formula

s/√N

What is S?

What is N?

in estimated standard error

S - sample standard deviation

N = sample size

Standard error shows us the reliability of

our sample mean as an estimator of the population mean

Central Limit therem part 3

If the population of scores is normally distributed, the sampling distribution of the mean will be normally distributed

What if population has a non-normal distribution?

The sampling distribution of the mean will approach the normal distribution as the sample size increases, regardless of the shape of the original population distribution

The more the population distribution differs from normal

 the greater the required size of N