Sampling error
is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
The distribution of sample means
is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.
- the distribution of sample means contains all the possible samples
Sampling distribution
A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Characteristics of the distribution sample mean
- The sample means should pile up around the population mean. Samples are not expected to be perfect but they are representative of the population.
As a result, most of the sample means should be relatively close to the population mean. - The pile of sample means should tend to form a normal-shaped distribution.
- In general, the larger the sample size, the closer the sample means should be to the population mean
central limit theorem
A mathematical theorem that specifies the characteristics of the distribution of sample means.
- provides a precise description of the distribution that would be obtained if you selected every possible sample, calculated every sample mean, and constructed the distribution of the sample mean.
For any population with mean μ
and standard deviation o, the distribution of sample means for sample size n will have a mean of
μ and a standard deviation of 0/square root of n and will approach a normal distribution as n approaches infinity.
First, it describes the distribution of sample means for any population, no matter what shape, mean, or standard deviation.
Second, the distribution of sample means “approaches” a normal distribution very rapidly.
describes: shape, central tendency, and variability
distribution of sample means is almost perfectly normal if either of the following two conditions is satisfied:
- The population from which the samples are selected is a normal distribution.
- The number of scores (n) in each sample is relatively large, around 30 or more.
expected value of M
The mean of the distribution of sample means is equal to the mean of the population of scores, μ, and is called the expected value of M.
The standard error serves the same two purposes for the distribution of sample means.
- The standard error describes the distribution of sample means. It provides a measure of how much difference is expected from one sample to another.
- Standard error measures how well an individual sample mean represents the entire distribution.
Standard error of M
The standard deviation of the distribution of sample means, oM , is called the standard error of M. The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean μ
The magnitude of the standard error is determined by two factors:
- the size of the sample - a large sample should be more accurate than a small sample -
- the standard deviation of the population from which the sample is selected - there is an inverse relationship between the sample size and the standard error: bigger samples have smaller error, and smaller samples have bigger error
the law of large numbers
The law of large numbers states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean.
standard error equation
OM = o/square root of n = square root of o^2/n
- As sample size (n) increases, the size of the standard error decreases. (Larger samples are more accurate.)
- When the sample consists of a single score , the standard error is the same as the standard deviation .
a z-score identifies the location with a signed number so that
- The sign tells whether the location is above (+) or below (−) the mean.
- The number tells the distance between the location and the mean in terms of the number of standard deviations.
z-score formula for sample mean
z = M-mu/oM(standard error)(using standard error equation)
