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Probability and Samples: The Distribution of Sample Means - Coggle Diagram
Probability and Samples: The Distribution of Sample Means
essential background material
Probability and the normal distribution
z-scores
Random Sampling
sampling error
the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
samples are variable
they are not all the same. If you take two separate samples from the same population, the samples will be different
They will contain different individuals, they will have different scores, and they will have different sample means
distribution of sample means
the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.
sampling distribution
a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Characteristics of the Distribution of Sample Means
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. Logically, most of the samples should have means close to μ, and it should be relatively rare to find sample means that are substantially different from μ. As a result, the sample means should pile up in the center of the distribution (around μ) and the frequencies should taper off as the distance between M and μ increases. This describes a normal-shaped distribution.
In general, the larger the sample size, the closer the sample means should be to the population mean, μ. Logically, a large sample should be better than a small sample because it is more representative. Thus, the sample means obtained with a large sample size should cluster relatively close to the population mean; the means obtained from small samples should be more widely scattered.
Central limit theorem
For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/n and will approach a normal distribution as n approaches infinity.
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.
standard error of M
standard error of M
The standard deviation of the distribution of sample means, σM,
The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ).
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=σM=σN=σ2n=σ2n
Three Different Distributions
original population of scores
The sample consists of a small set of scores for people who have been selected to represent the entire population
distribution of sample means
This is a theoretical distribution consisting of the sample means obtained from all the possible random samples of a specific size
original population of scores
This population contains the scores for thousands or millions of individual people, and it has its own shape, mean, and standard
Sampling Error.
The general concept of sampling error is that a sample typically will not provide a perfectly accurate representation of its population. More specifically, there typically is some discrepancy (or error) between a statistic computed for a sample and the corresponding parameter for the population.
Standard Error
These samples provide a fairly accurate representation of the population. On the other hand, some samples produce means that are out in the tails of the distribution, relatively far from the population mean.