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Chapter 07 Probability and Samples: The Distribution of Sample Means -…
Chapter 07 Probability and Samples: The Distribution of Sample Means
Sampling Error
is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
Sampling Error is the difference, or error between sample statistics and the corresponding population parameters,
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.
In most cases, especially for very large populations, it is possible to obtain many thousands, or even millions, of different samples from one population.
The huge set of possible samples forms a relatively simple and orderly pattern that makes it possible to predict the characteristics of a sample with some accuracy.
The ability to predict sample characteristics is based on the distribution of sample means
The ability to predict sample characteristics is based on the
Distribution of Sample Means
A Sampling Distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Because statistics are obtained from samples, a distribution of statistics is often referred to as a
Sampling Distribution
the distribution of sample means is an example of a sampling distribution. In fact, it often is called the sampling distribution of M.
The process of constructing the distribution of sample means. A sample of n scores is selected. Then the sample mean is computed and placed in a frequency distribution. This process is repeated over and over until all the possible random samples are obtained and the complete set of sample means is in the distribution.
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.
A mathematical proposition known as the Central Limit Theorem 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.
The central limit theorem describes the distribution of sample means by identifying the three basic characteristics that describe any distribution: shape, central tendency, and variability.
The distribution of sample means tends to be a normal distribution. .
This distribution 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.
A sample mean is “expected” to be near its population mean. When all of the possible sample means are obtained, the average value is identical to
μ
.
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 describes the distribution of sample means. It provides a measure of how much difference is expected from one sample to another.
When the standard error is small, all the sample means are close together and have similar values. If the standard error is large, the sample means are scattered over a wide range and there are big differences from one sample to another.
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.
The primary use for the distribution of sample means is to find the probability of selecting a sample with a specific mean.
It is possible to use a z-score to describe the exact location of any specific sample mean within the distribution of sample means.
The z-score tells exactly where the sample mean is located in relation to all the other possible sample means that could have been obtained.
The standard error provides a way to measure the “average,” or standard, distance between a sample mean and the population mean.
Sampling Error. The general concept of sampling error is that a sample typically will not provide a perfectly accurate representation of its population.