Chapter 7: Probability and Samples: The Distribution of Sample Means

Learning Outcomes

Define distribution of sampling means
The distribution of sampling means refers to the probability distribution of all possible sample means from a population.
Describe distribution by shape, expected value, and standard error
Distributions can be characterized by their shapes (e.g., normal), the expected value, and the variability captured by the standard error.
Describe location of sample mean M by z-score
The z-score indicates how far the sample mean is from the population mean in terms of standard deviations.
Determine probabilities corresponding to sample mean using z-scores and unit normal table
Utilizing the z-score enables the calculation of probabilities for sample means from a standard normal distribution.

Tools You Will Need

Random sampling (See Chapter 6)
Understanding how to select samples randomly.
Probability and the normal distribution (See Chapter 6)
Knowledge of how to calculate probabilities under the normal curve.
z-Scores (See Chapter 5)
Understanding how to compute and interpret z-scores.

7-1: Samples, Populations, and the Distribution of Sample Means

Each score's location in a sample or population can be represented using a z-score.
Researchers prefer to study samples rather than individual scores because a sample can provide a more representative estimate of a population.
The process for transforming a sample mean (M) into a z-score is established to facilitate statistical analyses.

Sampling Error

Definition: Sampling error is the natural discrepancy, or error, between a sample statistic and the corresponding population parameter.
- It does not imply that a mistake was made.
Variability in samples means that two samples from the same population are unlikely to be identical.

The Distribution of Sample Means

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Two samples drawn from the same population will likely differ, indicating variability in sample means.
The distribution of sample means comprises the collection of sample means for all potential random samples of a given size (n) from a population.

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Earlier chapters discussed distributions of individual scores; here we focus on a sampling distribution — a distribution of statistics (sample means, not individual scores).
The sample means will reflect the population's characteristics, but they constitute a unique distribution of their own.

Characteristics of the Distribution of Sample Means

Sample means cluster around the population mean.
The distribution of sample means tends to be approximately normal in shape.
As sample size increases, the sample means converge more closely to the population mean.

Figures

Figure 7.2: Frequency Distribution Histogram for a Population of N = 4 scores shows the frequency of scores across a limited number of samples.
Figure 7.3: The Distribution of Sample Means for n = 2 graphically illustrates how sample means vary.

7-2: Shape, Central Tendency, and Variability for the Distribution of Sample Means

It is possible to predict the distribution of sample means without conducting numerous samples using theoretical constructs.
The central limit theorem (CLT) helps specify the distribution's shape, central tendency, and variability.

The Central Limit Theorem

Applicability: The CLT applies to any population characterized by mean ( $μ$ ) and standard deviation ( $σ$ ).
As the sample size ( $n$ ) becomes larger, the distribution of sample means approaches a normal distribution irrespective of the original population's distribution.
For samples of size $n$ , the mean of the distribution of sample means is denoted as $μ_M$ , and the standard deviation is referred to as standard error.

The Shape of the Distribution of Sample Means

The distribution of sample means will be almost perfectly normal under two conditions:
1. The population from which samples are drawn is normally distributed, or
2. The number of scores ( $n$ ) is large (usually $n ext{≥} 30$ ).

The Mean of the Distribution of Sample Means

The mean of the distribution of sample means ( $μ_M$ ) equals the population mean ( $μ$ ).
This expected mean, $μ_M$ , indicates that the sample mean is an unbiased statistic, meaning it in theory equals the population mean.

7-3: The Standard Error of M

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The standard error of M ( $σ_M$ ) measures the extent to which an individual sample mean is representative of the population mean.
The variability of scores in a distribution is captured by the standard deviation, while the variability of sample means is modeled by the standard error of the sample.

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When the standard error of M is large, this suggests that sample means are widely dispersed around the population mean.
It provides an average measure of the expected distance between the sample mean ( $M$ ) and the population mean ( $μ$ ).

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The law of large numbers states that as the sample size increases, the probability that the sample mean closely resembles the population mean increases.
Population Variance: A smaller population variance suggests more consistent sample means that closely approximate the population mean.

Figures

Figure 7.4 displays the relationship between standard error and sample size, demonstrating how larger samples reduce standard error.

7-4: z-Scores and Probability for Sample Means

The primary application of the distribution of sample means is to calculate the probability associated with obtaining a sample mean of a specific value.
Proportions of the normal curve aid in representing these probabilities.

A z-Score for Sample Means

The sign of the z-score indicates whether the sample mean is above (+) or below (-) the population mean.
The z-score quantifies the distance of the sample mean from the population mean in units of standard deviations.
The formula for a z-score for sample means is given by: $z = \frac{M - μ<em>M}{σ</em>M}$ , where:
- $M$ = sample mean
- $μ_M$ = mean of the distribution of sample means
- $σ_M$ = standard error of the mean.

Figures

Figure 7.6 illustrates the distribution of sample means for $n = 16$ .
Figure 7.7 shows the middle 80% of the distribution of sample means along with its z-scores.

7-5: More about Standard Error

A discrepancy will typically exist between the sample mean and the actual population mean, known as sampling error.
The variability of this sampling error is assessed by the standard error of the mean.

Figures

Figure 7.8 presents an overview of a typical distribution of sample means.
Figure 7.9 demonstrates the distribution of means for different sample sizes (inputting $n = 1$ , $n = 4$ , and $n = 100$ ).

Reporting Standard Error

Standard error is reported differently across journals but commonly utilizes terms such as SE or SEM.
It is often included in tables alongside sample size (n) and means (M) for different experimental groups and may also be illustrated in graphs.

Figures

Figure 7.10 depicts mean scores (± SE) for treatment groups A and B.
Figure 7.11 shows the mean number of mistakes (± SE) for groups A and B across trials.

Looking Ahead to Inferential Statistics

Inferential statistics utilize sample data to draw broader conclusions about populations, accounting for the inevitable sampling error.
Variability and uncertainty arise from natural differences between samples and populations, affecting all inferential analyses.

Figures

Figure 7.12 outlines a research study including a population of weights for adult rats and treated samples.
Figure 7.13 illustrates the distribution of sample means for untreated rats based on an example in this chapter.

Learning Checks and Answers

Learning Check 1

A population has $μ = 60$ and $σ = 5$ with samples of size $n = 4$ having an expected value of:
Options: 5, 60, 30, 15
Correct Answer: 60

True/False Statements:

The shape of a distribution of sample means is always normal: False (Only if the population is normal or $n ≥ 30$ )
As sample size increases, the value of standard error decreases: True

Learning Check 2

For a random sample of $n = 16$ , from a population defined with $μ = 50$ and $σ = 16$ , the z-score for a sample mean of $M = 58$ is:
Options: z = 1.00, z = 2.00, z = 4.00, cannot determine
Correct Answer: z = 2.00

True/False Statements:

A sample mean with a z = 3.00 is a fairly typical, representative sample: False
The mean of the sample is always equal to the population mean: False

Questions and Clarifications

For any questions regarding concepts or equations, reach out for further clarification.