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3.7 Sampling Distributions of Sample Means

Section Overview

  • Gallery 3.7.1 – How Samp. Mean is Formed
  • Gallery 3.7.2 – Application of CLT
  • Example 3.7.1
  • Pop Quiz 3.7.1

The Sampling Distribution of the Sample Mean

Since we’ve already covered the sampling distributions of one and two proportions in the previous section, understanding the sampling distribution of the sample mean shouldn’t be too difficult. A sample mean is a statistic that is derived from taking the mean of the observations of a random sample of size nn. The sampling distribution of the sample mean is created when we take an infinite amount random samples of a fixed sample size and graph the frequency of a certain value of the sample mean. Gallery 3.7.1 shows the process of how a sampling distribution is derived from the population distribution.

Visible sample lists in Gallery 3.7.1, frame 1.
SampleComplete visible observationsFinal visible fragment at clipped edge
SRS 1 (red)15.23, 19.76, 28.42, 35.52, 40.13, 44.95, 47.23, 56.27, 58.3683
SRS 2 (purple)26.29, 32.44, 43.72, 48.79, 55.62, 61.76, 64.98, 74.42, 76.2379.
SRS 3 (yellow)18.88, 24.32, 31.77, 38.09, 42.81, 53.41, 63.62, 64.89, 66.3975
SRS 4 (green)10.49, 23.08, 27.36, 29.45, 38.86, 39.33, 47.92, 55.67, 61.2181
Source gap (S0319): Source data gap: The final number in each sample list is cut off by the gallery’s right edge. Only the visible fragments above are transcribed; no missing digits are inferred. Frame 2’s sample-mean labels are retained as visible: 42.9, 56.3, 48.0, 41.5.

Now that we’re a bit more familiar with the sampling distribution of the sample mean, let’s talk about some of its features. This time, we saved “Shape” for last because it discusses an important concept in Inferential Statistics.

Center – The mean of the sampling distribution the sample mean xˉ\bar x is equal to the population mean μ\mu.

Spread – The standard deviation of the sampling distribution of the sample mean is σn\frac{\sigma}{\sqrt n} for an SRS of size nn. This only applies if the population is at least 10 times as large as the sample, or 10n≤N10n\le N.

Shape – If the population distribution is Normal, then the sampling distribution of the sample mean is also Normal. Even if the original population distribution is non-Normal, then a rule called the Central Limit Theorem comes into play, where if n≥30n\ge30, then the sampling distribution of the sample mean is approximately Normal. This effect can be shown in Gallery 3.7.2, where increasing the sample size of a sampling distribution brings it closer and closer to approximating a Normal distribution.

Source gap (S0324): Source figure gap: All four Gallery 3.7.2 frames clip the vertical-axis label at the top. The source curves, sample-axis labels, and complete captions are retained.

The Sampling Distribution of the Difference Between Two Sample Means

Center – The mean of the sampling distribution of the difference between two sample means xˉ1−xˉ2\bar x_1-\bar x_2 is equal to the difference between population means μ1−μ2\mu_1-\mu_2.

Spread – The standard deviation of the sampling distribution of difference between two sample means is σ12n1+σ22n2\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} for an SRS of size nn. This only applies if the populations are at least 10 times as large as their respective samples, or 10n≤N10n\le N.

Shape – If both population distributions are Normal, then both sampling distributions are also Normal. If the population distributions are not normal, then the sampling distributions are approximately Normal if both sample sizes exceed 30 due to the Central Limit Theorem.

Example 3.7.1

A vending machine company, Pro Vending, takes an SRS of n=30n=30 days in a previous month for one of its vending machines and finds the mean revenue per day to be $26.15 and the standard deviation to be $4.73. It is thought that Pro Vending’s long-term revenue distribution is right-skewed. The company calculates that for its vending machines to be profitable, the revenue made per day must amount to at least $24.50. What is the probability that the average revenue per day meets or exceeds this amount?

Example 3.7.1 — recorded answer block on page 142

All three descriptions of the sampling distribution: Shape, Center, and Spread, will be used to find the probability.

Shape: We know that the sampling distribution of the sample mean closely follows a Normal distribution. This means we can use calculations similar to what we learned in Chapter 3.4, but only if the sample has at least 10 “successes” and 10 “failures,” where a “success” in this context is defined as a non-punctured bed. 400(0.97)=388≥10400(0.97)=388\ge10 and 400(0.3)=12≥10400(0.3)=12\ge10, so we are clear to use Normal calculations.

Center: We know that the mean of the sampling distribution of the sample proportion is p=0.97p=0.97.

Spread: We know that the standard deviation of the sampling distribution of the sample proportion is

Source gap (S0325): Source inconsistency and gap: The answer block beside the vending-machine prompt contains waterbed proportions and the non-punctured-bed definition from Example 3.6.1. The failure product is printed as 400(0.3) = 12. The Spread sentence stops after “is,” with no formula on this page. This distinct captured block is preserved separately from the vending-machine answer on pages 143–144.
Example 3.7.1 — recorded answer on pages 143–144

All three descriptions of the sampling distribution: Shape, Center, and Spread, will be used to find the probability.

Shape: Although the revenue distribution was said to be right-skewed, the sample size is at least 30, so the sampling distribution of the sample mean will be approximately Normal thanks to the Central Limit Theorem.

Center: We know that the mean of the sampling distribution of the sample mean is μ=$26.15\mu=\$26.15.

Spread: We know that the standard deviation of the sampling distribution of the sample mean is σn\frac{\sigma}{\sqrt n}.

However, we can only use this calculation if the population is at least 10 times as large as

Source gap (S0326): Source sentence gap: The left column stops at “as.” The next visible text in the right column starts a new sentence, “The revenue distribution was described as long-term.” No missing completion is supplied.

The revenue distribution was described as long-term, so we can assume that it contains values for at least 300 days. Therefore we have:

4.7330=0.86\frac{4.73}{\sqrt{30}}=0.86

Using these numbers, we can find the corresponding z-score for $24.50:

z=xˉ−μσ=24.50−26.150.86=−1.92z=\frac{\bar x-\mu}{\sigma}=\frac{24.50-26.15}{0.86}=-1.92

We are looking for the area to the right of z=−1.92z=-1.92 on a standard Normal distribution graph. According to the z-table, we get the corresponding probability of 0.0274. This is the area to the left of z=−1.92z=-1.92, however, so we subtract the probability from 1 to get:

1−0.0274=0.97261-0.0274=0.9726

This means that there is a 97% chance that the average revenue per day will meet or exceed $24.15.

Source gap (S0327): Source inconsistency: The final sentence uses $24.15; the prompt and z-score use $24.50. The original conclusion is retained. The equation 1 − 0.0274 = 0.9726 is repeated at the top of page 144 in the screenshots and is represented once above.

Pop Quiz 3.7.1

Question 1 of 5

The Central Limit Theorem states that if a sample size nn at least 30, then

Question 2 of 5

Scores on an exam were nearly Normal with mean 90 and standard deviation 9. Suppose a random sample of 400 students was chosen many times. The mean of the average scores from the random samples should be close to:

Question 3 of 5

Considering the scenario in question 2, the standard deviation of the average scores should be close to:

Question 4 of 5

Doughnuts that are machine-injected with jelly have a mean of 35.5 grams with a standard deviation of 0.5 grams. If four doughnuts are randomly selected each minute, then 95% of the observations should fall within:

Question 5 of 5

If the number of hours an air conditioner runs before failing has a right-skewed distribution, then:

Source gap (S0332): Source duplicate: Choices C and D are identical in the screenshot and remain identical here.
Source gap (S0332): Source answer gap: All five quiz questions and choices are captured, but no selected, checked, or feedback state is shown. No quiz answer key has been supplied.
Glossary and deeper reading · 8 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Sampling distribution

    A sampling distribution describes how a statistic varies across all possible samples drawn under a specified sampling design and sample size. It is a distribution of statistics rather than of individual observations. For example, the proportions from repeated random samples of 60 students form a sampling distribution.

    Read more: Sampling distribution in the captured Statistics reading →

  • Mean

    The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.

    Read more: Mean in the captured Statistics reading →

  • Statistical inference

    Statistical inference uses data and a study design to draw conclusions beyond the observed values. Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions. For example, a random sample can estimate a school-wide preference without surveying every student.

    Read more: Statistical inference in the captured Statistics reading →

  • Standard deviation

    Standard deviation measures spread around the mean using squared deviations and a square root. A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root. For example, a larger standard deviation of travel times indicates greater variability, in the original time units.

    Read more: Standard deviation in the captured Statistics reading →

  • Simple random sample

    A simple random sample of a fixed size gives every possible subset of that size the same chance of selection. Giving individuals equal selection chances alone does not establish that the design is an SRS. For example, randomly choosing three identifiers without replacement from ten gives an SRS when every three-identifier set is equally likely.

    Read more: Simple random sample in the captured Statistics reading →

  • Population

    A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.

    Read more: Population in the captured Statistics reading →

  • Sample

    A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.

    Read more: Sample in the captured Statistics reading →

  • Central limit theorem

    Editorial qualification: The captured n ≥ 30 statement is not a universal guarantee. Under the CLT’s assumptions, the accuracy of a Normal approximation depends on the population and sample size.

    For independent identically distributed observations with finite variance, the central limit theorem explains why the standardized sample mean approaches a Normal distribution as sample size increases. There is no universal sample size that makes every population sufficiently close to Normal; skewness and extreme tails affect the approximation. For example, averaging many independent waiting times can produce a less skewed distribution of averages than the original waiting-time distribution.

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