Sampling Distributions and Standard Error
A free Statistics and Data Analysis lesson from the “Inference and Conclusions” unit, with a worked example and practice problems including step-by-step solutions.
A sampling distribution shows how a statistic varies over many random samples. Standard error measures the typical sample-to-sample variation of that statistic. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Describe a sampling distribution
- Interpret standard error
- Connect sample size to variability
Worked example
Problem. A class takes many random samples of the same size and graphs the sample means. What is the graph showing?
- Worked Example: First identify exactly what the question is asking: A class takes many random samples of the same size and graphs the sample means. What is the graph showing?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- A sampling distribution shows many values of a statistic.
- Here each plotted value is a sample mean.
Answer: a sampling distribution
Practice problems
1. Practice case A: What values are plotted in a sampling distribution?
Choices: survey answer choices only · sample statistics from repeated samples · all individual raw data values · treatment names
Show solution
- Warm-up: First identify exactly what the question is asking: What values are plotted in a sampling distribution?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Each point comes from a sample statistic.
- The graph shows how that statistic varies.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: sample statistics from repeated samples
2. Practice case B: In a sampling distribution, standard error is the typical:
Choices: the treatment assignment · the survey wording · typical sample-to-sample variation of a statistic · the exact population value
Show solution
- Warm-up: First identify exactly what the question is asking: In a sampling distribution, standard error is the typical:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Standard error is about variability of statistics.
- It is not the same as an individual data value.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: typical sample-to-sample variation of a statistic
3. Practice case C: When sample size increases, standard error usually:
Choices: gets larger automatically · becomes a placebo · turns into a bar chart · gets smaller
Show solution
- Warm-up: First identify exactly what the question is asking: When sample size increases, standard error usually:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Larger samples vary less from sample to sample.
- That makes standard error smaller.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: gets smaller
4. Practice case D: Repeated sample means should center near:
Choices: the population parameter · the treatment group name · the largest sample size · the first response option
Show solution
- Warm-up: First identify exactly what the question is asking: Repeated sample means should center near:
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- An unbiased statistic centers around the parameter.
- Sampling variability spreads values around that center.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: the population parameter
5. Practice case E: Which phrase best separates a sampling distribution from raw data?
Choices: only one person's response · statistics from many samples, not individual cases · only treatment labels · only the population size
Show solution
- Warm-up: First identify exactly what the question is asking: Which phrase best separates a sampling distribution from raw data?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- A data distribution shows individual values.
- A sampling distribution shows statistics.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: statistics from many samples, not individual cases
Practice this interactively with instant feedback and an AI tutor.
Practice Sampling Distributions and Standard Error Take the free placement check