Sampling variability and sampling distributions

About this lesson

Algebra II › Statistical inference

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Sampling variability and sampling distributions

Warm-up

Three quick questions from earlier lessons.

Discover

Five classmates each survey 20 random students about a new rule. Will they all get the same percentage in favor?

  1. No, the percentages will vary
  2. Yes, if they all sample at random

Key idea

A statistic has a distribution of its own.

  1. Step 110 samples: a few values of p̂, scattered.
  2. Step 2100 samples: a shape appears.
  3. Step 31,000 samples (scaled): a mound centered near p — the sampling distribution.

The sampling distribution of a statistic is its distribution over all possible random samples of the same size. Its center tells us about bias; its spread (standard error) shrinks as the sample size grows.

Worked example

Imagine 1,000 samples of 50 students and the p̂ of each.

A histogram of p̂ values
Plot the 1,000 values.
Centered near the true p
Where it is centered.
Narrower: less variability
With samples of 200 instead.

Your turn, guided

Your turn: by what factor does SD(x̄) change when nn goes from 25 to 100?

It is halved: factor 12\tfrac{1}{2}
√25/√100.

Practice ★

  1. Many random samples of size 161 are taken from the same population. Which quantity changes from sample to sample?
    1. The sample proportion p̂ (a statistic)
    2. The population proportion p (a parameter)
    3. The sample size
  1. Compared with samples of size 73, samples of size 292 give sample means that…
    1. vary less around the population mean
    2. vary more
    3. are more biased

Watch out!

Practice ★★

  1. What is the sampling distribution of the sample mean for samples of size 25?
    1. The distribution of x̄ over all possible samples of size 25
    2. The distribution of the 25 values in one sample
    3. The distribution of the population

Exit ticket

Three questions to check you have got it.

  1. Many random samples of size 161 are taken from the same population. Which quantity changes from sample to sample?
    1. The sample proportion p̂ (a statistic)
    2. The population proportion p (a parameter)
    3. The sample size
  2. Many random samples of size 224 are taken from the same population. Which quantity changes from sample to sample?
    1. The sample proportion p̂ (a statistic)
    2. The population proportion p (a parameter)
    3. The sample size
  3. Many random samples of size 490 are taken from the same population. Which quantity changes from sample to sample?
    1. The sample proportion p̂ (a statistic)
    2. The population proportion p (a parameter)
    3. The sample size

Summary card

Sampling variability

  • Statistics vary from sample to sample.
  • Sampling distribution: a statistic over all samples.
  • Larger nn: less variability.

Watch out: data distribution vs sampling distribution.

End of the lesson

Next lesson: Point estimates and margins of error Back to the chapter

Answers: Sampling variability and sampling distributions

A

  1. The sample proportion p̂ (a statistic)

B

  1. vary less around the population mean

C

  1. The distribution of x̄ over all possible samples of size 25

D

  1. The sample proportion p̂ (a statistic)
  2. The sample proportion p̂ (a statistic)
  3. The sample proportion p̂ (a statistic)

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