Lesson 13What Makes a Good Sample?

Learning Goal

Let’s see what makes a good sample.

Learning Targets

  • I can determine whether a sample is representative of a population by considering the shape, center, and spread of each of them.

  • I know that some samples may represent the population better than others.

  • I remember that when a distribution is not symmetric, the median is a better estimate of a typical value than the mean.

Lesson Terms

  • mean
  • mean absolute deviation (MAD)
  • median
  • population
  • representative
  • sample

Warm Up: Number Talk: Division by Powers of 10

Problem 1

Find the value of each quotient mentally.

Activity 1: Selling Paintings

Your teacher will assign you to work with either means or medians.

Problem 1

A young artist has sold 10 paintings. Calculate the measure of center you were assigned for each of these samples:

  1. The first two paintings she sold were for $50 and $350.

  2. At a gallery show, she sold three paintings for $250, $400, and $1,200.

  3. Her oil paintings have sold for $410, $400, and $375.

Problem 2

Here are the selling prices for all 10 of her paintings:

  • $50

  • $200

  • $250

  • $275

  • $280

  • $350

  • $375

  • $400

  • $410

  • $1,200

Calculate the measure of center you were assigned for all of the selling prices.

Problem 3

Compare your answers with your partner. Were the measures of center for any of the samples close to the same measure of center for the population?

Activity 2: Sampling the Fish Market

Problem 1

The price per pound of catfish at a fish market was recorded for 100 weeks.

open applet in presentation mode

  1. What do you notice about the data from the dot plots showing the population and each of the samples within that population? What do you wonder?

  2. If the goal is to have the sample represent the population, which of the samples would be good? Which would be bad? Explain your reasoning.

Print Version

The price per pound of catfish at a fish market was recorded for 100 weeks.

  1. Here are dot plots showing the population and three different samples from that population. What do you notice? What do you wonder?

    Population

    A dot plot labeled dollars per pound of catfish. The numbers 1.6 through 2.8 in increments of zero point 2 are indicated. The data are as follows: 1 point 6 dollars per pound, 4 dots. 1 point 7 dollars per pound, 20 dots. 1 point 8 dollars per pound, 24 dots. 1 point 9 dollars per pound, 2 dots. 2 dollars per pound, 10 dots. 2 point 1 dollars per pound, 2 dots. 2 point 2 dollars per pound, 6 dots. 2 point 3 dollars per pound, 4 dots. 2 point 4 dollars per pound, 4 dots. 2 point 5 dollars per pound, 6 dots. 2 point 6 dollars per pound, 10 dots. 2 point 7 dollars per pound, 4 dots. 2 point 8 dollars per pound, 4 dots.

    Sample 1

    A dot plot labeled dollars per pound of catfish. The numbers 1.6 through 2.8 in increments of zero point 2 are indicated. The data, titled Sample 1, are as follows: 1 point 6 dollars per pound, 1 dots. 1 point 7 dollars per pound, 4 dots. 1 point 8 dollars per pound, 5 dots. 2 dollars per pound, 2 dots. 2 point 2 dollars per pound, 2 dots. 2 point 4 dollars per pound, 1 dot. 2 point 5 dollars per pound, 2 dots. 2 point 6 dollars per pound, 2 dots. 2 point 7 dollars per pound, 1 dot. 2 point 8 dollars per pound, 1dot.

    Sample 2

    A dot plot labeled dollars per pound of catfish. The numbers 1.6 through 2.8 in increments of zero point 2 are indicated. The data, titled Sample 2, are as follows: 1 point 6 dollars per pound, 2 dots. 1 point 7 dollars per pound, 6 dots. 1 point 8 dollars per pound, 8 dots. 2 dollars per pound, 4 dots.

    Sample 3

    A dot plot labeled dollars per pound of catfish. The numbers 1.6 through 2.8 in increments of zero point 2 are indicated. The data, titled Sample 3, are as follows: 1 point 8 dollars per pound, 3 dots. 2 dollars per pound, 1 dot. 2 point 2 dollars per pound, 3 dots. 2 point 3 dollars per pound, 1 dot. 2 point 4 dollars per pound, 2 dots. 2 point 5 dollars per pound, 3 dots. 2 point 6 dollars per pound, 4 dots. 2 point 7 dollars per pound, 2 dots. 2 point 8 dollars per pound, 1 dot.

  2. If the goal is to have the sample represent the population, which of the samples would work best? Which wouldn’t work so well? Explain your reasoning.

Are you ready for more?

Problem 1

When doing a statistical study, it is important to keep the goal of the study in mind. Representative samples give us the best information about the distribution of the population as a whole, but sometimes a representative sample won’t work for the goal of a study!

For example, suppose you want to study how discrimination affects people in your town. Surveying a representative sample of people in your town would give information about how the population generally feels, but might miss some smaller groups. Describe a way you might choose a sample of people to address this question.

Activity 3: Auditing Sales

Problem 1

An online shopping company tracks how many items they sell in different categories during each month for a year. Three different auditors each take samples from that data. Use the samples to draw dot plots of what the population data might look like for the furniture and electronics categories.

  1. Auditor 1’s sample

    A dot plot for “monthly sales of furniture online in hundreds.” The numbers 66 through 74 are indicated. The data titled "Auditor ones sample" are as follows: 67 hundred, 1 dot. 70 hundred, 1 dot. 73 hundred, 1 dot.

    Auditor 2’s sample

    A dot plot for “monthly sales of furniture online in hundreds.” The numbers 66 through 74 are indicated. The data titled "Auditor two's sample" are as follows: 70 hundred, 3 dots.

    Auditor 3’s sample

    A dot plot for “monthly sales of furniture online in hundreds.” The numbers 66 through 74 are indicated. The data titlted "Auditor three's sample" are as follows: 71 hundred, 2 dots. 73 hundred, 1 dot.

    Population

    A blank number line for “monthly sales of furniture online in hundreds.” The numbers 66 through 74 are indicated.

  2. Auditor 1’s sample

    A dot plot for “monthly sales of electronics online in thousands.” The numbers 38 through 43 are indicated. The data titled "Auditor ones sample" are as follows: 39 thousand, 1 dot. 41 thousand, 1 dot. 43 thousand, 1 dot.

    Auditor 2’s sample

    A dot plot for “monthly sales of electronics online in thousands.” The numbers 38 through 43 are indicated. The data titled "Auditor two's sample" are as follows: 41 thousand, 1 dot. 43 thousand, 2 dots.

    Auditor 3’s sample

    A dot plot for “monthly sales of electronics online in thousands.” The numbers 38 through 43 are indicated. The data titled "Auditor three's sample" are as follows: 40 thousand, 1 dot. 41 thousand, 1 dot. 43 thousand, 1 dot.

    Population

    A blank number line for “monthly sales of electronics online in thousands.” The numbers 38 through 43 are indicated.

Lesson Summary

A sample that is representative of a population has a distribution that closely resembles the distribution of the population in shape, center, and spread.

For example, consider the distribution of plant heights, in cm, for a population of plants shown in this dot plot. The mean for this population is 4.9 cm, and the MAD is 2.6 cm.

A dot plot for “height in centimeters.” The numbers 1 through 11 are indicated. The data are as follows: 1 centimeter, 5 dots; 2 centimeters, 7 dots; 3 centimeters, 8 dots; 4 centimeters, 8 dots; 5 centimeters, 5 dots; 6 centimeters, 3 dots; 7 centimeters, 2 dots; 8 centimeters, 2 dots; 9 centimeters, 1 dot; 10 centimeters, 3 dots; 11 centimeters, 5 dots.

A representative sample of this population should have a larger peak on the left and a smaller one on the right, like this one. The mean for this sample is 4.9 cm, and the MAD is 2.3 cm.

A dot plot for “height in centimeters.” The numbers 1 through 11 are indicated. The data are as follows: 1 centimeter, 1 dot; 2 centimeters, 2 dots; 3 centimeters, 4 dots; 4 centimeters, 4 dots; 5 centimeters, 2 dots; 6 centimeters, 1 dot; 7 centimeters, 1 dot; 10 centimeters, 1 dot; 11 centimeters, 2 dots.

Here is the distribution for another sample from the same population. This sample has a mean of 5.7 cm and a MAD of 1.5 cm. These are both very different from the population, and the distribution has a very different shape, so it is not a representative sample.

A dot plot for “height in centimeters.” The numbers 1 through 11 are indicated. The data are as follows: 3 centimeters, 1 dot; 4 centimeters, 3 dots; 5 centimeters, 3 dots; 6 centimeters, 2 dots; 7 centimeters, 1 dot; 8 centimeters, 2 dots; 9 centimeters, 1 dot.