Unit 8, Lesson 17
More about Sampling Variability (Optional)
Grade 7
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To Copy (from Blackline Masters)
Reaction Times Cutouts, Spanish
Students calculate the mean of a sample collected in an earlier lesson to compare with their partners. Students experience firsthand that different samples from the same population can produce different results. In later activities students will use the data they have collected here to develop a deeper understanding of sampling variability.
Arrange students in groups of 2 so that different partners are used from the ones used in the earlier activity analyzing reaction times of 12th graders for a track meet.
Remind students that the numbers came from a survey of all 120 seniors from a certain school. The numbers represent their reaction time in seconds during an activity in which they clicked a button as soon as they noticed that a square changed color. Those 120 values are the population for this activity.
Give students 2 minutes of quiet work time followed by partner work time. Follow with a whole-class discussion.
Earlier, you worked with the reaction times of twelfth graders to see if they were fast enough to help out at the track meet. Look back at the sample you collected.
The purpose of the discussion is for students to think about how the data they collected relates to the population data.
Some questions for discussion:
MLR8: Discussion Supports
To Gather
Sticky notes
In this activity, a dot plot is created by the class that includes all of the calculated sample means. Students then compare that display to the data from the entire population to better understand the information that can be gained from a sample mean (MP7). In the discussion that follows the activity, students look at similar displays of sample means for samples of different sizes (MP8). Students should see that larger samples tend to more accurately estimate the population mean than smaller samples.
Display the basis for a dot plot of sample means for all to see.
Allow students 5 minutes to create the class dot plot and complete the first set of questions.
Ask students to include their sample means from the previous task to the display for all to see (since the samples originated in pairs, there will be many repeats. It is OK to include the repeats or ensure only 1 of the original partners includes their point on the display). They may do this by adding a dot to the display on the board, placing a sticky note in the correct place, or passing a large sheet of paper around the class.
After students have had a chance to answer the first set of questions, pause the class to display the dot plot of the the reaction times for the entire population of 120 seniors for all to see.
Provide students 5 minutes to complete the problems. Follow with a whole-class discussion.
Your teacher will display a blank dot plot.
Plot your sample mean from the previous activity on your teacher's dot plot.
What do you notice about the distribution of the sample means from the class?
Where is the center?
Is there a lot of variability?
Is it approximately symmetric?
The population mean is 0.445 seconds. How does this value compare to the sample means from the class?
Pause here so your teacher can display a dot plot of the reaction times of the population.
What do you notice about the distribution of the population?
Where is the center?
Is there a lot of variability?
Is it approximately symmetric?
Compare the two displayed dot plots.
Based on the distribution of sample means from the class, do you think the mean of a random sample of 20 items is likely to be:
Explain or show your reasoning.
Display the dot plot of sample means for all to see throughout the unit. It may be helpful to refer to this display when viewing dot plots of sample means in later lessons.
The purpose of the discussion is for students to understand that the centers of the two dot plots in the activity are close, if not the same, but the sample means have less variability, and the shapes of the population distribution and the sample mean distribution are probably different. Additionally, students should use the dot plots shown here to see that larger samples have sample means that are still centered around the population mean, but have less variability than the means from smaller samples. Therefore, larger samples should provide better estimates of the population mean than smaller samples do.
Items for further discussion:
For a different set of data representing the wingspan in centimeters of a certain species of bird, compare these dot plots.
Here is a dot plot of the population data.
Here is a dot plot of the sample means for 100 different random samples each of size 10.
Here is a dot plot of the sample means for 100 different random samples each of size 30.
“What do you notice about the distribution of the sample means as the sample size increases?” (The variability decreases, but the center stays in about the same place.)
None
In this activity, students continue to look at how variability in the sample means can be used to think about the accuracy of an estimate of a population mean. If the means from samples tend to be very spread out, then there is reason to believe that the mean from a single sample may not be especially close to the mean for the population. If the means from samples tend to be tightly grouped, then there is reason to believe that the mean from a single sample is a good estimate of the mean for the population (MP7).
Arrange students in groups of 2.
Remind students about the context of earlier lessons in which they examined ages of people watching various shows.
Tell students that, although the dot plots seem to have a similar shape, attention should be given to the scale of the horizontal axis.
Allow students 3–5 minutes of quiet work time followed by partner discussion. Then follow with a whole-class discussion.
Here are the mean ages for 100 different samples of viewers from each show.
A dot plot for “sample means for Trivia the Game Show” with the numbers 40 through 70, in increments of 2, is indicated. The data are as follows: 44 years old, 2 dots. 44 point 5 years old, 1 dot. 47 years old, 2 dots. 48 years old, 1 dot. 50 point 5 years old, 1 dot. 51 years old, 1 dot. 52 years old, 4 dots. 52 point 5 years old, 4 dots. 53 years old, 3 dots. 54 years old, 1 dot. 54 point 5 years old, 2 dots. 55 years old, 2 dots. 55 point 5 years old, 3 dots. 56 years old, 6 dots. 56 point 5 years old, 2 dots. 57 years old, 6 dots. 57 point 5 years old, 1 dot. 58 years old, 6 dots. 58 point 5 years old, 7 dots. 59 years old, 9 dots. 59 point 5 years old, 2 dots. 60 years old, 4 dots. 60 point 5 years old, 2 dots. 61 years old, 4 dots. 61 point 5 years old, 3 dots. 62 years old, 2 dots. 63 years old, 3 dots. 63 point 5 years old, 1 dot. 64 years old, 1 dot. 64 point 5 years old, 2 dots. 65 years old, 1 dot. 65 point 5 years old, 1 dot. 66 years old, 2 dots. 66 point 5 years old, 1 dot. 67 point 5 years old, 1 dot. 68 years old, 1 dot.
A dot plot for “sample means for Science Experiments YOU Can Do” with the numbers 9 through 14, in increments of zero point 5, indicated. The data are as follows: 9 point 8 years old, 1 dot. 10 point 2 years old, 1 dot. 10 point 3 years old, 1 dot. 10 point 4 years old, 1 dot. 10 point 6 years old, 1 dot. 10 point 7 years old, 2 dots. 10 point 8 years old, 1 dot. 10 point 9 years old, 3 dots. 11 years old, 1 dot. 11 point 1 years old, 1 dot. 11 point 2 years old, 2 dots. 11 point 3 years old, 1 dot. 11 point 4 years old, 3 dots. 11 point 5 years old, 1 dot. 11 point 6 years old, 5 dots. 11 point 7 years old, 5 dots. 11 point 8 years old, 4 dots. 11 point 9 years old, 8 dots. 12 years old, 6 dots. 12 point 1 years old, 7 dots. 12 point 2 years old, 3 dots. 12 point 3 years old, 4 dots. 12 point 4 years old, 9 dots. 12 point 5 years old, 2 dots. 12 point 6 years old, 8 dots. 12 point 7 years old, 3 dots. 12 point 8 years old, 3 dots. 12 point 9 years old, 5 dots. 13 years old, 1 dot. 13 point 1 years old, 1 dot. 13 point 3 years old, 1 dot. 13 point 4 years old, 1 dot. 13 point 5 years old, 1 dot. 13 point 6 years old, 1 dot. 13 point 8 years old, 2 dots.
A dot plot for “sample means for Learning to Read” with the numbers 5 through 7, in increments of zero point 2, indicated. The data are as follows: 5 years old, 1 dot. 5 point 1 years old, 1 dot. 5 point 2 years old, 1 dot. 5 point 3 years old, 4 dots. 5 point 4 years old, 4 dots. 5 point 5 years old, 6 dots. 5 point 6 years old, 8 dots. 5 point 7 years old, 5 dots. 5 point 8 years old, 10 dots. 5 point 9 years old, 10 dots. 6 years old, 10 dots. 6 point 1 years old, 10 dots. 6 point 2 years old, 9 dots. 6 point 3 years old, 4 dots. 6 point 4 years old, 8 dots. 6 point 5 years old, 4 dots. 6 point 6 years old, 2 dots. 6 point 7 years old, 2 dots. 6 point 8 years old, 1 dot.
For each show, use the dot plot to estimate the population mean.
Trivia the Game Show
Science Experiments YOU Can Do
Learning to Read
The purpose of the discussion is to think about what the information in the given dot plots tells us about the accuracy of an estimate of a population mean based on the sample mean from a single sample.
Consider asking these questions for discussion:
A population of hummingbirds has a mean weight of 11.6 grams. This dot plot shows the weights of 18 hummingbirds selected from the population that have the same mean weight as the population. The mean weight is indicated with a triangle.
<p>A dot plot labeled “hummingbird weights in grams.” The numbers 8 through 16 are indicated. A triangle at approximately 11.6 is indicated. The data are as follows: 8 grams, 1 dot; 9 grams, 1 dot; 10 grams, 2 dots; 11 grams, 2 dots; 12 grams, 4 dots; 13 grams, 3 dots; 14 grams, 3 dots; 15 grams, 1 dot.</p>
20 more samples, each with 5 hummingbirds, are selected from the original population. For each sample, the mean of the 5 birds is calculated and plotted on this dot plot. The triangle represents the population mean of 11.6 grams. Notice that the sample means are fairly close to the population mean, but that they are not exactly the same.
<p>A dot plot for "means of samples of size 5" with the numbers 8 through 16 indicated. The data are as follows: 10, 1 dot. 10 point 3, 3 dots. 10 point 9, 1 dot. 11, 3 dots. 11 point 2, 1 dot. 11 point 4, 2 dots. 11 point 6, 3 dots. 11 point 8, 1 dot. 12, 2 dots. 12 point 2, 2 dots. 12 point 8, 1 dot. There is a triangle located at 11.6.</p>
What is different if more hummingbirds are included in the samples? This dot plot shows the means of 20 samples of 10 hummingbirds, selected at random. Notice that the means for these samples are generally better estimates for the population mean because they tend to be even closer to the mean for the entire population.
<p>A dot plot labeled “means of samples of size 10.” The numbers 8 through 16 are indicated. The data are as follows: 10 point 9, 1 dot. 11, 1 dot. 11 point 1, 1 dot. 11 point 2, 1 dot. 11 point 3, 2 dots. 11 point 4, 2 dots. 11 point 5, 1 dot. 11 point 6, 3 dots. 11 point 7, 2 dots. 11 point 8, 1 dot. 11 point 9, 1 dot. 12 point 1, 1 dot. 12 point 3, 2 dots. 12 point 5, 1 dot. A triangle is located at 11 point 6.</p>
In general, as the sample size gets bigger, the mean of a sample is more likely to be closer to the mean of the population.
Help us improve by sharing suggestions or reporting issues.
The center of the dot plot created by the class can be thought of as a mean of sample means. The phrase can be difficult for students to think through, so remind them what the dot they added to the dot plot represents and how they calculated that value. Consider displaying the class dot plot for the rest of the unit to help students remember this example to understand similar dot plots.
Students may get confused about what the dot plots represent. Refer to the class dot plot of sample means from the reaction time activity to help them understand that a single dot on the dot plot represents a mean from a single sample. Each dot plot shows means from several different samples.