Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Once a random sample has been collected, the next step is to use it: estimating the mean word length in a novel, predicting the winner of a class election, or estimating how many fish live in a lake. Seventh graders use data from random samples to draw inferences about a population characteristic they cannot measure directly.
Just as important is asking how far off an estimate might be. If you take several random samples of the same size, each gives a slightly different estimate. Seeing that spread, whether by hand or with a simulation, shows students how much to trust any single estimate. When three samples of 20 words give mean lengths of 4.5, 5.1 and 4.8 letters, a reasonable claim is that the true mean is probably near 4.8 and unlikely to be far outside that range. When estimates from repeated samples bounce around a lot, students learn that larger samples would make predictions steadier.
Students often report a sample statistic as if it were the population value. Comparing several samples shows that each is only an estimate.
When two groups get different results from fair samples, students may assume one is wrong. Variation between random samples is natural and expected.
If 18 of 50 sampled students like a menu, the school estimate is 18/50 of the school, not 18 times the number of samples. Writing the proportion as an equation helps.
Three random samples of 20 words from a novel have mean word lengths of 4.5, 5.1 and 4.8 letters. Estimate the mean word length for the whole book and describe how far off the estimate might be.
Answer: About 4.8 letters per word, give or take roughly 0.3 letters.
Give every pair a different random sample from the same data set, such as words from the same page or names from the same list, and collect their estimates on a class dot plot. The plot of sample means is a powerful picture of sampling variation, and repeating it with larger samples shows the dots tightening.
Test questions commonly ask for a population estimate from a sample proportion, or show several sample results and ask which prediction is most reasonable. Students should practice phrasing conclusions with words like 'about' and 'likely'.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 288
18 out of 50 is 0.36. Then 0.36 × 800 = 288 students.
Answer: C) Candidate A probably has roughly 50% to 60% support
The samples vary, which is normal, and they cluster between about 48% and 60%. A range-based conclusion is the honest one.
Answer: 12.5
Add the estimates: 12 + 15 + 9 + 14 = 50. Divide by 4: 50 ÷ 4 = 12.5.
Answer: A) To see how much the estimates vary from sample to sample
Comparing several samples shows the spread of estimates, which tells you how far a single estimate might be from the truth.
Answer: B) 500
8 out of 40, or 1/5, of the catch is tagged, so the 100 tagged fish are about 1/5 of the lake. 100 × 5 = 500 fish.
Answer: 4.7
Mean = total letters ÷ number of words = 141 ÷ 30 = 4.7 letters.
A full lesson with slides, activities and an exit ticket on making inferences from random samples, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.SP.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn making inferences from random samples into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It is a conclusion about a whole population based on data from a sample, such as estimating the share of voters who support a candidate.
With random number generators, spreadsheets or free statistics applets, students can draw dozens of samples of the same size and plot their results to see the variation.