Classroom Resource

Statistics: Using Sampling to Count Trees

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This exercise was developed to complement the film The National Parks of Texas by Texas PBS & Villita Media. In this activity, students will learn about estimating the number of trees in a large area based on a smaller area. 

This is one way statisticians measure forests and other wide expanses of land. It's also a great way to illustrate how polling works. Scientists will interview a smaller sample size of Americans, rather than every single American, and then make estimations based on their results. In the same way, we counted smaller samples of trees, rather than all of the trees individually to get an estimate of how many trees are in the park total.

Note: The corresponding lesson plan can be found under the "Support Materials for Teachers" link on the right side of the page.

Mathematics (2019) Grade(s): 09-12 - Geometry with Data Analysis

MA19.GDA.10

Use statistics appropriate to the shape of the data distribution to compare and contrast two or more data sets, utilizing the mean and median for center and the interquartile range and standard deviation for variability.

Unpacked Content

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Vocabulary

  • Center
  • Median
  • Mean
  • Spread
  • Interquartile range
  • Standard deviation
  • Absolute mean deviation

Knowledge

Students know:
  • Techniques to calculate the center and spread of data sets.
  • Techniques to calculate the mean absolute deviation and standard deviation.
  • Methods to compare data sets based on measures of center (median, mean) and spread (interquartile range and standard deviation) of the data sets.

Skills

Students are able to:
  • Accurately find the center (median and mean) and spread (interquartile range and standard deviation) of data sets.
  • -Present viable arguments and critique arguments of others from the comparison of the center and spread of multiple data sets.
  • Explain their reasoning on how standard deviation develops from the mean absolute deviation.

Understanding

Students understand that:
  • Multiple data sets can be compared by making observations about the center and spread of the data.
  • The center and spread of multiple data sets are used to justify comparisons of the data.
  • Both the mean and the median are used to calculate the mean absolute and standard deviations
Mathematics (2019) Grade(s): 09-12 - Geometry with Data Analysis

MA19.GDA.11

Interpret differences in shape, center, and spread in the context of data sets, accounting for possible effects of extreme data points (outliers) on mean and standard deviation.

Unpacked Content

UP:MA19.GDA.11

Vocabulary

  • Outliers
  • Center
  • Shape
  • Spread
  • Mean
  • Standard deviation

Knowledge

Students know:

  • Techniques to calculate the center and spread of data sets.
  • Methods to compare attributes (e.g. shape, median, mean, interquartile range, and standard deviation) of the data sets.
  • Methods to identify outliers.

Skills

Students are able to:

  • Accurately identify differences in shape, center, and spread when comparing two or more data sets.
  • Accurately identify outliers for the mean and standard deviation.
  • Explain, with justification, why there are differences in the shape, center, and spread of data sets.

Understanding

Students understand that:

  • Differences in the shape, center, and spread of data sets can result from various causes, including outliers and clustering.
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.25

From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.

Unpacked Content

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Vocabulary

  • Normal distribution
  • Population Percentages
  • Empirical Rule
  • Normal curve
  • Mean
  • Standard deviation

Knowledge

Students know:
From a normal distribution,
  • Techniques to find the mean and standard deviation of data sets using technology.
  • Techniques to use calculators, spreadsheets, and standard normal distribution tables to estimate areas under the normal curve.

Skills

Students are able to:
  • From a normal distribution, accurately find the mean and standard deviation of data sets using technology.
  • Make reasonable estimates of population percentages from a normal distribution.
  • Read and use normal distribution tables and use calculators and spreadsheets to accurately estimate the areas under a normal curve.

Understanding

Students understand that:
Under appropriate conditions,
  • The mean and standard deviation of a data set can be used to fit the data set to a normal distribution.
  • Population percentages can be estimated by areas under the normal curve using calculators, spreadsheets, and standard normal distribution tables.
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.26

Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Unpacked Content

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Vocabulary

  • Sample surveys
  • Experiments
  • Observational studies
  • Randomization

Knowledge

Students know:
  • Key components of sample surveys, experiments, and observational studies.
  • Procedures for selecting random samples.

Skills

Students are able to:
  • Use key characteristics of sample surveys, experiments, and observational studies to select the appropriate technique for a particular statistical investigation.

Understanding

Students understand that:
  • Sample surveys, experiments, and observational studies may be used to make inferences made about the population.
  • Randomization is used to reduce bias in statistical procedures.
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.32

Produce a sampling distribution by repeatedly selecting samples of the same size from a given population or from a population simulated by bootstrapping (resampling with replacement from an observed sample). Do initial examples by hand, then use technology to generate a large number of samples.

Unpacked Content

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Vocabulary

  • Bootstrapping
  • Population mean
  • Approximately normal
  • Standard deviation
  • Confidence interval

Knowledge

Students know:
  • Techniques for producing a sampling distribution.
  • Properties of a normal distribution.

Skills

Students are able to:
  • Produce a sampling distribution.
  • Reach accurate conclusions regarding the population from the sampling distribution.
  • Accurately create and interpret a confidence interval based on observations from the sampling distribution.

Understanding

Students understand that:
  • The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable's distribution in the population.
  • A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.33

Use data from a randomized experiment to compare two treatments; limit to informal use of simulations to decide if an observed difference in the responses of the two treatment groups is unlikely to have occurred due to randomization alone, thus implying that the difference between the treatment groups is meaningful.

Unpacked Content

UP:MA19.A2.33

Vocabulary

  • Randomized experiment
  • Significant
  • Parameters

Knowledge

Students know:
  • Techniques for conducting randomized experiments.
  • Techniques for conducting simulations of randomized experiment situations.

Skills

Students are able to:
  • Design and conduct randomized experiments with two treatments.
  • Draw conclusions from comparisons of the data of the randomized experiment.
  • Design, conduct, and use the results from simulations of a randomized experiment situation to evaluate the significance of the identified differences.

Understanding

Students understand that:
  • Differences of two treatments can be justified by a significant difference of parameters from a randomized experiment.
  • Statistical analysis and data displays often reveal patterns in data or populations, enabling predictions.
Link to Resource
https://aptv.pbslearningmedia.org/resource/statistics-sampling-count-trees/stat…

CR Resource Type

Audio/Video

Resource Provider

PBS
Accessibility
License

License Type

Custom

Custom License Type (URL)

https://aptv.pbslearningmedia.org/help/terms-of-use/#.YD-bCNhKiUk

Comments or Notes

Note: The corresponding lesson plan can be found under the "Support Materials for Teachers" link on the right side of the page.

Approved Date

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