Course 3 Unit 4 - Samples and Variation

In Courses 1 and 2 of Core-Plus Mathematics, students learned to analyze data using plots and measures of center and variability. In Course 1 Unit 2, the normal distribution was briefly introduced. Probability distributions were studied in Course 1 Unit 8 and also in Course 2 Unit 8. In Course 3 Unit 1, students extended their ability to reason statistically. They learned how to design a good experiment to compare two different treatments. They also learned how to use randomization to produce a sampling distribution in order to decide if one treatment is more effective than another treatment. Finally, they learned the difference between sample surveys, experiments, and observational studies.
     Course 3 Unit 4, Samples and Variation, provides students with experience in thinking about and working with the variation in measurements. The first and second lessons involve students in thinking about the shape, mean, and standard deviation of normal and binomial distributions and in using the standard deviation as a unit of measure that can be used to describe location in a normal distribution. In the second lesson, they will explore the relationship between the normal and binomial distributions. The third lesson introduces students to control charts, which display samples of measurements taken over time.

Unit Overview
This unit extends student understanding of the measurement of variability, develops student ability to use the normal distribution as a model of variation, introduces students to the binomial distribution and its use in decision making, and introduces students to the probability and statistical inference involved in control charts used in industry for statistical process control. Topics included are the normal distribution, standardized scores, binomial distributions (shape, expected value, standard deviation), a normal approximation to a binomial distribution, odds, statistical process control, control charts, and the Central Limit Theorem.

CPMP-Tools CPMP-Tools has pre-entered data sets that are used in this unit. You will find them under Course 3, Data Analysis, Data, Unit 4 Samples and Variation. Custom tools related to binomial distributions and control charts are available under the Statistics menu. A sample screen showing a histogram of nickel weights with the theoretical normal distribution and the plus and minus 1, 2, and 3 standard deviations from the mean superimposed on the histogram is shown below.

Nickel weights histogram

Objectives of the Unit
  • Understand the standard deviation as a measure of variability and the normal distribution as a model of variability
  • Use the number of standard deviations from the mean (standardized value) as a measure of position
  • Construct binomial distributions and determine the probability of events in binomial situations
  • Use a random sample to decide whether a given proportion p is plausible as the proportion of successes in the population from which the sample came
  • Construct and interpret control charts
  • Understand the Central Limit Theorem and how it is applied to statistical process control

Sample Overview
The sample material for this unit is Investigation 1 of Lesson 1. This lesson introduces students to the normal distribution and its characteristics. In a normal distribution, most values are clustered near the mean with values trailing off at each end of the distribution. Also, the mean and median are the same, and the distribution is symmetric around the mean. In Lessons 2 and 3, students will learn how the normal distribution is used in making decisions.
     Also included in the sample are selected homework tasks for Lesson 1. Note that some of these tasks are ones that would be assigned after Investigations 2 and 3, not the sample investigation. Review Tasks 28 and 31 given in the sample are just-in-time Review tasks for Lesson 2. Review Tasks 26 and 29 are samples of the types of distributed practice built into each Review set.

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View the Unit Table of Contents and Sample Lesson Material
You will need the free Adobe Acrobat Reader software to view and print the sample material.

How the Statistics and Probability Strand Continues
As students finish Course 3 of the Core-Plus Mathematics curriculum, they will have learned a great deal about statistics and probability. Among the fundamental ideas studied were shapes of distributions, measures of center (mean and median), and measures of variation (standard deviation, interquartile range). Interpretation of graphical displays including histograms, box plots, and scatterplots was emphasized.
Other topics studied include correlation and least-squares regression, experimental design and the randomization test, simulation, the geometric distribution, the multiplication and addition rules of probability, and the normal distribution.
     As mentioned in the objectives, this unit includes standardized scores, the binomial distribution, control charts, and the Central Limit Theorem. This knowledge will be useful to them in science labs, in social science courses, and in interpreting the reports of surveys and experiments that they hear and read daily in the media.
     Students who have completed Course 3 are exceptionally well prepared to take Advanced Placement Statistics. The AP Statistics syllabus includes all of the topics listed above except control charts. (See the CPMP Courses 1-4 descriptions.)

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