Looking at One-Variable Data Sets

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Objective

SWBAT graph one-variable data sets and calculate summary statistics like mean, median and standard deviation.

Big Idea

Making graphs and calculating summary statistics helps us describe one-variable data sets.

Warm-Up

15 minutes

Students complete Warm up - Representing One-Variable Data Sets in order to practice making dot-plots, bar-charts and histograms.  My students have learned to make these charts in middle school so a quick review should get them back up to speed.  If not, there are some good free online videos available that I can point them towards.

After the warm-up, we review the previous day's assignment, WS Descriptive Stats Vocabulary.  My goal in assigning this packet was to help them acquire the vocabulary they will need to be able to discuss the statistical concepts in this unit.  Students will have had the opportunity to check their work on Edmodo, so the time we spend going over the homework is really about addressing specific questions they developed in checking their work.

Structured Discussion

30 minutes

When students have completed the warm-up, we will look at the posters from the previous day's presentations. As we review the posters, I explain that our study of statistics will be divided into two parts:

  • one-variable (univariate) statistics 
  • two-variable (bivariate) statistics

I point out which of the presentations involved a single variable and which ones involved two variables.  I let students know that our initial unit will focus on one-variable statistics.

To begin our study of univariate statistics, we review the warm-up work, in which students created a bell-shaped histogram, a left-skewed box plot and a right-skewed dot plot.  Students have made and described these plots in Algebra 1 (S-ID.1), so I anticipate that this will be familiar material for them. We discuss how the plots should look and my students take a few notes on the various ways that the distribution of a single variable can be portrayed (dot plot, box plot, histogram, stem plot) and the shapes that the distributions can take (bell-shaped, skewed, bimodal, uniform).  

Returning to the data used for the Warm up, I ask students to calculate the mean and median of each of the three data sets.  If necessary, I review how to calculate mean and median and provide some notes on this.

We discuss the relationship between the mean and median for symmetric and skewed distributions. We explore the following pattern:

  • for symmetric distributions, mean = median
  • for right skewed distributions mean > median
  • for left-skewed distributions mean < median

 

 

Introduction to Standard Deviation

30 minutes

Standard deviation can be a difficult topic.  The formula is complicated-looking and my students do not always see the need for a measure of dispersion in the first place.  To address this, I present them with a data set, WS Quiz, Test, Project Scores, with three three variables.  The distribution of these data sets (quiz, test and project scores) have the following characteristics:

  • The mean and range of the quiz and project scores are the same
  • The standard deviation of the quizzes is much higher than the standard deviation of the project scores.
  • The mean test score for tests is lower that the quiz scores, but the standard deviation is the same.

Because the quiz and project scores have the same mean and range but a different standard deviation, comparing these distributions makes a nice introduction to standard deviation. 

The procedure for making plots of one-variable distributions on the TI-Nspire graphing calculators is spelled out in WS Quiz, Test, Project Scores.  I have typed the data from the worksheet into a spreadsheet, TI NSpire Data: Test Quiz Project, that I send to them through the Navigator system. I tell my students to follow the steps on the worksheet and answer the questions about the distributions printed on the back. 

When they have completed this work, we come together to discuss it.  Through this discussion, I introduce the concept of standard deviation and the method of computing it.  We work a few examples of calculating standard deviation by hand using variables from our class data set. 

 

Wrap Up and Assignment

15 minutes

To reinforce the big picture of the day's work, I ask students to describe three distributions presented graphically in WS Describing Univariate Data Sets.  I want my students to remember that the calculations and procedures we are working with are all about being able to describe the distribution of values that have a context.

For homework, students will compare the distributions of two variables in WS Comparing Distributions.  In this worksheet, students will compare age distributions of attendees at two events [MP2].