Alliance Stat Class Understanding Measures of Center and Spread

06/28/10 What is the median? How do you define and how do you find the value? What happens if you have an odd number of data values? What happens if you have an even number of data values? Example: find the median for the following sets of values:

Using the back to back stemplot of the battery data from your assignment Find the median for each battery type. Practice finding the Median

What is the mean? Can you define or tell me what it means without saying the word average?

How do you determine the mean? Versus How do you interpret the mean?

Level A Activity Length of First Name A Conceptual Activity for:Developing an Understanding of the Mean as the “Fair Share” valueDeveloping a Measure of Variation from “Fair Share”

A Statistical Question How long are the first names of students in class?Nine students were asked what was the length of their first name.Each student represented her/his name length with a collection snap cubes.

Snap Cube Representation for Nine Name Lengths

How might we examine the data on the name length for these nine children?

Ordered Snap Cube & Numerical Representations of Nine Name Lengths

Notice that the name lengths vary. What if we used all our name lengths and tried to make all names the same length, in which case there is no variability. How many people would be in each name length?

How can we go about creating these new groups? We might start by separating all the names into one large group.

All 43 letters in the students’ names

Create Nine “New” Groups

Original Question: What if we used all our name lengths and tried to make all names the same length, in which case there is no variability. How many people would be in each name length?

06/28/10 This is the value of the mean number of letters in the 9 students names. What is the formula to find the mean? ∑x = ____ n How does this formula compare with how we found the mean through the idea of fair share?

A New Problem What if the fair share value or the mean number of letters for nine children is 6? What are some different snap cube representations that might produce a fair share value of 6? How many total number of cubes do you need?

Snap Cube Representation of Nine Families, Each of Size 6

Two Examples with Fair Share Value of 6. Which group is “closer” to being “fair?”

How might we quantity “how close” a group of name lengths is to being fair?

Steps to Fair One step occurs when a snap cube is removed from a stack higher than the fair share value and placed on a stack lower than the fair share value.A measure of the degree of fairness in a snap cube distribution is the “number of steps” required to make it fair. Note -- Fewer steps indicates closer to fair

Number of Steps to Make Fair: 8 Number of Steps to Make Fair: 9

Students completing Level A understand: the notion of “fair share” for a set of numeric datathe fair share value is also called the mean valuethe algorithm for finding the meanthe notion of “number of steps” to make fair as a measure of variability about the meanthe fair share/mean value provides a basis for comparison between two groups of numerical data with different sizes (thus can’t use total)

Data About Us Investigation 3: What do we mean by Mean? Data Distributions Investigation 2: Making Sense of Measures of Center 2.1 The mean as an Equal Share 2.2 The Mean as a Balance Point in a Distribution

Level B Activity The Name Length Problem How long are the first names of students in class? A Conceptual Activity for:Developing an Understanding of the Mean as the “Balance Point” of a DistributionDeveloping Measures of Variation about the Mean

Level B Activity How long is your first name? Nine children were asked this question. The following dot plot is one possible result for the nine children:

Do These Distributions Balance? Why?

In which group do the data (name length) vary (differ) more from the mean value of 6?

In Distribution 1, the Total Distance from the Mean is 16. In Distribution 2, the Total Distance from the Mean is 18. Consequently, the data in Distribution 2 differ more from the mean than the data in Distribution 1.

The SAD is defined to be: The Sum of the Absolute Deviations Note the relationship between SAD and Number of Steps to Fair from Level A: SAD = 2xNumber of Steps

Note the total distance for the values below the mean of 6 is 8, the same as the total distance for the values above the mean. Hence the distribution will “balance” at 6 (the mean)‏

An Illustration where the SAD doesn’t work!

Finding SAD Since both points are 4 from the mean SAD = 8 Since all 8 are 1 from the mean SAD = 8 Find the “average” distance from mean 2 data points: SAD/ 2 = 4 8 data points: SAD / 8 = 1 We now have found MAD = Mean Absolute Deviation

Adjusting the SAD for group sizes yields the: MAD = Mean Absolute Deviation

Summary of Level BMean as the balance point of a distributionMean as a “central” pointVarious measures of variation about the mean.

Other measures of variability Range = Highest – Lowest MAD – Mean Absolute Deviation Standard Deviation

XX – X barAbsolute Deviation 22-6 = = – 6 = – 6 = – 6 = – 6 = – 6 = – 6 = 44 Sum of Absolute Deviations = 18 Mean Absolute Deviation (MAD) = 18/9 = 2

Interpret MAD On “average” how far is the data from the mean. Example: Find MAD Find the mean 2.Complete table 3. Find the mean of the absolute deviations XX – x barAbsolute deviations

06/28/10 What is the sum of the x – xbar column? Find the sum of the (x – x bar) column. Divide by 4 MAD = _______ Interpret this value XX – x barAbsolute deviations 3030 – 47 = – 47 = – 47 = – 47= 1616

Another Measure of Spread: Standard Deviation This is the most common measure of Variability

06/28/10 What is the sum of the x – xbar column? Find the sum of the (x – x bar)^2 column. Divide by 4 Take the square root Standard Deviation = _______ Interpret this value XX – x barSquared Deviations 3030 – 47 = – 47 = – 47 = – 47= 16256

Steps to find standard deviation 1.Find the mean 2.Complete the table. 2.Divide by n 3.Take the square root 4.This is the standard deviation What does standard deviation mean? “Average” distance a data point is from the mean. XX – x barSquared Deviations

Practice Measures of Variability Set of Data Length of first and middle names in my family 14, 9, 10, 12 Find and interpret Standard Deviation Ages (x)

Practice Standard Deviation 14, 9, 10, 12 The mean = Find the sum of the squared deviations = Divide by 4 = Square root = 1.92 Ages (x)X – x barSquared deviations 1414– = – = – = – =