AP Statistics Exploring Data Describing Quantitative Data with Numbers

EdTech 541 Angie Kruzich September 2014

Learning Objectives

MEASURE center using mean & median

CALCULATE mean

DETERMINE median

COMPARE mean & median

CONSTRUCT a boxplot

Measuring Center: The Mean

The most common measure of center is the ordinary arithmetic average, or mean, , (pronounced “x-bar”).

x

Calculate mean by adding all data values and dividing by number of observations.

If the n observations are x1, x2, x3, …, xn, then:

x sum of observations

n

x1 x2 ... xn

n

Mean Definition

In mathematics, the capital Greek letter Σ (sigma) is short for “add them all up.” Therefore, the mean formula can also be written:

x xi

n

More Mean

Measuring Center: The Median Another common measure of center is

the median. The median describes the midpoint of a distribution.

Median Definition

It is the midpoint of a distribution such

that half of the observations are smaller and the other half larger.

Finding Median

1. Arrange numbers from smallest to largest.

2. The Median is the number in the middle, unless…

Odd versus Even Numbers of Data

Interactive Quiz

Obtain an Nspire classroom calculator

Log on

Your teacher will be sending you a document

Quiz Measuring Center

Calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers.

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

Quiz Measuring Center

On page 1.1 finish entering the data in the spreadsheet.

Press control right/left

arrow to change pages on calculator.

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

Quiz Measuring Center

Read the instructions on page 1.2.

Quiz Measuring Center

On page 1.3 use the calculator page provided to calculate the mean.

Watch your formatting!

Quiz Measuring Center

On page 1.4 and 1.5 enter your final solutions.

Press control arrow up when you are done.

0 5

1 005555

2 0005

3 00

4 005

5

6 005

7

8 5

Key: 4|5

represents a

New York

worker who

reported a 45-

minute travel

time to work.

M 20 25

2 22.5 minutes

Quiz Median Solution

To calculate the median:

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

Quiz Mean Solution

To calculate the mean:

x 10 30 5 25 ... 40 45

20 31.25 minutes

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

If mean and median are close together, then distribution is roughly symmetric.

If mean and median are exactly the same, distribution is exactly symmetric.

Comparing the Mean and the Median

In a skewed distribution, the mean is usually farther out in the long tail than is the median.

Comparing the Mean and the Median

The mean and median measure center in different ways.

Don’t confuse the “average” value of a variable with its “typical” value.

Comparing the Mean and the Median

The Five Number Summary

The mean and median tell us little about the tails of a distribution.

The five-number summary of a distribution consists of:

What are Quartiles?

Constructing Boxplots Also known as box-and-whisker plots. The five number summary gives us values to

construct a boxplot: Minimum Q1 M Q3 Maximum

Constructing Boxplots

Consider our NY travel times data. In your groups, discuss & construct a

boxplot for the data on your Nspires.

Constructing Boxplots

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

M = 22.5 Q3= 42.5 Q1 = 15 Min=5

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85

Max=85

Constructing Boxplots

Summary

• Mean is average

• Median is middle

• How to compare mean and median

• How to construct a boxplot

Resources Images • Slide 3 – Courtesy of Math is Fun

http://www.mathsisfun.com/definitions/mean.html

• Slide 6 – Courtesy of W3.org

http://www.w3.org/2013/11/w3c-highlights/

• Slide 7 – Courtesy of Knowledge Center

http://knowledgecenter.csg.org/kc/content/stats-101-mean-versus-median

• Slide 8 – Courtesy of Sparkle Box

http://www.sparklebox.co.uk/6771-6780/sb6779.html#.VCigcRaK18E

• Slide 10 – Courtesy of Underwood Distributing

http://www.underwooddistributing.com/shop/shop?page=shop.browse&category_id=109

• Slide 11 – Courtesy of Streetsblog USA

http://usa.streetsblog.org/2008/01/10/does-times-square-have-too-many-people-or-just-too-many-cars/

Images • Slide 18 – Courtesy of Profit of Education

http://profitofeducation.org/?p=2152

• Slide 19 and 20 – Courtesy of Data Analysis for Instructional Leaders

https://www.floridaschoolleaders.org/general/content/NEFEC/dafil/lesson2-5.htm

• Slide 21 – Courtesy of Penn State

https://onlinecourses.science.psu.edu/stat100/node/11

• Slide 23 and 25 – Courtesy of GCSE Math Notes

http://astarmathsandphysics.com/gcse-maths-notes/gcse-maths-notes-five-figure-summaries-and-boxplots.html

Reference Starnes, D., Yates, D., & Moore, D. (2011). The practice of statistics. New York,

New York: W.H. Freeman and Company.

Resources