AP Statistics Exploring DataDescribing Quantitative Data with Numbers
EdTech 541Angie KruzichSeptember 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 51 0055552 00053 004 00556 00578 5
Key: 4|5 represents a New York worker who reported a 45-minute travel time to work.
M 20 25
222.5 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
2031.25 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 SummaryThe 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 BoxplotsAlso 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
TravelTime0 10 20 30 40 50 60 70 80 90
Collection 5 Box Plot
M = 22.5 Q3= 42.5Q1 = 15Min=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
ResourcesImages• 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 Boxhttp://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 Statehttps://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
ReferenceStarnes, D., Yates, D., & Moore, D. (2011). The practice of statistics.
New York, New York: W.H. Freeman and Company.
Resources