1. AP Statistics Exploring Data Describing Quantitative Data with Numbers EdTech 541 Angie Kruzich September 2014

2. Learning Objectives MEASURE center using mean & median CALCULATE mean DETERMINE median COMPARE mean & median CONSTRUCT a boxplot

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

4. Calculate mean by adding all data values and dividing by number of observations. If the n observations are x 1, x 2, x 3, …, x n, then: Mean Definition

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

6. Measuring Center: The Median Another common measure of center is the median. The median describes the midpoint of a distribution.median

7. Median Definition It is the midpoint of a distribution such that half of the observations are smaller and the other half larger.

8. Finding Median 1.Arrange numbers from smallest to largest. 2.The Median is the number in the middle, unless…

9. Odd versus Even Numbers of Data

10. Interactive Quiz  Obtain an Nspire classroom calculator  Log on  Your teacher will be sending you a document

11. Quiz Measuring Center Calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers. 103052540201015302015208515651560 4045

12. Quiz Measuring Center On page 1.1 finish entering the data in the spreadsheet. Press control right/left arrow to change pages on calculator. 103052540201015302015208515651560 4045

13. Quiz Measuring Center Read the instructions on page 1.2.

14. Quiz Measuring Center On page 1.3 use the calculator page provided to calculate the mean. Watch your formatting!

15. Quiz Measuring Center On page 1.4 and 1.5 enter your final solutions. Press control arrow up when you are done.

16. 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. Quiz Median Solution To calculate the median: 103052540201015302015208515651560 4045

17. Quiz Mean Solution To calculate the mean: 103052540201015302015208515651560 4045

18. 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

19. In a skewed distribution, the mean is usually farther out in the long tail than is the median. Comparing the Mean and the Median

20. 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

21. 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:

22. What are Quartiles?

23. Constructing Boxplots Also known as box-and-whisker plots. The five number summary gives us values to construct a boxplot: Minimum Q 1 M Q 3 Maximum

24. Constructing Boxplots

25. Consider our NY travel times data. In your groups, discuss & construct a boxplot for the data on your Nspires. Constructing Boxplots 103052540201015302015208515651560 4045

26. M = 22.5 Q 3 = 42.5 Q 1 = 15 Min=5 103052540201015302015208515651560 4045 510 15 20 2530 40 4560 6585 Max=85 Constructing Boxplots

27. Summary Mean is average Median is middle How to compare mean and median How to construct a boxplot

28. 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&c ategory_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/

29. 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/lesso n2-5.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