1. Summarizing and Displaying Measurement Data
Statistics lecture 2
Summarizing and Displaying
Measurement Data

2. Thought Question 1
If a study shows that daily use of a certain expensive exercise machine resulted in an average loss of 10 pounds, what more would you want to know about the numbers than just the average?

3. Thought Question 2
Imagine you wanted to compare the cost of living in two different cities. You get local papers and write down the rental costs of 50 apartments in each place. How would you summarize the values in order to compare the two places?

4. Goals for Lecture 2
Realize that summarizing important features of a list of numbers gives more information than just the unordered list.
Understand the concept of the shape of a set of numbers.
Learn how to make stemplots and histograms
Understand summary measures like the mean and standard deviation

5. Height in centimeters of Doig’s students
170, 163, 178, 163, 168, 165, 170, 155, 191, 178, 175, 185, 183, 165, 165, 180, 185, 165, 168, 152, 178, 183, 157, 165, 183, 157, 170, 168, 163, 165, 180, 163, 140, 163, 163, 163, 165, 178, 150, 170, 165, 165, 157, 165, 173, 160, 163, 165, 178, 173, 180, 196, 185, 175, 160, 168, 193, 173, 183, 165, 163, 175, 168, 160, 208, 157, 180, 170, 155, 173, 178, 170, 157, 163, 163, 180, 170, 165, 170, 170, 180, 168, 155, 175, 168, 147, 191, 178, 173, 170, 178, 185, 152, 170, 175, 178, 163, 175, 175, 165, 175, 175, 157, 163, 165, 160, 178, 152, 160, 170, 170, 160, 157,

6. Height in centimeters of Doig’s students (sorted)
208, 196, 193, 191, 191, 185, 185, 185, 185, 183, 183, 183, 183, 180, 180, 180, 180, 180, 180, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 175, 175, 175, 175, 175, 175, 175, 175, 175, 173, 173, 173, 173, 173, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 168, 168, 168, 168, 168, 168, 168, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 160, 160, 160, 160, 160, 160, 157, 157, 157, 157, 157, 157, 157, 155, 155, 155, 152, 152, 152, 150, 147, 140

7. Three Useful Features of a Set of Data
The Center
The Variability
The Shape

8. The Center
Mean (average): Total of the values, divided by the number of values
Median: The middle value of an ordered list of values
Mode: The most common value
Outliers: Atypical values far from the center

9. Example: 2006 Baseball Salaries
Average: $2,827,104
Median: $950,000
Mode: $327,000 (also the minimum)
Outlier: $21.7 million (Alex Rodriguez of the NY Yankees)

10. The Variability Some measures of variability:
Maximum and minimum: Largest and smallest values
Range: The distance between the largest and smallest values
Quartiles: The medians of each half of the ordered list of values
Standard deviation: Think of it as the average distance of all the values from the mean.

11. What is “normal”? Don’t consider the average to be “normal”
Variability is normal
Anything within about 3 standard deviations of the mean is “normal”

12. Measuring Variability
125 Highest Upper quartile 110 Interquartile 100 Median Range Lower quartile Lowest R A N G E

13. Standard Deviation: How to Compute
Data: 90, 90, 100, 110, 110
Mean: 100
Deviations from mean: -10, -10, 0, 10, 10
Devs squared: 100, 100, 0, 100, 100
Sum of squared devs: 400
Sum of sq devs/(n-1): 400/4=100 (variance)
Square root of variance: 10
Therefore, the standard deviation is 10

14. Standard Deviation: How to Compute
Data: 50, 60, 100, 140, 150
Mean: 100
Deviations from mean: -50, -40, 0, 40, 50
Devs squared: 2500, 1600, 0, 2500, 1600
Sum of squared devs: 8200
Sum of sq devs/(n-1):8200/4=2050 (variance)
Square root of variance: 45.3
Therefore, the standard deviation is 45.3

15. The Shape a stemplot (also called stem-and-leaf) a histogram
The shape of a list of values will tell you important things about how the values are distributed.
To visualize the shape of a list of values, plot them using:
a stemplot (also called stem-and-leaf)
a histogram
or a smooth line (next lecture)

16. How to Make a Stemplot
Divide the range into equal units, so that the first few digits can be used as the stems. (Ideally, 6-15 stems.)
Attach a leaf, made of the next digit, to represent each data point. (Ignore any remaining digits.)

17. How to Make a Stemplot Ages in years:
42.2, 22.7, 21.2, 65.4, 29.3, 22.3, 21.5, 20.7, 29.4, 23.1, 22.9, 21.5, 21.4, 21.3, 21.3, 21.2, 21.2, 21.1, 20.8, 30.2, 25.7, 24.5, 23.2, 22.3, 22.2, 22.2, 22.2, 22.1, 21.9, 21.8, 21.7, 21.7, 21.6, 21.4, 21.3, 21.2, 21.2, 21.2, 21.2, 21.2, 21.1, 21.1, 20.8, 20.7, 20.7, 20.1, 20.0, 19.5, 35.8, 26.1, 22.3, 22.2, 21.8, 21.5, 20.4, 47.5, 45.5, 30.6, 28.1, 27.4, 26.5, 24.1, 23.3, 23.3, 22.9, 22.9, 22.6, 22.4, 22.4, 22.3, 22.3, 22.0, 21.9, 21.9, 21.8, 21.7, 21.7, 21.7, 21.6, 21.6, 21.6, 21.5, 21.5, 21.5, 21.4, 21.2, 21.2, 21.2, 21.1, 21.1, 21.0, 20.9, 20.9, 20.8, 20.8, 20.8, 20.8, 20.8, 20.6, 20.6, 20.6, 20.5, 20.5, 20.5, 20.5, 20.4, 20.4, 20.3, 20.2, 19.9, 19.6, 63.2, 55.0

18. How to Make a Stemplot
19 | 20 | 21 | 22 | 23 |

19. How to Make a Stemplot
19 | 5 20 | | | | 12

20. Stemplot of class ages up to 3019 |
569 20 21 22 23
1233 24 15 25 7 26 27 4 28 1 29 34 30

21. Another Age Stemplot (Each Stem = 5 Years)
2| (20-24) 2| (25-29) 3| (30-34) 3| (35-39) 4| (40-44) 4| (45-49) 5| (50-54)

22. Another Age Stemplot
2| | |01 3|6 4|2 4|57 5| 5|5 6|3 6|5

23. Histogram Shows the shape of a set of values, similar to a stemplot
More useful for large data sets because you don’t have to enter every value
X-axis: Range of possible values
Y-axis: The count of each possible value

24. Height in Inches of Doig’s Students

25. Shape: Symmetric Data Set

26. Shape: Right-Skewed Data Set
(15-19)

27. Shape: Left-Skewed Data Set
(15-19)

28. Unimodal or Bimodal?

29. Bimodal! But why?

30. Height by Gender

31. Measuring Variability
125 Highest Upper quartile 110 Interquartile 100 Median Range Lower quartile Lowest R A N G E

32. Median Lower quartile Upper quartile Lowest value Highest value
Five-Number Summary
Median Lower quartile Upper quartile Lowest value Highest value

33. Five-Number Summary of MCO302 Height in Centimeters
Lowest 140
First quartile 163
Median 168
Third quartile 178
Highest 208

34. Heights
Women: 140, 150, 152, 152, 155, 155, 155, 157, 157, 157, 157, 157, 157, 157, 160, 160, 160, 160, 160, 160, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 168, 168, 168, 168, 168, 168, 168, 168, 168, 168, 168, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 173, 173, 173, 173, 175, 175, 175, 175, 175, 175, 178, 178, 180, 180, 180, 208
Men: 147, 152, 163, 165, 168, 170, 170, 170, 173, 175, 175, 175, 178, 178, 178, 178, 178, 178, 178, 178, 180, 180, 180, 183, 183, 183, 183, 185, 185, 185, 185, 191, 191, 193, 196

35. Five-Number Summary by Gender
Women Men Lowest First quartile Median Third quartile Highest

36. Ages of Death
Presidents: 67, 90, 83, 85, 73, 80, 78, 79, 68, 71, 53, 65, 74, 64, 77, 56, 66, 63, 70, 49, 56, 71, 67, 71, 58, 60, 72, 67, 57, 60, 90, 63, 88, 78, 46, 64, 81, 93
Vice-Presidents: 90, 83, 80, 73, 70, 51, 68, 79, 70, 71, 72, 74, 67, 54, 81, 66, 62, 63, 68, 57, 66, 96, 78, 55, 60, 66, 57, 71, 60, 85, 76, 8, 77, 88, 78, 81, 64, 66, 70

37. Age of Death: Five-Number Summary
Presidents Vice-Presidents Lowest age Lower quartile Median age Upper quartile Highest age 93 98

38. Perguntas?