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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?

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?

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

Height in centimetersofDoig’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,

Height in centimetersofDoig’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

Three Useful Featuresof a Set of Data

- The Center
- The Variability
- The Shape

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

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)

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.

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”

A

N

G

E

Measuring Variability125 Highest

120

110 Upper quartile

110 Interquartile

100 Median Range

90

90 Lower quartile

80

75 Lowest

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

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

The Shape

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)

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.)

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

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)

Another Age Stemplot

2|000000000000001111111111111111111111111111111

11111111122222222222222222222222222223333333334

2|56677899

3|01

3|6

4|2

4|57

5|

5|5

6|3

6|5

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

Height in InchesofDoig’s Students

Shape: Symmetric Data Set

Shape: Right-Skewed Data Set

(15-19)

Shape: Left-Skewed Data Set

(15-19)

A

N

G

E

Measuring Variability125 Highest

120

110 Upper quartile

110 Interquartile

100 Median Range

90

90 Lower quartile

80

75 Lowest

Five-Number Summaryof MCO302 Height in Centimeters

- Lowest 140
- First quartile 163
- Median 168
- Third quartile 178
- Highest 208

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

Five-Number Summaryby Gender

Women Men

Lowest 140 147

First quartile 163 174

Median 165 178

Third quartile 170 183

Highest 208 196

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

Age of Death:Five-Number Summary

Presidents Vice-Presidents

Lowest age 46 51

Lower quartile 63 64

Median age 69 70

Upper quartile 78 79

Highest age 93 98

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