Summarizing and DisplayingMeasurement Data

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?

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?

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

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,

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

The CenterThe VariabilityThe Shape

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

Average: $2,827,104 Median: $950,000 Mode: $327,000 (also the minimum) Outlier: $21.7 million (Alex Rodriguez of the

NY Yankees)

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.

Don’t consider the average to be “normal” Variability is normal Anything within about 3 standard deviations

of the mean is “normal”

125 Highest120110 Upper quartile110

Interquartile100 Median Range 90 90 Lower quartile 80 75 Lowest

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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

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 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)

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.)

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

19 |20 |21 |22 |23 |

19 | 520 | 012344421 | 011111222222222222 | 0122223 | 12

19 | 569

20 | 01234445555666777888888899

21 | 011111222222222223334445555556666777778889999

22 | 012222333334467999

23 | 1233

24 | 15

25 | 7

26 | 15

27 | 4

28 | 1

29 | 34

30 | 26

2| (20-24)2| (25-29)3| (30-34)3| (35-39)4| (40-44)4| (45-49)5| (50-54)

2|000000000000001111111111111111111111111111111111111111222222222222222222222222222233333333342|566778993|013|64|24|575|5|56|36|5

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

Total

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55 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

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(15-19)

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55 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

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55 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

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55 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 75 76 77 89

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Count of Q3

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Q3

125 Highest120110 Upper quartile110

Interquartile100 Median Range 90 90 Lower quartile 80 75 Lowest

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Median

Lower quartile Upper quartile

Lowest value Highest value

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

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

Women MenLowest 140 147First quartile 163 174Median 165 178Third quartile 170 183Highest 208 196

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

Presidents Vice-PresidentsLowest age 46 51Lower quartile 63 64Median age 69 70Upper quartile 78 79Highest age 93 98