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Statistics lecture 2

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

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Statistics lecture 2

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  1. 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 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,

  6. 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

  7. Three Useful Featuresof 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. R A N G E Measuring Variability 125 Highest 120 110 Upper quartile 110 Interquartile 100 Median Range 90 90 Lower quartile 80 75 Lowest

  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 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 | 0123444 21 | 0111112222222222 22 | 01222 23 | 12

  20. Stemplot of class ages up to 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|000000000000001111111111111111111111111111111 11111111122222222222222222222222222223333333334 2|56677899 3|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 InchesofDoig’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. R A N G E Measuring Variability 125 Highest 120 110 Upper quartile 110 Interquartile 100 Median Range 90 90 Lower quartile 80 75 Lowest

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

  33. Five-Number Summaryof 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 Summaryby Gender Women Men Lowest 140 147 First quartile 163 174 Median 165 178 Third quartile 170 183 Highest 208 196

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

  38. Perguntas?

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