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Section 6A Characterizing a Data Distribution pages 380 - 391

Section 6A Characterizing a Data Distribution pages 380 - 391. Definition -The distribution of a variable (or data set) describes the values taken on by the variable and the frequency (or relative frequency) of these values.

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Section 6A Characterizing a Data Distribution pages 380 - 391

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  1. Section 6ACharacterizing a Data Distribution pages 380 - 391

  2. Definition -The distribution of a variable (or data set) describes the values taken on by the variable and the frequency (or relative frequency) of these values. ex1/381 Eight grocery stores sell the PR energy bar for the following prices: $1.09, $1.29, $1.35, $1.79, $1.49, $1.59, $1.39, $1.29

  3. How do we characterize a data distribution? Average - Mean- Median- Mode- Effect of an Outlier- Confusion Shape of a Distribution - Number of Peaks- Symmetry or Skewness- Variation more in section 6B

  4. What do we mean by AVERAGE? The mean is what we most commonly call the average value. The median is the middle value in the sorted data set (or halfway between the two middle values.) The mode is the most common value (or group of values).

  5. ex1/381 Eight grocery stores sell the PR energy bar for the following prices: $1.09, $1.29, $1.35, $1.79, $1.49, $1.59, $1.39, $1.29 median: $1.09, $1.29, $1.29, $1.35, $1.39, $1.49, $1.59, $1.79 median: $1.37 mode: $1.09, $1.29, $1.29, $1.35, $1.39, $1.49, $1.59, $1.79 mode: $1.29

  6. 17/389 High temperatures (oF) during a 15 day period in Alaska in March: 15, 11, 10, 9, 0, 2, 4, 5, 5, 7, 10, 12, 15, 18, 19 median: 0, 2, 4, 5, 5, 7, 9, 10, 10, 11, 12, 15, 15, 18, 19 median: 10 (oF) mode: 0, 2, 4, 5, 5, 7, 9, 10, 10, 11, 12, 15, 15, 18, 19 modes: 5, 10, 15trimodal

  7. 17/389 High temperatures (oF) during a 15 day period in Alaska in March: 15, 11, 10, 9, 0, 2, 4, 5, 5, 7, 10, 12, 15, 18, 19 Mean – balancing pointMedian – middle pointMode – high point(s)

  8. How do we characterize a data distribution? Average - Mean- Median- Mode- Effect of an Outlier- Confusion Shape of a Distribution - Number of Peaks- Symmetry or Skewness- Variation more in section 6B

  9. median: 0, 0, 0, 0, 3500000 median: $0 mode: 0, 0, 0, 0, 3500000 mode: $0 The Effect of an Outlier Definition: An outlier is a data value that is much higher or much lower than almost all other values. ex/382 Five graduating seniors on a college basketball team receive the following first-year contract offers to play in the National Basketball Association: $0, $0, $0, $0, $3,500,000 ??? Including an outlier can pull the mean significantly upward or downward.Including an outlier does not affect the median.Including an outlier does not affect the mode.

  10. The Effect of an Outlier ex2/383 A track coach wants to determine an appropriate heart rate for her athletes during their workouts. In the middle of the workout, she reads the following heart rates (beats/min) from five athletes: 130, 135, 140, 145, 325, median: 130, 135, 140, 145, 325 median: 140 bpm mode: none _____________________________________________Cleary 325 is an outlier. Clearly 325 is a mistake (faulty heart monitor?) Throw out the outlier? median: 130, 135, 140, 145 median: 137.5 bpm mode: none

  11. How do we characterize a data distribution? Average - Mean- Median- Mode- Effect of an Outlier- Confusion Shape of a Distribution - Number of Peaks- Symmetry or Skewness- Variation more in section 6B

  12. Confusion about “Average” ex3/383 A newspaper surveys wages for assembly workers and reports an average of $22 per hour. The workers at one large firm immediately request a pay raise, claiming that they work as hard as other companies but their average wage is only $19. The management rejects their request, telling them that they are overpaid because their average wage, in fact is $23 per hour. Can they both be right? median: $19 mean: $23 salaries: $19, $19, $19, $19, outlier salaries: $19, $19, $19, $19, $39

  13. Confusion about “Average” ex3/383 A newspaper survey wages for assembly workers and reports an average of $22 per hour. The workers at one large firm immediately request a pay raise, claiming that they work as hard as other companies but their average wage is only $19. The management rejects their request, telling them that they are overpaid because their average wage, in fact is $23 per hour. Can they both be right? median: $23 mean: $19 salaries: outlier, $20, $23, $23, $23 salaries: $6, $20, $23, $23, $23

  14. Confusion about “Average” ex4/383 All 100 first-year students at a small college take three courses in the Core Studies Program. The first two courses are taught in large lectures, with all 100 students in a single class. The third course is taught in ten classes of 10 students each. The students claim that the mean size of their Core Studies classes is 70. The administrators claim that the mean class size is only 25 students. Explain. Students say my average class size is: mean class size per student Administrators say the average Core Studies class size is: mean number of students per class

  15. How do we characterize a data distribution? Average - Mean- Median- Mode- Effect of an Outlier- Confusion Shape of a Distribution - Number of Peaks- Symmetry or Skewness- Variation more in section 6B

  16. Shape of a DistributionUse a smooth curve

  17. Shape of a DistributionNumber of Peaks

  18. How do we characterize a data distribution? Average - Mean- Median- Mode- Effect of an Outlier- Confusion Shape of a Distribution - Number of Peaks- Symmetry or Skewness- Variation more in section 6B

  19. Mode = Mean = Median SYMMETRIC Shape of a DistributionSymmetry and Skewness A distribution is symmetric if its left half is a mirror image of its right half.(note positioning of mean, median, and mode.)

  20. Mean Mode Median SKEWED LEFT (negatively) Shape of a DistributionSymmetry and Skewness A distribution is left-skewed if its values are more spread out on the left (outliers?).(note positioning of mean, median, and mode.)

  21. Mean Mode Median SKEWED RIGHT (positively) Shape of a DistributionSymmetry and Skewness A distribution is right-skewed if its values are more spread out on the right (outliers?).(note positioning of mean, median, and mode.)

  22. Shape of a DistributionSymmetry and Skewness ex6/387 Do you expect the distribution of heights of 100(20) women to be symmetric, left-skewed, or right-skewed? Explain. ex6/387 Do you expect the distribution of speeds of cars on a road where a visible patrol car is using radar to be symmetric, left-skewed, or right skewed. Explain.

  23. How do we characterize a data distribution? Average - Mean- Median- Mode- Effect of an Outlier- Confusion Shape of a Distribution - Number of Peaks- Symmetry or Skewness- Variation more in section 6B

  24. Low variation Moderate variation High variation Shape of a DistributionVariation Variation describes how widely data values are spread out about the center of distribution. ex7/388 How would you expect the variation to differ between times in the Olympic marathon and times in the New York Marathon? Explain.

  25. Shape of a DistributionNumber of Peaks, Symmetry/Skewness, Variation • number of peaks • symmetric, left-skewed, or right-skewed • small or large variation. 27/389The exam scores on a 100-point exam where 50 students got an A, 20 students got a B, and 5 students got a C. ex5/385The heights of all students at Virginia Tech. ex5/385The numbers of people with a particular last digit (0 through 9) in their Social Security Number.

  26. How do we characterize a data distribution? Average - Mean- Median- Mode- Effect of an Outlier- Confusion Shape of a Distribution - Number of Peaks- Symmetry or Skewness- Variation more in section 6B

  27. Homework Pages 388-391 #14,16,18,20, 28,29,30,31

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