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2.4 Describing Distributions Numerically – cont.

2.4 Describing Distributions Numerically – cont. Describing Symmetric Data. Symmetric Data Body temp. of 93 adults. Recall: 2 characteristics of a data set to measure. center measures where the “middle” of the data is located variability measures how “spread out” the data is.

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2.4 Describing Distributions Numerically – cont.

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  1. 2.4 Describing Distributions Numerically – cont. Describing Symmetric Data

  2. Symmetric DataBody temp. of 93 adults

  3. Recall: 2 characteristics of a data set to measure • center measures where the “middle” of the data is located • variability measures how “spread out” the data is

  4. Measure of Center When Data Approx. Symmetric • mean (arithmetic mean) • notation

  5. Connection Between Mean and Histogram • A histogram balances when supported at the mean. Mean x = 140.6

  6. Mean: balance pointMedian: 50% area each halfright histo: mean 55.26 yrs, median 57.7yrs

  7. Properties of Mean, Median 1. The mean and median are unique; that is, a data set has only 1 mean and 1 median (the mean and median are not necessarily equal). 2. The mean uses the value of every number in the data set; the median does not.

  8. Example: class pulse rates • 53 64 67 67 70 76 77 77 78 83 84 85 85 89 90 90 90 90 91 96 98 103 140

  9. 2010 n = 845  = $3,297,828 median = $1,330,000 max = $33,000,000 2014 n = 848  = $3,932,912 median = $1,456,250 max = $28,000,000 2010, 2014 baseball salaries

  10. Disadvantage of the mean • Can be greatly influenced by just a few observations that are much greater or much smaller than the rest of the data

  11. Mean, Median, Maximum Baseball Salaries 1985 - 2014

  12. Skewness: comparing the mean, and median • Skewed to the right (positively skewed) • mean>median

  13. Skewed to the left; negatively skewed • Mean < median • mean=78; median=87;

  14. Symmetric data • mean, median approx. equal

  15. Describing Symmetric Data (cont.) • Measure of center for symmetric data: • Measure of variability for symmetric data?

  16. Example • 2 data sets: x1=49, x2=51 x=50 y1=0, y2=100 y=50

  17. 49 51 On average, they’re both comfortable 0 100

  18. Ways to measure variability 1. range=largest-smallest ok sometimes; in general, too crude; sensitive to one large or small obs.

  19. Previous Example

  20. The Sample Standard Deviation, a measure of spread around the mean • Square the deviation of each observation from the mean; find the square root of the “average” of these squared deviations

  21. Calculations … Women height (inches) Mean = 63.4 Sum of squared deviations from mean = 85.2 (n − 1) = 13; (n − 1) is called degrees freedom (df) s2 = variance = 85.2/13 = 6.55 inches squared s = standard deviation = √6.55 =2.56 inches

  22. 1. First calculate the variance s2. We’ll never calculate these by hand, so make sure to know how to get the standard deviation using your calculator, Excel, or other software. Mean ± 1 s.d. 2.Then take the square root to get the standard deviation s.

  23. Population Standard Deviation

  24. Remarks 1. The standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

  25. Remarks (cont.) 2. Note that s and s are always greater than or equal to zero. 3. The larger the value of s (or s ), the greater the spread of the data. When does s=0? When does s =0?

  26. Remarks (cont.) 4. The standard deviation is the most commonly used measure of risk in finance and business • Stocks, Mutual Funds, etc. 5. Variance • s2 sample variance • 2 population variance • Units are squared units of the original data • square $, square gallons ??

  27. Remarks 6):Why divide by n-1 instead of n? • degrees of freedom • each observation has 1 degree of freedom • however, when estimate unknown population parameter like m, you lose 1 degree of freedom

  28. Remarks 6) (cont.):Why divide by n-1 instead of n? Example • Suppose we have 3 numbers whose average is 9 • x1= x2= • then x3 must be • once we selected x1 and x2, x3 was determined since the average was 9 • 3 numbers but only 2 “degrees of freedom”

  29. Computational Example

  30. class pulse rates

  31. Example #1 #2 #3 #4 32 33 38 37 41 35 39 42 44 45 39 45 47 50 40 46 50 52 56 47 53 54 57 48 56 58 58 50 59 59 61 67 68 64 62 68 • x 50 50 50 50 • s 10.6 10.6 10.6 10.6 • m 50 52 56 47

  32. Boxplots: same mean, standard deviation

  33. More Boxplots of the 4 data sets

  34. Review: Properties of s and s • s and s are always greater than or equal to 0 when does s = 0? s = 0? • The larger the value of s (or s), the greater the spread of the data • the standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

  35. Summary of Notation

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