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Chapter Three Averages and Variation

Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College. Chapter Three Averages and Variation. Measures of Central Tendency. Mode Median Mean. The Mode. the value or property that occurs most frequently in the data. Find the mode:.

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Chapter Three Averages and Variation

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  1. Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Lynn SmithGloucester County College Chapter Three Averages and Variation

  2. Measures of Central Tendency • Mode • Median • Mean

  3. The Mode the value or property that occurs most frequently in the data

  4. Find the mode: 6, 7, 2, 3, 4, 6, 2, 6 The mode is 6.

  5. Find the mode: 6, 7, 2, 3, 4, 5, 9, 8 There is no mode for this data.

  6. The Median the central value of an ordered distribution

  7. To find the median of raw data: • Order the data from smallest to largest. • For an odd number of data values, the median is the middle value. • For an even number of data values, the median is found by dividing the sum of the two middle values by two.

  8. Find the median: Data: 5, 2, 7, 1, 4, 3, 2 Rearrange: 1, 2, 2, 3, 4, 5, 7 The median is 3.

  9. Find the median: Data: 31, 57, 12, 22, 43, 50 Rearrange: 12, 22, 31, 43, 50, 57 The median is the average of the middle two values =

  10. The Mean The mean of a collection of data is found by: • summing all the entries • dividing by the number of entries

  11. Find the mean: 6, 7, 2, 3, 4, 5, 2, 8

  12. Sigma Notation The symbol  means “sum the following.” is the Greek letter (capital) sigma.

  13. Sample mean “x bar” Population mean Greek letter (mu) Notations for mean

  14. Number of entries in a set of data • If the data represents a sample, the number of entries = n. • If the data represents an entire population, the number of entries = N.

  15. Sample mean

  16. Population mean

  17. Resistant Measure a measure that is not influenced by extremely high or low data values

  18. Mean Median The mean is less resistant. It can be made arbitrarily large by increasing the size of one value. Which is less resistant?

  19. Trimmed Mean a measure of center that is more resistant than the mean but is still sensitive to specific data values

  20. To calculate a (5 or 10%) trimmed mean • Order the data from smallest to largest. • Delete the bottom 5 or 10% of the data. • Delete the same percent from the top of the data. • Compute the mean of the remaining 80 or 90% of the data.

  21. 15, 17, 18, 20, 20, 25, 30, 32, 36, 60 Delete the top and bottom 10% New data list: 17, 18, 20, 20, 25, 30, 32, 36 10% trimmed mean = Compute a 10% trimmed mean:

  22. Measures of Variation • Range • Standard Deviation • Variance

  23. The Range the difference between the largest and smallest values of a distribution

  24. Find the range: 10, 13, 17, 17, 18 The range = largest minus smallest = 18 minus 10 = 8

  25. The standard deviation a measure of the average variation of the data entries from the mean

  26. Standard deviation of a sample mean of the sample n = sample size

  27. To calculate standard deviation of a sample • Calculate the mean of the sample. • Find the difference between each entry (x) and the mean. These differences will add up to zero. • Square the deviations from the mean. • Sum the squares of the deviations from the mean. • Divide the sum by (n  1) to get the variance. • Take the square root of the variance to get the standard deviation.

  28. The Variance the square of the standard deviation

  29. Variance of a Sample

  30. Find the standard deviation and variance x 30 26 22 4 0 4 16 0 16 ___ Sum = 0 78 32 mean= 26

  31. The variance = 32  2 =16

  32. The standard deviation s =

  33. Find the mean, the standard deviation and variance Find the mean, the standard deviation and variance x 4 5 5 7 4 1 0 0 2 1 1 0 0 4 1 mean = 5 25 6

  34. The mean, the standard deviation and variance Mean = 5

  35. Computation formula for sample standard deviation:

  36. To find Square the x values, then add.

  37. To find Sum the x values, then square.

  38. Use the computing formulas to find s and s2 x 4 5 5 7 4 x2 16 25 25 49 16 n = 5 (Sx) 2 = 25 2 =625 Sx2 = 131 SSx = 131 – 625/5 = 6 s2 = 6/(5 –1) = 1.5 s = 1.22 131 25

  39. Population Mean and Standard Deviation

  40. COEFFICIENT OF VARIATION: a measurement of the relative variability (or consistency) of data

  41. CV is used to compare variability or consistency • A sample of newborn infants had a mean weight of 6.2 pounds with a standard deviation of 1 pound. • A sample of three-month-old children had a mean weight of 10.5 pounds with a standard deviation of 1.5 pounds. • Which (newborns or 3-month-olds) are more variable in weight?

  42. To compare variability, compare Coefficient of Variation For newborns: For 3-month-olds: • Higher CV: more variable CV = 16% CV = 14% Lower CV: more consistent

  43. Use Coefficient of Variation To compare two groups of data, to answer: Which is more consistent? Which is more variable?

  44. CHEBYSHEV'S THEOREM For any set of data and for any number k, greater than one, the proportion of the data that lies within k standard deviations of the mean is at least:

  45. CHEBYSHEV'S THEOREM for k = 2 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 2) standard deviations of the mean? At least of the data falls within 2 standard deviations of the mean.

  46. CHEBYSHEV'S THEOREM for k = 3 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 3) standard deviations of the mean? At least of the data falls within 3 standard deviations of the mean.

  47. CHEBYSHEV'S THEOREM for k =4 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 4) standard deviations of the mean? At least of the data falls within 4 standard deviations of the mean.

  48. Using Chebyshev’s Theorem • A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6. • According to Chebyshev's Theorem, between what two values would at least 75% of the grades be?

  49. Mean = 77 Standard deviation = 6 At least 75% of the grades would be in the interval: 77 – 2(6) to 77 + 2(6) 77 – 12 to 77 + 12 65 to 89

  50. Mean and Standard Deviation of Grouped Data • Make a frequency table • Compute the midpoint (x) for each class. • Count the number of entries in each class (f). • Sum the f values to find n, the total number of entries in the distribution. • Treat each entry of a class as if it falls at the class midpoint.

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