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Chapter Three

Chapter Three. Averages and Variation. Measures of Central Tendency. Mode Median Mean. The Mode. the value that occurs most frequently in a data set. Find the mode:. 6, 7, 2, 3, 4, 6, 2, 6. The mode is 6. Find the mode:. There is no mode for this data. 6, 7, 2, 3, 4, 5, 9, 8.

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Chapter Three

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  1. Chapter Three Averages and Variation

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

  3. The Mode • the value that occurs most frequently in a data set

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

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

  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 values pick the middle value. or • For an even number of values compute the average of the middle two values

  8. Find the median: Rearrange: 1, 2, 2, 3, 4, 5, 7 Data: 5, 2, 7, 1, 4, 3, 2 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. Finding the median for a large data set For an ordered data set of n values: Position of the middle value =

  11. The Mean • An average that uses the exact value of each entry • Sometimes called the arithmetic mean

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

  13. Find the Mean: 6, 7, 2, 3, 4, 5, 2, 8

  14. Sigma Notation • The symbol S means “sum the following.” • Sis the Greek letter (capital) sigma.

  15. Sample mean Population mean Notations for mean Greek letter (mu)

  16. 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.

  17. Sample mean

  18. Population mean

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

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

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

  22. 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.

  23. 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:

  24. Weighted Average • An average where more importance or weight is assigned to some of the numbers

  25. Weighted Average If x is a data value and w is the weight assigned to that value Weighted average =

  26. Calculating a Weighted Average In a pageant, the interview is worth 30% and appearance is worth 70%. Find the weighted average for a contestant with an interview score of 90 and an appearance score of 80.

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

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

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

  30. The Standard Deviation • a measure of the average variation of the data entries from the mean

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

  32. 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.

  33. The Variance • the square of the standard deviation

  34. Variance of a Sample

  35. Find the standard deviation and variance x 30 26 22 4 0 -4 16 0 16 ___ Sum = 0 78 32 Mean = 26

  36. The variance = 32 ¸ 2 =16

  37. The standard deviation s =

  38. 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

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

  40. Computation Formulas for Sample Variance and Standard Deviation:

  41. To find Sx2 • Square the x values, then add.

  42. To find (Sx) 2 Sum the x values, then square.

  43. Use the computing formulas to find s and s2 x 4 5 5 7 4 x2 16 25 25 49 16 131 25

  44. Population Mean

  45. Population Standard Deviation

  46. Coefficient Of Variation: • A measurement of the relative variability (or consistency) of data.

  47. 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 pound. • Which (newborns or 3-month-olds) are more variable in weight?

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

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

  50. 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:

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