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2.4 Measures of Variation

2.4 Measures of Variation. The Range of a data set is simply: Range = (Max. entry) – (Min. entry). Deviation. The deviation of an entry, x , is the difference between the entry and the mean, , of the data set. Mean = Deviation of x = x - . Population Variance.

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2.4 Measures of Variation

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  1. 2.4 Measures of Variation The Range of a data set is simply: Range = (Max. entry) – (Min. entry)

  2. Deviation • The deviation of an entry, x, is the difference between the entry and the mean, , of the data set. • Mean = • Deviation of x = x - 

  3. Population Variance • We are not going to be talking much about the Population Variance. We will be talking more about the Sample Variance. • Population Variance is found by: • Find the mean of the population  (note the symbol) • Find the deviation of each point by subtracting the mean from each data point • Square the differences • Add all the squares up • Divide by the total number of data points in the population • Population Variance:

  4. Population Standard Deviation • The Population Standard Deviation is the square root of the Population Variance.

  5. Sample Variance • We will be talking mostly about the Sample Variance. • Why? • Sample Variance is found by: • Find the mean of the sample: • Find the deviation by subtracting the mean of the sample from each data point • Square the differences • Add all the squares up • Divide by the total number of data points in the sample minus 1. • Sample Variance:

  6. Sample Standard Deviation • The Sample Standard Deviation is the square root of the Sample Variance.

  7. Example • Find the standard deviation of the following sample:

  8. Example • Find the standard deviation of the following sample:

  9. Example • Find the standard deviation of the following sample:

  10. Example • Find the standard deviation of the following sample:

  11. Example • Find the standard deviation of the following sample: What will be the sum of this column?

  12. Example • Find the standard deviation of the following sample: What will be the sum of this column? It will always be zero

  13. Example • Find the standard deviation of the following sample:

  14. Example • Find the standard deviation of the following sample:

  15. Standard Deviation • The TI calculators can calculate both standard deviations quickly: • Stats • Calc • 1-Var Stats • Enter the list you want to use • Enter

  16. Standard Deviation • This gives: • The mean of the data: • The sum of all of the data: • The sum of the squares of all the data: • Sample standard deviation: • Population standard deviation: • The number of data points: • The smallest data point value: minX • Etc.

  17. Standard Deviation • What does Standard Deviation represent? • It is a measure of the distance from the mean. • It is a measure of how far the data is from the mean. • It is a measure of the spread of data. • The larger the Standard Deviation, the more spread out the data is.

  18. Standard Deviation • Calculate the mean, range, and standard deviations for 8 units at a value of 7: • Mean = 7 • Range = 0 • Population and Sample Standard Deviations = 0, why? • There is no spread in the data. It is all the exact same number

  19. Standard Deviation • Calculate the mean, range, and standard deviations for 4 units each at 6 and 8: • Mean = 7 • Range = 2 • Population Standard deviation = 1, why? • The data is an average of one unit from the mean • Sample Standard Deviation = 1.069, why? • We are dividing by (n-1)

  20. Standard Deviation • Calculate the mean, range, and sample standard deviation for 2 units each at 4, 6, 8 and 10: • Mean = 7 • Range = 6 • Sample Standard deviation = 2.39 and Population Standard Deviation = 2.236, why not 2? • Even though the data is an average of 2 units from the mean, the standard deviations are not exactly 2 because we are working with the square of the distances.

  21. Standard Deviation Summary • Standard deviation is the square root of variance • Population standard deviation has an “n” in the denominator • Sample standard deviation has an “n – 1” in the denominator • Both standard deviations is a measure of the spread of data • The more the spread, the larger the standard deviation

  22. Standard Deviation in a Normal Curvefrom http://allpsych.com/researchmethods/images/normalcurve.gif

  23. Standard Deviation in a Normal Curvefrom http://www.comfsm.fm/~dleeling/statistics/normal_curve_diff_sx.gif

  24. Class Work • Pg 79, # 16, 18, 24

  25. Homework • Page 78, # • 5 – 9 all, • 13 – 21 odd, • 22 • 25 & 26 • Total of 13 problems

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