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Averages and Variability

Averages and Variability. FSE 200. Outline. Measures of Central Tendency Mean Median Mode Descriptive Statistics Range Standard Deviation Variance. Salkind , Chapter 2. Computing and Understanding averages. Example Data Set.

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Averages and Variability

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  1. Averages and Variability FSE 200

  2. Outline • Measures of Central Tendency • Mean • Median • Mode • Descriptive Statistics • Range • Standard Deviation • Variance

  3. Salkind, Chapter 2 Computing and Understanding averages

  4. Example Data Set The following are the number of calls ran per year for Anywhere Fire Department.

  5. Measures of Central Tendency The AVERAGE is a single score that represents a set of scores Averages are also known as “Measures of Central Tendency” Three different ways to describe the distribution of a set of scores… • Mean – typical average score • Median – middle score • Mode – most common score

  6. Computing the Mean Formula for computing the mean “X bar” is the mean value of the group of scores “” (sigma) tells you to add whatever follows it X is each individual score in the group The n is the sample size

  7. Computing the Mean Example 5 students scored the following on their quizzes: 79, 83, 65, 98, and 86 The average (X-bar) is the sum of the scores (ΣX) divided by the number of students (n) The average quiz score for this group of students was 82.2

  8. Using the AVERAGE function Select the cell for the AVERAGE function Create a formula to average the three values • =(A1+A2+A3)/3 OR type the AVERAGE function • =AVERAGE(A1:A3)

  9. More Excel Arithmetic Mean Sum of the deviation is equal to zero Geometric Mean GEOMEAN uses multiplication instead of addition Moving Mean More accurate…good for unique distributions Weighted Mean Accounts for the frequency of a score’s occurrence

  10. Weighted Mean Example Using Excel to Compute a Weighted Mean

  11. Weighted Mean Example The Computation of a Weighted Mean

  12. Computing the Median Median = point/score at which 50% of scores fall above and 50% fall below No standard formula • Rank order scores from highest to lowest or lowest to highest • Find the “middle” score BUT… • What if there are two middle scores? • What if the two middle scores are the same?

  13. Using the MEDIAN function • Select the cell and type the MEDIAN function • =MEDIAN(A2:A7)

  14. Computing the Mode Mode = most frequently occurring score No formula • List all values in the distribution • Tally the number of times each value occurs • The value occurring the most is the mode Democrats = 90 Republicans = 70 Independents = 140 – the MODE!! • When two values occur the same number of times -- Bimodal distribution

  15. Using the MODE function =MODE(A2:A20)

  16. Descriptive Statistics Toolpak The Descriptive Statistics Dialog Box

  17. Descriptive Statistics Toolpak The New and Improved Descriptive Statistics Output

  18. Salkind, Chapter 3 Descriptive Statistics

  19. Why Variability Is Important • Variability is how different the scores are from one particular score • Spread • Dispersion • What is the score of interest here? • The MEAN!! So…variability is really a measure of how each score in a group of scores differs from the mean of that set of scores.

  20. Measures of Variability • Three types of variability examine the amount of spread or dispersion in a group of scores • Range • Standard Deviation • Variance • Typically report the average and the variability together to describe a distribution

  21. Computing the Range • Range is the most general estimate of variability • Two types: • Exclusive Range • R = h - l • Inclusive Range • R = h – l + 1

  22. Computing Standard Deviation • Standard deviation (SD) is the most frequently reported measure of variability • SD = average amount of variability in a set of scores

  23. Using Excel’s STDEV Function Data for the STDEV Function

  24. Using Excel’s STDEV Function Using the STDEV Function

  25. Why n – 1? • The standard deviation is intended to be an estimate of the POPULATION standard deviation • We want it to be an unbiased estimate • Subtracting 1 from n artificially inflates the SD, making it larger • In other words, we want to be conservative in our estimate of the population

  26. Why n – 1? Comparing the STDEV and STDEVP Functions

  27. Things to Remember… • Standard deviation is computed as the average distance from the mean • The larger the standard deviation, the greater the variability • Like the mean, standard deviation is sensitive to extreme scores • If s = 0, then there is no variability among scores; they must all be the same value

  28. Computing Variance • Variance = standard deviation squared • So…what do these symbols represent? Does the formula look familiar?

  29. Using Excel’s VAR Function Computing the Variance

  30. Standard Deviation or Variance • Although the formulas are quite similar, the two are also quite different • Standard deviation is stated in original units • Variance is stated in units that are squared • Which do you think is easier to interpret???

  31. Acknowledgement The majority of the content of these slides were from the Sage Instructor Resources Website

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