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Measures of Variability

Measures of Variability

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Measures of Variability

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  1. Measures of Variability Overheads 4

  2. Quiz:Week 4 Understanding Check The quiz is located under “Course Materials” You may want to stop the video and complete the quiz now. You are NOT required to submit this quiz.

  3. Quiz: Question 1 1. What is the independent variable in Problem 1? • Type A and B Blood • Survival Rate • 80% chance of survival • Blood Type

  4. Quiz: Question 2 2. What is the dependent variable in Problem 1? • Blood Type • Survival Rate • 80% chance of survival • Type A and B Blood

  5. Quiz: Question 3 3. Which of the following would be most appropriately measured on an ordinal scale? • Type of Religion (Catholic, Protestant, Jewish…) • Gender (Male, Female) • Academic Ranking (Freshman, Sophomore, Junior, Senior) • Hair Color (Blonde, Brunette, Redhead…)

  6. Quiz: Question 4 4. For Data 1, what is the mode of Group 1? • 3.00 • 4.00 • 5.00 • 6.00

  7. Quiz: Question 5 4. For Data 1, what is the median of the entire data set? • 3.00 • 4.00 • 5.00 • 6.00

  8. Quiz: Question 6 4. What is the mean of Group 2? • 3.40 • 4.60 • 4.00 • 5.00

  9. Quiz: Question 7 7. What is the value of this symbol for Data 1? • 40.00 • 10.00 • 18.00 • 8.00

  10. Measures of Variability • How do we best describe the spread or dispersion of the scores in our distribution? • It depends on which measure of central tendency we consider. • Measures of Variability, some examples • For the Mode: Range • For the Median: Interquartile Range • For the Mean: Variance or Standard deviation

  11. X Figure 1

  12. X Figure 2

  13. X Figure 3

  14. Mathematical Measures of Variability • Nominal data: Range • X(max) – (X(min) • The best we can do to describe spread around a mode. • Notice, only two scores go into the calculation of the range. • Ordinal data: (Semi-) Interquartile range • Only takes the location of the scores (top half/bottom half) into account, not the value of the scores. Interquartile range Semi-interquartile range

  15. Figure 4: Interquartile Range 25% 25% Q1 Q2 Q3

  16. Mathematical Measures of Variability • Nominal data: Range • X(max) – (X(min) • The best we can do to describe spread around a mode. • Notice, only two scores go into the calculation of the range. • Ordinal data: (Semi-) Interquartile range • Only takes the location of the scores (top half/bottom half) into account, not the value of the scores. Interquartile range Semi-interquartile range

  17. Figure 5: Semi-Interquartile Range 25% 25% Q1 Q2 Q3

  18. Q1 Q2 Q3 Normal Distribution Q3 – Q2 =Q2 – Q1

  19. Negatively Skewed Distribution Q3 – Q2 <Q2 – Q1 - + Q1 Q2 Q3

  20. Positively Skewed Distribution Q3 – Q2 >Q2 – Q1 - + Q1 Q2 Q3

  21. Deviation Scores • A deviation score is a measure of how far away from the mean each score falls. • The farther away a score is (meaning that it is in the tails of the distribution) the larger its deviation score. • If you added up all of the deviation scores they would equal zero. • This only happens when you calculate deviations around the mean.

  22. The Mean Sample Mean Population Mean

  23. Deviation Scores • A deviation score is a measure of how far away from the mean each score falls. • The farther away a score is (meaning that it is in the tails of the distribution) the larger its deviation score. • If you added up all of the deviation scores they would equal zero. • This only happens when you calculate deviations around the mean.

  24. Deviation Scores x1 = +5.00 x2 = - 2.00 X2 X1

  25. Squared deviation scores. • Since the deviation scores always add up to 0.00, then it doesn’t make sense to average them (since that would only give you 0.00/N). • So…instead we square the deviation scores before taking the average of the deviation scores. • This value is called the sums of squares.

  26. Step 1 X  x Raw Deviation Score Score

  27. Step 2 X  x  x2 Raw Deviation Squared Score Score Deviation Score

  28. Step 3 X  x  x2  Raw Deviation Squared Sum of Score Score Deviation Squared Score Deviation Scores

  29. Formulas 2 Sum of the Sum of the Sum of the Squared Squared Scores Deviation Scores Scores Squared (Sum of Squares)

  30. Least squares criterion. • We already said that one property about the mean is that the sum of the deviation scores around the mean is equal to zero. • The other property of the mean is the least squares criterion. • If you square all of the deviation scores around the mean and add them together you will get a smaller value than if you did the same thing about any other number in a distribution (e.g., the median, the model).

  31. Example of Least Squares Criterion

  32. Variance • Variance of a population is the average of the squared deviations around the population mean (mu). Sum of Squared Deviations (Sums of Squares) Sample size

  33. Calculations using the definitional formula for a population. (σ2) • After you calculate the squared deviations and you add then together (the sums of squares) then you divide that value by your sample size.

  34. Variance side note: Bias • Variance for a population, σ2 • Variance for a sample, s2 : This adjustment is made because the sample produces a “biased” estimate of the population value

  35. Calculations using the definitional formula for a sample variance (s2). • After you calculate the squared deviations and you add then together (the sums of squares) then you divide that value by your sample size.

  36. Standard Deviation • Standard deviations are difficulty to interpret since they are usually utilized as a mathematically pleasing measure of variability. • Simply the square root of the variance. • They are useful for some inferences we will make later using the normal curve and probabilities. Population Standard Deviation Sample Standard Deviation

  37. Definitional vs Computational • Definitional Formula • Population • Definitional Formula • Sample • Computational Formula • Population • Computational Formula • Sample

  38. Visual Displays and Variability • We want graphs or plots that can show us how spread out our data are. • If scores are VERY spread out, then there is a lot of diversity amongst the individuals of our population (sample). • If this is true, then using a measure of central tendency

  39. What does Variability look like? • We use various graphical displays to examine the variability of the scores • Frequencies of continuous variables • Histogram • Stem-and-leaf • Grouped data • Boxplots (Box and whisker plots) • Bar charts, pie charts, etc.

  40. Histogram Example

  41. Boxplot Example

  42. In-Class Problem Set & Solution Located under “Course Materials”

  43. Writing a Results Section For the online course you are not required to write a Results section