1. Measures of Central Tendency and Variability Chapter 5:113-123 Using Normal Curves For Evaluation

2. Types of Curves... The Normal Curve The Normal Curve :

3. Normal Means “Average” …Sort of In a Normal Distribution, most of the scores are found closest to the middle They’re “average” Either “tail” represents rare scores They’re “special”

4. When “Average” isn’t Good Enough Representative “Normal” “Typical” Not Outstanding or Extreme

5. Statistical Measures of Central Tendency Mean: The calculated “average” Median: The middle of the ordered scores Mode: The most frequently occurring score(s) The mean is the measure of choice if You want to do further statistical analysis.

6. The Mean X = Σx i / N Considered more precise and stable than the median or mode Can be used in additional statistical analysis Don’t use with nominal or ordinal data

7. The Median In an ORDERED set of scores The Median score is exactly in the middle Median = Mdn Mdn = (Number of scores +1)/ 2 That tells us where the Mdn score is found…

8. Like so: Set of scores: 5, 6, 3, 7, 4, 9, 2 Order the scores: 2, 3, 4, 5, 6, 7, 9 Find the position of the median Score: Mdn = (N+1) / 2 Mdn = (7+1) / 2 = 4 The median score is the 4 th score: 2, 3, 4, 5, 6, 7, 9

9. Comparing the Median and Mean Scores: Mdn = 5 X = 36/7 = 5.14 Make a conclusion about this set of scores

10. The Mode: The most frequently occurring score(s) Gives a quick BUT ROUGH sense of the typical score… Can you think of a situation when the MODE is not the mean or median, but is a better description of what the typical student in your group is like? (HINT: Lab 1)

11. Pull-Up Scores X = 4.8 pull-ups The mode is usually used to describe the most typical score in NOMINAL data: Eg. Nebraska is the most common birth-state of WSC students

12. Did you hear the one about the two statisticians who went pheasant hunting together?

13. The Point Please The “Cluster” of a set of scores is one thing Spread may actually be more important for interpretation

14. What is the Standard Deviation? The appropriate measure of the variability of a set of scores, when the mean is used as the measure of central tendency. The average deviation of any randomly chosen score from the mean

15. Using the Mean and Median to determine “Normalcy” 50% of the scores fall above and below the Median score It will be exactly in the middle of the range of scores When the Mean = Median, the curve is NORMAL When Mean > Median it is skewed Right When Mean < Median it is skewed left…like so

16. Curve “Skewness” Mean Median More than ½ the scores Are above the mean: Skewed Left More than ½ the scores Are below the mean: Skewed Right

17. Why the Fuss About Normal Curves? Whole populations will always be distributed in a “Normal” arrangement For a SAMPLE of that population to accurately reflect the population, the sample MUST BE NORMAL – or conclusions won’t be valid

18. Example: Population: PE Majors Sample: PE Majors at WSC, graduating in 2002 Measurement: Mean Starting Salary Results: $78,000 –Believe it?

19. WSC PE Graduates: Salary <$20k $25-29K>$40K 2 6 8 N = 20 Range: $12,500 - $350,000 Mean: $78,000 SD: +/- $52,000 This guy plays For the NBA and Makes $350K!!

20. The Truth: If we through out the NBA player, the mean is then $29,050 With the NBA player in there…the mean is “skewed to the right” of the true average of the “typical” graduate… BUYER BEWARE!

21. Evaluating Individual Scores Normal Curves Z-Scores Comparing Apples to Oranges…

22. Use of “Group” Statistics Compare different groups Evaluate individuals within the group

23. QUESTION: “What if your roommate came home and said, “I got a 95 on my test!” ?

25. What does his score mean? There were 200 possible The highest score was only 101 The mean was 98 The range was 95-101

26. Individuals want to know what their scores mean. They want some kind of a judgment so they can make decisions.

27. Types of Norm Referenced Evaluations Percentile Rank: mathematically tedious, defined as the percent of the scores below an individuals score Z-Scores: Calculating how many standard deviations a score is from the mean

28. A Word About Percentile Ranks: Compares your score to the rest of the “group” Norm-Referenced Evaluation BUT WHAT GROUP? National Norms: ACT scores, President’s Fitness Test Local Norms: Developed from at least 100 local scores

29. Calculating Z-Scores Find the mean and standard deviation of a set of scores Z i = ( X i - X)/ s The value of Z is a multiple +/- of the standard deviation

30. What the heck does that mean? Z-Scores reflect a score’s relationship to the rest of the scores....

31. Let’s Jump to Conclusions -Z = below average +Z = above average Value of Z = how many standard deviations (How far below) 68% of the scores will be within 1 standard deviation....

32. Let’s Evaluate your Roommate’s Score by Z-Score Let’s Evaluate your Roommate’s Score by Z-Score: Mean = 98 SD = 1.5 X R = 95 Z R = (X R – X)/ SD Z = (95-98)/1.5 = -2 Your roommate’s score is 2 standard deviations below average!

33. Conclusions: His score was only better than ~2.5% of all students (that’s bad) How did I get there?

34. Graphing the Data: 98 96.5 99.5 95 101 68% 95% 2.5%

35. Summary Means and Standard Deviations describe groups of scores Normal curves have predictable dimensions Z-Scores convert raw scores into multiples of the standard deviation

36. Summary cont. Finally: Using Z-scores to evaluate (give meaning to) an individual’s score is a type of Norm Referenced Evaluation Z-Scores can only be used in “Normal” groups

37. Assignment: Problems Calculating Z Scores: Determine the Mean Determine the SD The Z score for ANY INDIVIDUAL in that group is calculated: Z i = (X i – X)/ SD