Measures of Central Tendency and Variability

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

2. Types of Curves... 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 = Σxi / 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 4th 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” More than ½ the scores Are above the mean: Skewed Left More than ½ the scores Are below the mean: Skewed Right Mean Median

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

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 knowwhat 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 • Zi = (Xi - 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 yourRoommate’s Score by Z-Score: • Mean = 98 • SD = 1.5 • XR = 95 • ZR = (XR – 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: 68% 2.5% 99.5 96.5 101 95 98 95%

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: • Zi = (Xi – X)/ SD