1 Lecture 3 Outline 1. Chebyshev’s Theorem 2. The Empirical Rule 3. Measures of Relative Standing 4. Examples

2 Chebyshev’s Theorem Applies to any data set. At least ¾ of the observations in any data set will fall within 2s (2 standard deviations) of the mean: ( – 2s, + 2s). At least 8/9 of the observations in any data set will fall within 3s of the mean: ( – 3s, + 3s).  k >1, at least 1 – (1/k 2 ) of the observations will fall within ks of the mean. (Works with ,  too) Lecture 3

3 The Empirical Rule Applies to distributions that are mound-shaped and symmetric 68% of the observations will fall within +/- 1s of the mean 95% of the observations will fall within +/- 2s of the mean 99.7% (essentially all) of the observations will fall within +/- 3s of the mean Works with ,  too. Lecture 3

4 The Empirical Rule Example: IQ scores o Population mean:  = 100 o Population std. dev:  = 15 o The empirical rule tells us that 99.7% of IQ scores fall in the range 55 to 145. o Do you see why? Lecture 3

5 Measures of Relative Standing A.Percentiles B.Standard scores (Z scores) Lecture 3

6 Measures of Relative Standing Measures of relative standing tell us something about a given score by reporting how it relates to other scores. For example, one such measure tells us what proportion of scores in a data set are smaller than a given score. Lecture 3

7 Measures of Relative Standing A. Percentiles With observations in a set arranged in order (smallest to largest), P th percentile is a number such that P% of the observations fall below it. In a set of 200 observations, if a number X is larger than 150 of the observations, then X is at the 75 th percentile (150/200 = 75%). Lecture 3

8 Measures of Relative Standing B. Standard Scores (aka Z scores) i. For a sample:Z = X – s ii. For a population:Z = X –   Z-scores can be positive or negative. A (raw) score below the mean has a negative Z score. Lecture 3

9 2 Important qualities of the Z score 1. Z-scores are like a ruler – they measure distances. o Z scores give the distance between any score X and the mean, expressed in standard deviation units. 2. Values expressed in Z scores can be compared, regardless of their original units. Lecture 3

10 Z scores are like a ruler Compare these situations:  = 100 and  = 10  = 100 and  = =  + 2  120 =  + ½  A score of 120 is more impressive on the left, where it is 2 standard deviations above the mean (vs. one-half s on the right). Lecture 3

11 Z-Scores are unit-free Values expressed in Z scores can be compared, regardless of their original units. o For example, suppose you had exam grades for 100 students in Psych 281 and also knew how many hours each student studied for that exam. o You could compare any student’s Z-score for their grade with the Z-score for their # of hours of studying. Lecture 3

12 Z-Scores Suppose we have the following data for our class: Mean grade for the exam:70 Standard deviation:10 Mean # hours studying for exam:8 Standard deviation:2 Lecture 3

13 Z-Scores BillBobBen Grade Z # hours6126 Z Lecture 3

14 Example – Assignment 2, Q.1 The distribution of a sample of 100 test scores is symmetrical and mound-shaped, with a mean of 50 and a variance of 144. a. Approximately how many scores are equal to or greater than 74? Lecture 3

15 Lecture 3 Example – Assignment 2, Q Since s 2 = 144, s = 12, and 74 = + 2s What percentage of scores falls above the red line?

16 Example – Assignment 2, Q. 1 Since 74 = + 2s, we have p = % of scores are ≥ 74. Since there are 100 scores, 2.5% = approx. 3 scores. Lecture 3

17 Example – Assignment 2, Q. 1 b. What score corresponds to the 75 th percentile? To answer this, we use interpolation. o A score at the 84 th percentile is 1 s above the mean. o = 50 and s = 12, so 62 is one s above the mean – which is the 84 th percentile. (Why?) Lecture 3

18 Example – Assignment 2, Q.1 Remember that the distribution is mound- shaped and symmetric. By the Empirical Rule, 68% of the distribution is between -1 s and +1 s around the mean – so half of that (34%) is between the mean and 1 s above the mean. Lecture 3

19 Example – Assignment 2, Q. 1 The 84 th percentile is 1 s above the mean by the Empirical Rule. A score of 62 is 1 s above the mean because the mean is 50 and s = 12. Therefore 84 th percentile = 62 Lecture 3

20 Lecture 3 Example – Assignment 2, Q.1 X = 50X = 62 34% 84 th percentile 50% + 34% = 84% 50%

21 Example – Assignment 2, Q. 1 Now we know which score is at the 50 th percentile (the mean score – by definition in a symmetric distribution). We also know which score is at the 84 th percentile. Now we can answer our question: which score is at the 75 th percentile? Lecture 3

22 Example – Assignment 2, Q. 1 X%ile50 a Lecture ∆ ∆ =

23 Example – Assignment 2, Q. 1 ∆ = ∆ = 25 (12) = th percentile:X = = ~ 59 Lecture 3 Interpolation

24 Example – Assignment 2, Q. 1 c. If the lowest and highest scores in this sample are 14 and 89, respectively, what is the range of the scores in standard deviation units? Z 1 = 14 – 50Z 2 = 89 – = -3.0 = 3.25 Range = -3.0 to 3.25 = 6.25 Lecture 3

25 Example – Assignment 2, Q. 1 d. Assume the 100 scores have same mean and variance but now have a strongly skewed distribution. At least how many of the scores fall between 32 and 68 in this distribution? By Chebyshev’s Theorem: Z 1 = 32 – 50 = –1.5Z 2 = 68 – 50 = Lecture 3

26 Example – Assignment 2, Q. 1 k = – (1/k 2 ) = 1 – (1/1.5 2 ) = 1 – 1/2.25 = % or ~ 56% of scores lie between 32 and 68. Lecture 3