Describing Location in a Distribution

1. Describing Location in a Distribution • 2.1 Measures of Relative Standing • and Density Curves • YMS3e

2. 6 | 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 569 9 | 03 Her score is “above average”... but how far above average is it? Sample Data • Consider the following test scores for a small class: Jenny’s score is noted in red. How did she perform on this test relative to her peers?

3. Standardized Value: “z-score” If the mean and standard deviation of a distribution are known, the “z-score” of a particular observation, x, is: Standardized Value • One way to describe relative position in a data set is to tell how many standard deviations above or below the mean the observation is.

4. Calculating z-scores • Consider the test data and Julia’s score. According to Minitab, the mean test score was 80 while the standard deviation was 6.07 points. Julia’s score was above average. Her standardized z-score is: Julia’s score was almost one full standard deviation above the mean. What about Kevin: x=

5. 6 | 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 569 9 | 03 Calculating z-scores Julia: z=(86-80)/6.07 z= 0.99 {above average = +z} Kevin: z=(72-80)/6.07 z= -1.32 {below average = -z} Katie: z=(80-80)/6.07 z= 0 {average z = 0}

6. Statistics Chemistry Comparing Scores • Standardized values can be used to compare scores from two different distributions. • Statistics Test: mean = 80, std dev = 6.07 • Chemistry Test: mean = 76, std dev = 4 • Jenny got an 86 in Statistics and 82 in Chemistry. • On which test did she perform better? Although she had a lower score, she performed relatively better in Chemistry.

7. 6 | 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 569 9 | 03 Percentiles • Another measure of relative standing is a percentile rank. • pth percentile: Value with p % of observations below it. • median = 50th percentile {mean=50th %ile if symmetric} • Q1 = 25th percentile • Q3 = 75th percentile Jenny got an 86. 22 of the 25 scores are ≤ 86. Jenny is in the 22/25 = 88th %ile.

8. Chebyshev’s Inequality: In any distribution, the % of observations within k standard deviations of the mean is at least Chebyshev’s Inequality • The % of observations at or below a particular z-score depends on the shape of the distribution. • An interesting (non-AP topic) observation regarding the % of observations around the mean in ANY distribution is Chebyshev’s Inequality.

9. Density Curve: An idealized description of the overall pattern of a distribution. Area underneath = 1, representing 100% of observations. Density Curve • In Chapter 1, you learned how to plot a dataset to describe its shape, center, spread, etc. • Sometimes, the overall pattern of a large number of observations is so regular that we can describe it using a smooth curve.

10. Density Curves • Density Curves come in many different shapes; symmetric, skewed, uniform, etc. • The area of a region of a density curve represents the % of observations that fall in that region. • The median of a density curve cuts the area in half. • The mean of a density curve is its “balance point.”

11. Example • Pretend you are rolling a die. The numbers 1,2,3,4,5,6 are the possible outcomes. In 120 rolls, how many of each number would you expect to roll? • Calculator can do a simulation: • Clear L1 in your calc. Use random integer generator to generate 120 random whole numbers between 1 and 6 then store in L1 • RandInt (1, 6, 120) STO-> L1 • Set viewing window: X (1,7) by Y (-5,25). • Specify a histogram using the data in L1 • Repeat simulation several times. 2nd Enter will recall/reuse the previous command. In theory we should expect a uniform outcome...

12. 2.1 Summary • We can describe the overall pattern of a distribution using a density curve. • The area under any density curve = 1. This represents 100% of observations. • Areas on a density curve represent % of observations over certain regions. • An individual observation’s relative standing can be described using a z-score or percentile rank.

13. Homework • Thursday • 2.1-4, 7-8, 9-12 • Friday • 15, 18, 19

14. 2.2 Normal Distributions • Normal Curves: symmetric, single-peaked, bell-shaped. and median are the same. Size of the will affect the spread of the normal curve.

19. Example • Scores on the SAT verbal test in recent years follow approximately the N (505, 110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT? • 1. State the problem and draw a picture. Shade the area we’re looking for. • 2. Find the Z score with the table • 3. Convert to raw score.

20. Assessing Normality • Method 1: Construct a histogram, see if graph is approximately bell-shaped and symmetric. Median and Mean should be close. Then mark off the -2, -1, +1, +2 SD points and check the 68-95-99.7 rule.

21. Normal Probability Plot • Method 2: Construct Normal Probability Plot • 1. Arrange the observed data values from smallest to largest. Record what percentile of the data each value occupies (example, the smallest observation in a set of 20 is at the 5% point, the second is at 10% etc.) • Use Table A to find the Z’s at these same percentiles (example -1.645 is @ 5%, -1.28 is @10% • Plot each data point against the corresponding Z (x-values on the horizontal axis, z-scores on the vertical axis)

22. rkgnt • Normal w/Outliers Right Skew Normal Interpretation: draw your X = Y line with a straight edge- points shouldn’t vary too much

23. Constructing Probability Plot on Calculator

24. Homework

25. Case Closed • The New SAT • Chapter 2

26. %Between 600 and 700≈.9452-.7673≈.1779 %Below 700≈.9452 %Below 600≈.7673 ≈N(516, 115) 700 600 516 SAT Writing Scores I: Normal Distributions • 1. SAT Writing Scores are N(516, 115) • What percent are between 600 and 700?

27. 0.65 ≈N(516, 115) ? 516 SAT Writing Scores I: Normal Distributions • 1. SAT Writing Scores are N(516, 115) • What score would place a student in the 65th Percentile?

28. ≈N(491,110) 502 491 Male Writing Scores II: Comparing Observations • 2. Male scores are N(491,110) • Female scores are N(502,108) • a) What % of males earned scores below 502?

29. ≈N(502,108) 491 502 Female Writing Scores II: Comparing Observations • 2. Male scores are N(491,110) • Female scores are N(502,108) • b) What % of females earned scores above 491?

30. 85th %-ile for Females ≈N(491,110) 491 614.32 Male Writing Scores II: Comparing Observations • 2. Male scores are N(491,110) • Female scores are N(502,108) • c) What % of males earned scores above the 85th %-ile of female scores?

31. III:Determining Normality • 3a. Did males or females perform better? The male and female scores are very similar. Both have roughly symmetric distributions with no outliers. The median for females is slightly higher (580 vs 570), but the male average is slightly higher (584.6 vs 580). Both have similar ranges, but the males had slightly more variability in the middle 50%.

32. III:Determining Normality • 3b. How do the male scores compare with National results? The males at this school did much better than the overall national mean (584.6 vs. 516). Their scores were also more consistent as evidenced by a lower standard deviation (80.08 vs 115).

33. III:Determining Normality • 3c. Are the male and female scores approximately Normal? The Normal Quantile Plots for both the male and female scores are approximately linear. Therefore, there is evidence that their scores are approximately Normal.