1. Describing Location in a Distribution
Text
2.1 Measures of Relative Standing
and Density Curves
YMS3e
AP Stats at CSHNYC
Ms. Namad

2. Sample Data
Consider the following test scores for a small class: 79 81 80 77 73 83 74 93 78 75 67 86 90 85 89 84 82 72
Jenny’s score is noted in red. How did she perform on this test relative to her peers?
6 | 7
7 | 2334
7 |
8 |
8 | 569
9 | 03
Her score is “above average”...
but how far above average is it?

3. 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.
Standardized Value: “z-score”
If the mean and standard deviation of a distribution are known, the “z-score” of a particular observation, x, is:

4. Calculating z-scores
Consider the test data and Julia’s score. 79 81 80 77 73 83 74 93 78 75 67 86 90 85 89 84 82 72
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. Calculating z-scores 6 | 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 56979 81 80 77 73 83 74 93 78 75 67 86 90 85 89 84 82 72
Julia: z=(86-80)/6.07
z= {above average = +z}
6 | 7
7 | 2334
7 |
8 |
8 | 569
9 | 03
Kevin: z=(72-80)/6.07
z= {below average = -z}
Katie: z=(80-80)/6.07
z= {average z = 0}

6. Comparing Scores Statistics Chemistry
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?
Statistics
Chemistry
Although she had a lower score, she performed relatively better in Chemistry.

7. 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
6 | 7
7 | 2334
7 |
8 |
8 | 569
9 | 03
Jenny got an 86.
22 of the 25 scores are ≤ 86.
Jenny is in the 22/25 = 88th %ile.

8. 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.
Chebyshev’s Inequality:
In any distribution, the % of observations within k standard deviations of the mean is at least

9. Density Curve 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.
Density Curve:
An idealized description of the overall pattern of a distribution.
Area underneath = 1, representing 100% of observations.

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. 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.

18. 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.

19. 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 rule.

20. 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 %,
Plot each data point against the corresponding Z (x- values on the horizontal axis, z-scores on the vertical axis is what I do, either is fine)

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

22. Constructing Probability Plot on Calculator
Students in Mr. Pryor’s stats class
X values on horizontal axis 79 81 80 77 73 83 74 93 78 75 67 86 90 85 89 84 82 72

23. Case Closed
The New SAT
Chapter 2
AP Stats at CSHNYC
Ms. Namad

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

25. I: Normal Distributions
1. SAT Writing Scores are N(516, 115)
What score would place a student in the 65th Percentile?
Table A
Standard Normal probabilities (continued) z
0.00
0.01
...
0.07
0.08
0.09
0.0
0.500
0.5040
0.5279
0.5319
0.5359
0.3
0.6179
0.6217
0.6443
0.6480
0.6517
0.4
0.6554
0.6591
0.6808
0.6844
0.6879
Table A
Standard Normal probabilities (continued) z
0.00
0.01
...
0.07
0.08
0.09
0.0
0.500
0.5040
0.5279
0.5319
0.5359
0.3
0.6179
0.6217
0.6443
0.6480
0.6517
0.4
0.6554
0.6591
0.6808
0.6844
0.6879
Table A
Standard Normal probabilities (continued) z
0.00
0.01
...
0.07
0.08
0.09
0.0
0.500
0.5040
0.5279
0.5319
0.5359
0.3
0.6179
0.6217
0.6443
0.6480
0.6517
0.4
0.6554
0.6591
0.6808
0.6844
0.6879
516
SAT Writing Scores
≈N(516, 115) ?
0.65

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

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

28. 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?
85th %-ile for Females
614.32
491
Male Writing Scores
≈N(491,110)

29. 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%.

30. 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).

31. 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.