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

Mar 26, 2015

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

Describing Location in a Distribution

Describing Location in a Distribution

2.1 Measures of Relative Standingand Density Curves

YMS3e

AP Stats at CSHNYCMs. Namad

2.1 Measures of Relative Standingand Density Curves

YMS3e

AP Stats at CSHNYCMs. Namad

Text

Page 2: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Sample DataSample DataConsider the following test scores for a small class:

79 81 80 77 73 83 74 93 78 80 75 67 73

77 83 86 90 79 85 83 89 84 82 77 72Jenny’s score is noted in red. How did she perform on this test relative to her peers?

6 | 77 | 23347 | 57778998 | 001233348 | 5699 | 03

Her score is “above average”...but how far above average is it?

Page 3: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Standardized ValueStandardized ValueOne 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:

z=x−mean

standard deviation

Page 4: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Calculating z-scoresCalculating z-scoresConsider the test data and Julia’s score.

79 81 80 77 73 83 74 93 78 80 75 67 73

77 83 86 90 79 85 83 89 84 82 77 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:

z=x−80

6.07=

86 − 80

6.07= 0.99

Julia’s score was almost one full standard deviation above the mean. What about Kevin: x=

Page 5: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Calculating z-scoresCalculating z-scores79 81 80 77 73 83 74 93 78 80 75 67 73

77 83 86 90 79 85 83 89 84 82 77 72

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 | 77 | 23347 | 57778998 | 001233348 | 5699 | 03

Page 6: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Comparing ScoresComparing ScoresStandardized values can be used to compare scores from two different distributions.

Statistics Test: mean = 80, std dev = 6.07Chemistry Test: mean = 76, std dev = 4Jenny got an 86 in Statistics and 82 in Chemistry.On which test did she perform better?

Statistics

z=86 − 80

6.07= 0.99

Chemistry

z=82 − 76

4=1.5

Although she had a lower score, she performed relatively better in Chemistry.

Page 7: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

PercentilesPercentilesAnother 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.

6 | 77 | 23347 | 57778998 | 001233348 | 5699 | 03

Page 8: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Chebyshev’s InequalityChebyshev’s InequalityThe % 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

%within k std dev ≥ 1−1

k 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Page 9: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Density CurveDensity CurveIn 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.

Page 10: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Density CurvesDensity CurvesDensity 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.”

Page 11: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

ExampleExample• 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...

Page 12: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

2.1 Summary2.1 SummaryWe 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.

z=x−mean

standard deviation

Page 13: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

2.2 Normal Distributions2.2 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.

σμ

Page 14: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.
Page 15: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.
Page 16: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.
Page 17: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.
Page 18: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

ExampleExample• 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.

Page 19: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Assessing NormalityAssessing 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.

Page 20: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Normal Probability Plot

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 is what I do, either is fine)

Page 21: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

• rkgnt

• Normal w/Outliers Right Skew Normal

Interpretation: draw your X = Y line with a straight edge- points shouldn’t vary too much

Page 22: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Constructing Probability Plot on Calculator

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 80 75 67 73

77 83 86 90 79 85 83 89 84 82 77 72

Page 23: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

Case Case ClosedClosedCase Case

ClosedClosedThe New SATThe New SAT

Chapter 2Chapter 2

AP Stats at CSHNYCAP Stats at CSHNYCMs. NamadMs. Namad

The New SATThe New SATChapter 2Chapter 2

AP Stats at CSHNYCAP Stats at CSHNYCMs. NamadMs. Namad

Page 24: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

I: Normal Distributions1. SAT Writing Scores are N(516, 115)

What percent are between 600 and 700?

z700 =700 − 516

115

=184

115=1.6

z600 =600 − 516

115

=84

115= 0.73

516SAT Writing Scores

≈N(516, 115)

600700

%Between 600 and 700≈.9452-.7673≈.1779

%Below 700≈.9452

%Below 600≈.7673

Page 25: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

I: Normal Distributions1. SAT Writing Scores are N(516, 115)

What score would place a student in the 65th Percentile?

516SAT Writing Scores

≈N(516, 115)

?

0.65

? = mean + 0.39(s)

? = 516 + 0.39(115)

? = 516 + 44.85

? = 560.85

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

z0.65 ≈ 0.39

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

Page 26: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

II: Comparing Observations2. Male scores are N(491,110)

Female scores are N(502,108)

a) What % of males earned scores below 502?

491Male Writing Scores

≈N(491,110)

502

z=502 − 491

110z = 0.1

%below ≈ .5398

Page 27: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

II: Comparing Observations2. Male scores are N(491,110)

Female scores are N(502,108)

b) What % of females earned scores above 491?

502Female Writing Scores

≈N(502,108)

491

z=491− 502

108z = −0.101

%below ≈ .4602

%above =1− .4602 = .5398

Page 28: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

II: Comparing Observations2. 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?

491Male Writing Scores

≈N(491,110)

z.85 =1.04

score = 502 +1.04(108)

score = 614.32

85th %-ile for Females

614.32

z=614.32 − 491

110z =1.12

%below = .8686

%above = .1314

Page 29: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

III:Determining Normality3a. Did males or females perform better?

Writing400 450 500 550 600 650 700 750 800

SATs Box Plot

Writing400 450 500 550 600 650 700 750 800

SATs Box Plot

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%.

Page 30: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

III:Determining Normality3b. 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).

1

2

3

4

5

6

7

8

9

Male400 450 500 550 600 650 700 750 800

SATs Histogram

1

2

3

4

5

6

7

8

9

Male400 450 500 550 600 650 700 750 800

SATs HistogramSATs

Male

584.5833348

80.0786411.558356

39S1 = ( )meanS2 = ( )countS3 = ( )stdDevS4 = ( )stdErrorS5 = ( )missing( )count

SATs

Male

584.5833348

80.0786411.558356

39S1 = ( )meanS2 = ( )countS3 = ( )stdDevS4 = ( )stdErrorS5 = ( )missing( )count

Page 31: Describing Location in a Distribution 2.1 Measures of Relative Standing and Density Curves YMS3e AP Stats at CSHNYC Ms. Namad 2.1 Measures of Relative.

III:Determining Normality3c. 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.

Normal Quantile = -7.3 + 0.0125Male

-2

-1

0

1

2

Male400 450 500 550 600 650 700 750 800

SATs Normal Quantile Plot

Normal Quantile = -7.3 + 0.0125Male

-2

-1

0

1

2

Male400 450 500 550 600 650 700 750 800

SATs Normal Quantile Plot

Normal Quantile = -7.4 + 0.0127Female

-2.5

-2.0-1.5

-1.0-0.5

0.0

0.51.0

1.52.0

2.5

Female400 450 500 550 600 650 700 750 800

SATs Normal Quantile Plot

Normal Quantile = -7.4 + 0.0127Female

-2.5

-2.0-1.5

-1.0-0.5

0.0

0.51.0

1.52.0

2.5

Female400 450 500 550 600 650 700 750 800

SATs Normal Quantile Plot