Describing Location in a Distribution

Describing Location in a Distribution

Describing Location in a Distribution

2.1 Measures of Relative Standingand Density Curves

YMS3e

2.1 Measures of Relative Standingand Density Curves

YMS3e

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 72

Jenny’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?

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

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

6.070.99

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

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

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

6.070.99

Chemistry

z 82 76

41.5

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

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

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 11

k 2

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.

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.”

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...

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

HomeworkHomework

• Thursday

• 2.1-4, 7-8, 9-12

• Friday

• 15, 18, 19

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.

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.

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.

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)

• rkgnt

• Normal w/Outliers Right Skew Normal

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

Constructing Probability Plot on Calculator

Constructing Probability Plot on Calculator

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

HomeworkHomework

Case Case ClosedClosedCase Case

ClosedClosedThe New SATThe New SAT

Chapter 2Chapter 2The New SATThe New SAT

Chapter 2Chapter 2

I: Normal Distributions•1. SAT Writing Scores are N(516, 115)

What percent are between 600 and 700?

z700 700 516

115

184

1151.6

z600 600 516

115

84

1150.73

516SAT Writing Scores

≈N(516, 115)

600700

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

%Below 700≈.9452

%Below 600≈.7673

I: Normal Distributions•1. 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

II: Comparing Observations• 2. 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

II: Comparing Observations• 2. 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

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?

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

III:Determining Normality• 3a. 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%.

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).

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

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.

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