Chapters 2 and 3 : Frequency Distributions, Histograms, Percentiles and Percentile Ranks and their Graphical Representations Note: we’ll be skipping book.

Chapters 2 and 3 : Frequency Distributions, Histograms,Percentiles and Percentile Ranks and their Graphical Representations

Note: we’ll be skipping book sections:

2.4 (apparent and real limits) 2.8, 2.9 (percentile and percentile ranks for grouped data)

Chapter 2: Frequency Distributions, Histograms,Percentiles and Percentile Ranks

How can we represent or summarize a list of values?

frequency distribution: shows the number of observations for the possible categories or score values in a set of data. Can be done on any scale (nominal, ordinal, interval, or ratio).

Often represented as a bar graph (Chapter 3).

Example of a frequency distribution for nominal scale data:

2008 Auto sales by country:

Japan: 11,563,629

China: 9,345,101

US: 8,705,239

Germany: 6,040,582

South Korea: 3,806,682

Brazil: 3,220,475

Japan China US Germany South Korea Brazil0

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Japan: 11,563,629

China: 9,345,101

US: 8,705,239

Germany: 6,040,582

South Korea: 3,806,682

Brazil: 3,220,475

Car sales drawn as a histogram

Distribution of all M&M's.

This histogram shows the proportion of members for each category.

We need to bin the raw scores into a set of class intervals. How do we decide these class intervals?

Be sure the intervals don’t overlap, have the same width, and cover the entire range of scores.

Use around 10 to 20 intervals.

Use a ‘sensible’ width (like 5, and not 2.718285)

Make the lower score a multiple of the width (e.g. if the width is 5, a lower score should be 50, not 48)

If a score lands on the border, put it in the lower class interval.

Ice Dancing , compulsory dance scores, 4 Winter

Olympics

111.15108.55

106.6103.33100.06

97.3896.6796.1292.7589.6285.3684.5883.8983.1280.47

80.379.3176.7374.2572.0168.8763.7359.64

Making histograms from interval and ratio data

Ice Dancing , compulsory dance scores,

Winter OlympicsLet’s use a class interval width of 5 points, with a lowest score of 55.

110-115105-110100-10595-10090-9585-9080-8575-8070-7565-7060-6555-60

Class Intervals Frequency (f)

122312522111

n=23

Count the number of scores in each bin to get the frequency

111.15108.55

106.6103.33100.06

97.3896.6796.1292.7589.6285.3684.5883.8983.1280.47

80.379.3176.7374.2572.0168.8763.7359.64

110-115105-110100-10595-10090-9585-9080-8575-8070-7565-7060-6555-60


122312522111

Histogram of Ice Dancing Scores (frequency)

55 60 65 70 75 80 85 90 95 100 105 110 1150

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Ice Dancing Score

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Relative frequency

110-115105-110100-10595-10090-9585-9080-8575-8070-7565-7060-6555-60


122312522111

n=23

Divide by the total number of scores to get relative frequency in proportion

Relative frequency

(%)

4.358.708.70

13.044.358.70

21.748.708.704.354.354.35

.0435

.0870

.0870

.1304

.0435

.0870

.2174

.0870

.0870

.0435

.0435

.0435

Relative frequency

(prop)

Then multiply by 100 to get relative frequency in percent

111.15108.55

106.6103.33100.06

97.3896.6796.1292.7589.6285.3684.5883.8983.1280.47

80.379.3176.7374.2572.0168.8763.7359.64

Relative frequency histogram of Ice Dancing Scores (frequency)

110-115105-110100-10595-10090-9585-9080-8575-8070-7565-7060-6555-60

Class IntervalsRelative

frequency (%)4.358.708.70

13.044.358.70

21.748.708.704.354.354.35

55 60 65 70 75 80 85 90 95 100 105 110 1150

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interval width 10 interval width 5

interval width 3 interval width 1

Choosing your class intervals can have an influence on the way your histogram looks

These three graphs have the same class intervals on the same scores!

60 70 80 90 100 1100

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60 70 80 90 100 110012345

Ice Dancing Score

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When possible, include zero on your y-axis.

Not like this

As of March 27 March 31 Goal

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As of March 27 March 31 Goal0

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When possible, include zero on your y-axis:When possible, include zero on your y-axis.

Not like this

Like this

“Fox News Apologizes For Obamacare Graphic, Corrects Its 'Mistake‘”

Percentile ranks and percentile point:

Percentile Point: A point on the measurement scale below which a specific percentage of scores fall.

Percentile Rank: The percentage of cases that fall below a given point on the measurement scale.

Percentile ranks are always between zero and 100.

Growth charts convert percentile points to percentile ranks

At 30 mos. P95 = 36lbs


Ice Dancing , compulsory dance scores, Winter Olympics

Class interval f rel f(%)

Cumu-lative f

Cumu-lative %

110-115 1 4.35 23 100105-110 2 8.7 22 95.65100-105 2 8.7 20 86.9695-100 3 13.04 18 78.2690-95 1 4.35 15 65.2285-90 2 8.7 14 60.8780-85 5 21.74 12 52.1775-80 2 8.7 7 30.4370-75 2 8.7 5 21.7465-70 1 4.35 3 13.0460-65 1 4.35 2 8.755-60 1 4.35 1 4.35

What is the percentile rank for a percentile point of 100?In other words,What proportion of scores fall below a score of 100?

78.26% of the scores fall below 100

The number 78.26 is the percentile rankThe number 100 is the corresponding percentile point We write P78.26 =100


21.74% of the scores are below 75or

P21.74 = 75 or

100-21.74=78.26% of the scores are above 75.


Cumu-lative f

Cumu-lative %

110-115 1 4.35 23 100105-110 2 8.7 22 95.65100-105 2 8.7 20 86.9695-100 3 13.04 18 78.2690-95 1 4.35 15 65.2285-90 2 8.7 14 60.8780-85 5 21.74 12 52.1775-80 2 8.7 7 30.4370-75 2 8.7 5 21.7465-70 1 4.35 3 13.0460-65 1 4.35 2 8.755-60 1 4.35 1 4.35


60 65 70 75 80 85 90 95 100 105 110 1150

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Class interval

Cumu-lative %

110-115 100105-110 95.65100-105 86.9695-100 78.2690-95 65.2285-90 60.8780-85 52.1775-80 30.4370-75 21.7465-70 13.0460-65 8.755-60 4.35

The Cumulative Percentage Curve

21.74% of the scores fall below a score of 75

The number 21.74 is the percentile rankThe number 75 is the corresponding percentile point We write P21.74 = 75

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Class interval

Cumu-lative %

110-115 100105-110 95.65100-105 86.9695-100 78.2690-95 65.2285-90 60.8780-85 52.1775-80 30.4370-75 21.7465-70 13.0460-65 8.755-60 4.35


78.26% of the scores fall below a score of 100

The number 78.26is the percentile rankThe number 100 is the corresponding percentile point We write P78.26 = 100

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Cumu-lative %

110-115 100105-110 95.65100-105 86.9695-100 78.2690-95 65.2285-90 60.8780-85 52.1775-80 30.4370-75 21.7465-70 13.0460-65 8.755-60 4.35


50% of the scores fall below a score of about 84

The number 50 is the percentile rankThe number 84 is an estimate of the percentile point We write P50 = 84

60 65 70 75 80 85 90 95 100 105 110 115

Cumulative frequency distribution


What is the percentile point for a percentile rank of 21.74%?

Answer: 75 points (21.75% of the scores fall below 75)


Cumu-lative f

Cumu-lative %

110-115 1 4.35 23 100105-110 2 8.7 22 95.65100-105 2 8.7 20 86.9695-100 3 13.04 18 78.2690-95 1 4.35 15 65.2285-90 2 8.7 14 60.8780-85 5 21.74 12 52.1775-80 2 8.7 7 30.4370-75 2 8.7 5 21.7465-70 1 4.35 3 13.0460-65 1 4.35 2 8.755-60 1 4.35 1 4.35

Cumulative frequency distribution

110-115105-110100-10595-10090-9585-9080-8575-8070-7565-7060-6555-60


122312522111


Cumulativeproportion

2322201815141275321

1.00.96.87.78.65.61.52.30.22.13.09.04

Cumulativefrequency

Cumulativepercent

100968778656152302213

84

What is the percentile point for a percentile rank of 50? (Or what is P50?)

We know it’s between 80 and 85, since 52% fall below 85 and 30% fall below 80.

Here’s how to calculate the percentile rank for each raw score:note this is different from the book!

Ice Dancing, compulsory dance scores, Winter Olympics

Rank order Subtract 1/2 Divide by n (23) Multiply by 100

The percentile point for a percentile rank of 50 is 84.58

( P50 = 84.58)

111.15 23 22.5 0.98 98108.55 22 21.5 0.93 93

106.6 21 20.5 0.89 89103.33 20 19.5 0.85 85100.06 19 18.5 0.80 80

97.38 18 17.5 0.76 7696.67 17 16.5 0.72 7296.12 16 15.5 0.67 6792.75 15 14.5 0.63 6389.62 14 13.5 0.59 5985.36 13 12.5 0.54 5484.58 12 11.5 0.50 5083.89 11 10.5 0.46 4683.12 10 9.5 0.41 4180.47 9 8.5 0.37 37

80.3 8 7.5 0.33 3379.31 7 6.5 0.28 2876.73 6 5.5 0.24 2474.25 5 4.5 0.20 2072.01 4 3.5 0.15 1568.87 3 2.5 0.11 1163.73 2 1.5 0.07 759.64 1 0.5 0.02 2

Score

Here’s how to calculate the percentile rank for each raw score:


Rank order Subtract 1/2 Divide by 23 Multiply by 100

The percentile point for a percentile rank of 80 is 100.6

(P80 = 100.6)

111.15 23 22.5 0.98 98108.55 22 21.5 0.93 93

106.6 21 20.5 0.89 89103.33 20 19.5 0.85 85100.06 19 18.5 0.80 80

97.38 18 17.5 0.76 7696.67 17 16.5 0.72 7296.12 16 15.5 0.67 6792.75 15 14.5 0.63 6389.62 14 13.5 0.59 5985.36 13 12.5 0.54 5484.58 12 11.5 0.50 5083.89 11 10.5 0.46 4683.12 10 9.5 0.41 4180.47 9 8.5 0.37 37

80.3 8 7.5 0.33 3379.31 7 6.5 0.28 2876.73 6 5.5 0.24 2474.25 5 4.5 0.20 2072.01 4 3.5 0.15 1568.87 3 2.5 0.11 1163.73 2 1.5 0.07 759.64 1 0.5 0.02 2

Score

How do we calculate the percentile point for all the other ranks?


Example: What is the percentile point for the percentile rank of 90%?

We know it’s between 106.6 and 108.55

In fact, it’s ¼ of the way between 106.6 and 108.55 (90-89)/(93-89) = 1/4

That means that P90 = 106.6 + 1/4(108.55-106.6) = 107.09

111.15 23 22.5 0.98 98108.55 22 21.5 0.93 93

106.6 21 20.5 0.89 89103.33 20 19.5 0.85 85100.06 19 18.5 0.80 80

97.38 18 17.5 0.76 7696.67 17 16.5 0.72 72

Score

How do we calculate the percentile point for other ranks?


Example, what is the percentile point for the percentile rank of P75?

We know it’s ¾ of the way between 96.67 and 97.38

96.67 + 3/4(97.38-96.67) = 97.2

111.15 23 22.5 0.98 98108.55 22 21.5 0.93 93

106.6 21 20.5 0.89 89103.33 20 19.5 0.85 85100.06 19 18.5 0.80 80

97.38 18 17.5 0.76 7696.67 17 16.5 0.72 72

Score

How do we calculate the percentile point for other ranks?


Example, what is the percentile score for the percentile rank of P25?

We know it’s 1/4 of the way between 76.73 and 79.31

76.73 + 1/4(79.31-76.73) = 77.37

80.47 9 8.5 0.37 3780.3 8 7.5 0.33 33

79.31 7 6.5 0.28 2876.73 6 5.5 0.24 2474.25 5 4.5 0.20 2072.01 4 3.5 0.15 1568.87 3 2.5 0.11 1163.73 2 1.5 0.07 759.64 1 0.5 0.02 2

Score


Example, what is the percentile point for the percentile rank of 81?

111.15 23 22.5 0.98 98108.55 22 21.5 0.93 93

106.6 21 20.5 0.89 89103.33 20 19.5 0.85 85100.06 19 18.5 0.80 80

97.38 18 17.5 0.76 7696.67 17 16.5 0.72 72

Score

General formula for calculating percentile points:

1) Make a chart like the one above2) Find the two rows that fall above and below the percentile rank3) Let PH and PL be the high and low cumulative percentiles (85 and 80 in this example)4) Let SH and SL be the high and low scores (103.33 and 100.06 in this example)5) If p is the percentile rank (81 in our example), then the percentile point is:

PLPH

PLpSLSHSL )( 71.100

8085

8081)06.10033.103(06.100

Going the other way: from percentile ranks to percentile points

Example: What is the percentile rank for the percentile point of 103.33?

Rank order Subtract 1/2 Divide by 23 Multiply by 100111.15 23 22.5 0.98 98108.55 22 21.5 0.93 93

106.6 21 20.5 0.89 89103.33 20 19.5 0.85 85100.06 19 18.5 0.80 80

97.38 18 17.5 0.76 7696.67 17 16.5 0.72 72

Score

This is easy, since 103.33 is one of the scores. The percentile rank is 85%.

85% of the scores fall below 103.33


Example: What is the percentile rank for the percentile point of 100?

Rank order Subtract 1/2 Divide by 23 Multiply by 100111.15 23 22.5 0.98 98108.55 22 21.5 0.93 93

106.6 21 20.5 0.89 89103.33 20 19.5 0.85 85100.06 19 18.5 0.80 80

97.38 18 17.5 0.76 7696.67 17 16.5 0.72 72

Score

This is not as easy, since 100 is not one of the scores. We do know that it is between 76 and 80. In fact, we know it must be really close to 80, since P80 is 100.06

Here’s how to do it. After finding the two rows that bracket the percentile point, if S is the percentile point, then the percentile rank is:

SLSH

SLSPLPHPL )( 91.79

38.9706.100

38.97100)7680(76

79.91% o the scores fall below 100

Class interval f58-61 161-64 064-67 067-70 170-73 273-76 076-79 179-82 582-85 485-88 288-91 191-94 394-97 0

97-100 0

9493929187868584848382818180807773736859

Raw Test Scores

Another Example: integer valued dataScores on Professor Flans’ Midterm (n = 20)

We’ll choose a class interval width of 3. An odd number for width is good for integer data because the middle value will be a whole number.

Remember, scores that land on the border are assigned to the lower class interval.

So 85 lands in the interval82-85.

60 63 66 69 72 75 78 81 84 87 90 93 96 990

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Class interval f58-61 161-64 064-67 067-70 170-73 273-76 076-79 179-82 582-85 485-88 288-91 191-94 394-97 0

97-100 0

Bins labeled by the centers of the class intervals

58-61 61-64 64-67 67-70 70-73 73-76 76-79 79-82 82-85 85-88 88-91 91-94 94-97 97-1000

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You can also show the whole interval on the x-axis labels


Class Interval frequency97-100 094-97 091-94 388-91 185-88 282-85 479-82 576-79 173-76 070-73 267-70 164-67 061-64 058-61 1

Relative frequency(%)

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155

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Cumulativefrequency

202020171614105442111

cumulative frequency %

1001001008580705025202010555

Class Interval

Cumulative frequency%

97-100 10094-97 10091-94 10088-91 8585-88 8082-85 7079-82 5078-79 2573-76 2070-73 2067-70 1064-67 561-64 558-61 5

The Cumulative Percentage Curve for Professor Flans’ Midterm

61 64 67 70 73 76 79 82 85 88 91 94 97 1000

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(%)

About 50% of the scores fall below 82. (So P50 is about 82)

Estimate the percentile point for a percentile rank of 50%

Estimating percentile points and percentile ranks from the cumulative percentage curve

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Estimate the percentile point for a percentile rank of 90%

90% of the scores fall below a score of about 92. (P90 is about 92)

61 64 67 70 73 76 79 82 85 88 91 94 97 100

94 20 19.5 0.975 97.593 19 18.5 0.925 92.592 18 17.5 0.875 87.591 17 16.5 0.825 82.587 16 15.5 0.775 77.586 15 14.5 0.725 72.585 14 13.5 0.675 67.584 13 12.5 0.625 62.584 12 11.5 0.575 57.583 11 10.5 0.525 52.582 10 9.5 0.475 47.581 9 8.5 0.425 42.581 8 7.5 0.375 37.580 7 6.5 0.325 32.580 6 5.5 0.275 27.577 5 4.5 0.225 22.573 4 3.5 0.175 17.573 3 2.5 0.125 12.568 2 1.5 0.075 7.559 1 0.5 0.025 2.5

Calculating percentile points from raw data.

What is the percentile point for a percentile rank of 50%?

Rank order Subtract 1/2 Divide by 20 Multiply by 100Test score

It’s between 82 and 83

pLpH

PLpSLSHSL )(

5.825.475.52

5.4750)8283(82

P50 = 82.5

94 20 19.5 0.975 97.593 19 18.5 0.925 92.592 18 17.5 0.875 87.591 17 16.5 0.825 82.587 16 15.5 0.775 77.586 15 14.5 0.725 72.585 14 13.5 0.675 67.584 13 12.5 0.625 62.584 12 11.5 0.575 57.583 11 10.5 0.525 52.582 10 9.5 0.475 47.581 9 8.5 0.425 42.581 8 7.5 0.375 37.580 7 6.5 0.325 32.580 6 5.5 0.275 27.577 5 4.5 0.225 22.573 4 3.5 0.175 17.573 3 2.5 0.125 12.568 2 1.5 0.075 7.559 1 0.5 0.025 2.5

What is the percentile point for a percentile rank of 90%?


It’s between 92 and 93

pLpH

PLpSLSHSL )(

5.925.875.92

5.8790)9293(92

It’s exactly halfway between 92 and 93

Calculating percentile points from raw data.


94 20 19.5 0.975 97.593 19 18.5 0.925 92.592 18 17.5 0.875 87.591 17 16.5 0.825 82.587 16 15.5 0.775 77.586 15 14.5 0.725 72.585 14 13.5 0.675 67.584 13 12.5 0.625 62.584 12 11.5 0.575 57.583 11 10.5 0.525 52.582 10 9.5 0.475 47.581 9 8.5 0.425 42.581 8 7.5 0.375 37.580 7 6.5 0.325 32.580 6 5.5 0.275 27.577 5 4.5 0.225 22.573 4 3.5 0.175 17.573 3 2.5 0.125 12.568 2 1.5 0.075 7.559 1 0.5 0.025 2.5


Example, what is the percentile rank for the percentile point of 90?

SLSH

SLSpLpHpL )(

25.818791

8790)5.775.82(5.77

It’s between 77.5 and 82.5

81.25% of the scores fall below 90 points

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60 63 66 69 72 75 78 81 84 87 90 93 96 990

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Frequency histogram Frequency polygon

More stuff about frequency distributions:

Properties of frequency distributions

‘normal’ or bell-shaped

positively skewedNegatively skewed

Example of a negatively skewed distribution

300 350 400 450 500 550 600 650 700 750 8000

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GRE quant scores

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Example of positively skewed distribution: Household annual income

•P0-89 (bottom 90%) — income below $104,696 (average income, $30,374*) •P90-100 (top 10%) — income above $104,696 (average income, $269,658*) •P90-95 (next 5%) — income between $104,696 and $148,423 (average income, $122,429*) •P95-99 (next 4%) — income between $148,423 and $382,593 (average income, $210,597*) •P99-100 (top 1%) — income above $382,593 (average income, $1,243,516*) •P99.5-100 (top 0.5%) — income above $597,584 (average income, $2,022,315*) •P99.9-100 (top 0.1%) — income above $1,898,200 (average income, $6,289,800*) •P99.99-100 (top .01%) —income above $10,659,283 (average income, $29,638,027*)

Household income distribution as of 2006:

http://www.wealthandwant.com/issues/income/income_distribution.html

So the ‘top 1%’ can be described as:

P99 = $382,593

Shift in central tendency

0 20 40 60 80 100Scores

Shift in variability

0 20 40 60 80 100Scores

Two (of many) ways that frequency distributions differ

Chapters 2 and 3 : Frequency Distributions, Histograms, Percentiles and Percentile Ranks and their Graphical Representations Note: we’ll be skipping book.

Documents

frequency distributions

proportion relative

relative frequency prop

histogram slide

class interval width

ratio data slide

grouped data slide

lower class interval