Statistical Methods in Computer Science

Statistical Methods in Computer Science © 2006-now Gal Kaminka / Ido Dagan 1

Statistical Methods in Computer Science

Descriptive StatisticsData 1:

Frequency Distributions

Ido Dagan


Concrete Theory: Relates Variables to Each Other

Examples: Mathematically accurate

Memory = 2*sizeof(input) + 3 Runtime = 500 + 30*sizeof(input) + 20

Asymptotically correct Memory = O(sizeof(input)) in worst case, Runtime = O(log (sizeof(input))) in best case Accuracy is proportional to run-time

Qualitative User performance is increased with reduced cognitive load number of bugs discovered is monotonically decreasing, but

positive, if the same programmer is used, otherwise, it increases


Behavior Parameters/Variables(typical of Computer Science)

Hardware parameters CPU model and organization, cache organization, latencies in the

system

System parameters Memory availability, usage CPU running time (sometimes approximated by world-clock time) Communication bandwidth, usage Program characteristics

requires floating-point, heavy disk usage, integer math, graphics

large heap, large stack, uses non-local information, ...


Scales of Measurements Nominal (also called categorical): No order, just labels

e.g., “Algorithm Name” Ordinal (also called rank): Order, but not numerical

Difference between ranks is not necessarily the same e.g., ranks in (hierarchical/military) organization

Interval: Difference between values has same meaning everywhere e.g., temperature in Celsius (rise of 10 degrees is the same

everywhere) But 100C is not twice as hot as 50C, and 0C is not lack of heat

Ratio: Interval + Fixed zero point e.g., robot position, memory usage, run-time


Scale Hierarchy

Nominal < Ordinal < Interval < Ratio

Propositions that are true for some level, are true above it But not necessarily the other way around

e.g., we can calculate the mean (average) value for numerical variables But not for nominal and ordinal

e.g., we can calculate the most frequent value for all variables

http://en.wikipedia.org/wiki/Levels_of_measurement

“Numerical”


Variables

Discrete: Can take on only certain values: symbols, exact numbers

For ordinal, interval and ratio scales, this means there will be gaps e.g., User satisfaction surveys, memory usage

Continuous: Can take on any value within its range: no gaps e.g., run-time, CPU temperature, robot velocity and position In practice: limited by measurement accuracy

Up to researcher to determine needed accuracy


Data• The collection of values that a variable X took during

the measurement


Describing Data

Our task: Describe the data we have collected Find ways to characterize it, represent it Find properties that are true of the data


Data Distribution

The collection of data is called the sample distribution We will investigate distributions:

Find values that “best” represent a distribution Measure their dispersion, range, shape Identify extraordinary values in a distribution Find visual representations for a distribution

Remember hierarchy: Nominal < Ordinal < Interval < Ratio Think about how the following techniques apply


Frequency Distribution

Examine the frequency of values

f(x) = # of times variable took on value x.


Frequency Distribution

Examine the frequency of values


Student Grade Grade fX1 60 82 2X2 43 75 1X3 57 60 2X4 82 57 1X5 75 43 1X6 32 32 1X7 82 Total 8X8 60

?


Frequency DistributionExamine the frequency of values


Student GradeX1 60X2 43X3 57X4 82X5 75X6 32X7 82X8 60

Convention (Ordinal/Numerical): Sort by value

Grade f82 281 080 0

...... 075 1

...... 060 2

...... 057 1

...... ...Total 8


Grouped Frequency Distributions In ordinal/numerical variables, possible to group

values together Create Grouped Frequency Distributions

Score f Score f96 1 78 495 0 77 294 0 76 193 0 75 192 0 74 091 1 73 190 1 72 289 3 71 088 2 70 387 2 69 186 6 68 285 2 67 184 2 66 083 1 65 082 3 64 081 2 63 080 2 62 1

Score f96-100 191-95 186-90 1481-85 1076-80 1171-75 466-70 761-65 2

N= 50

Width (i) =5


Grouped Frequency Distributions In ordinal/numerical variables, possible to group

values together Create Grouped Frequency Distributions

Score f Score f96 1 78 495 0 77 294 0 76 193 0 75 192 0 74 091 1 73 190 1 72 289 3 71 088 2 70 387 2 69 186 6 68 285 2 67 184 2 66 083 1 65 082 3 64 081 2 63 080 2 62 1

Score f95-99 190-94 285-89 1580-84 1075-79 1070-74 665-69 460-64 2

N= 50

i=5

Score f96-100 191-95 186-90 1481-85 1076-80 1171-75 466-70 761-65 2

N= 50

i=5

Warning: Loss of Information


Real and Apparent Limits

Continuous values are more difficult to divide into intervals Score of 95 falls within 95-99, not within 90-94 But what about temperature of 94.87 ? 94 < 94.87 < 95 !

By convention, the real limits of a score are within ½ the measurement resolution If our resolution is 0.1, then limits are within 0.05 If our resolution 100, then limits are within 50

Note: we break convention only for exceptional cases e.g., age: “I am 35” is true of [35.0 .. 36.0)


Real/Apparent Limits

For example: Resolution of 0.01. Interval 95..99 really covers values

94.995 to 99.005 Apparent limits: 95..99 Real limits: 94.995 to 99.005

Resolution of 10: 740-800 really covers values 735 to 805.


Relative Frequency Distributions

A frequency count can be misleading Algorithm X was fastest on 60,000 trials: Is this good? 100,000 people voted for candidate A: Is she the winner?

Relative frequency distributions: translate f into percentage or ratio rel f (proportion) = f/N rel f (%) = 100 * f/N

Warning: Can be misleading, if ignoring count magnitude 50% of all test cases succeeded (with only two cases…)


Relative Frequency Distributions

Score f (%)95-99 290-94 485-89 3080-84 2075-79 2070-74 1265-69 860-64 4

N= 100

i=5

Score f95-99 190-94 285-89 1580-84 1075-79 1070-74 665-69 460-64 2

N= 50

i=5

f/N

Example:


Cumulative Frequency Distribution For ordinal/numerical variables Where values are with respect to others: How many

below or above

Cumulative frequency distribution


Cumulative Frequency DistributionBased on the cumulative distribution, can answer

question such as: What percentage of scores fall below 80? How many scores below 95?


Percentiles, Percentile Ranks (Value of) Percentile X: Value for which X percent of values are lower

e.g. baby height We use P

x to denote the Xth percentile, e.g., P

98 is in range 90-94.

Percentile rank of value X: the percent of values that fall below X. e.g., percentile rank of the interval 65-69 is 12.


Computing Percentiles, P. RanksHow do we compute percentiles and percentile ranks

from grouped data? What is the score which defines the top 20% of scores? Is it between 84 and 85?

Score f f (%)

95-99 1 2

90-94 2 4

85-89 15 30

80-84 10 20

75-79 10 20

70-74 6 12

65-69 4 8

60-64 2 4N= 50 100


Computing Percentiles

We want to compute P80

. 80% of 50 cases = 40 cases.

We look under the cum f heading. 32 of the 40 scores are less than 84.5 (real limit).





We look under the cum f heading. 32 of the 40 scores are less than 84.5 (real limit).i =5

Score cum f cum %95-99 50 10090-94 49 9885-89 47 9480-84 32 6475-79 22 4470-74 12 2465-69 6 1260-64 2 4






We need 8 more.i =5

Score cum fcum %95-99 50 10090-94 49 9885-89 47 9480-84 32 6475-79 22 4470-74 12 2465-69 6 1260-64 2 4


The interval 85-89 contains 47-32 = 15 cases. real limit 84.5

These are spread over width of 5 (= 89.5-84.5).

Assume scores are evenly distributed within interval 8 more cases ==>

8/15 * 5 = 2.67 (linear interpolation)

P80

= 84.5 + 2.67 = 87.17

Computing Percentiles We want to compute P

80. 80% of 50 cases = 40 cases.


We need 8 more.


Computing Percentile Ranks We want to compute the percentile rank of 86 Lies in the interval 85-89, real limits 84.5 – 89.5. 86-84.5 = 1.5 score points. Width of interval = 5. Assuming uniform spread of scores in interval:

1.5/5 = 0.3 ==> 30% of scores in interval (0.3*15 = 4.5)

Score cum fcum %95-99 50 10090-94 49 9885-89 47 9480-84 32 6475-79 22 4470-74 12 2465-69 6 1260-64 2 4


Computing Percentile Ranks We want to compute the percentile rank of 86 Lies in the interval 85-89, real limits 84.5 – 89.5. 86-84.5 = 1.5 score points. Width of interval = 5. 1.5/5 = 0.3 ==> 30% of scores in

interval (0.3*15 = 4.5)

So we have 32 scores up to 84.5 4.5 scores from 84.5 to 86. Total: 4.5 + 32 = 36.5 scores. 36.5/50 = 73%. This is the percentile rank

of 86.

Score cum fcum %95-99 50 10090-94 49 9885-89 47 9480-84 32 6475-79 22 4470-74 12 2465-69 6 1260-64 2 4


Frequency Distributions and Scales


Displaying Frequency Distributions:Nominal Data


Displaying Frequency Distributions:Ordinal/Numerical Data

Histogram

Score f f (%)

95-99 1 2

90-94 2 4

85-89 15 30

80-84 10 20

75-79 10 20

70-74 6 12

65-69 4 8

60-64 2 4N= 50 100 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

0

2

4

6

8

10

12

14

16

2

4

6

10 10

15

2

1

Scores Histogram


Displaying Frequency Distributions:Ordinal/Numerical Data

Histogram: Different Grouping

Score f f (%)

95-99 1 2

90-94 2 4

85-89 15 30

80-84 10 20

75-79 10 20

70-74 6 12

65-69 4 8

60-64 2 4N= 50 100 60-69 70-79 80-89 90-99

0

5

10

15

20

25

30

6

16

25

3

Scores Histogram


Lying with Visuals

60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

0

120108 110 111 112 111 110 109 108

Scores Histogram

60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99

107

115

108

110

111

112

111

110

109

108

Scores Histogram


Characteristics of Distributions

Shape, Central Tendency, Variability

Different Central Tendency

Different Variability

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Statistical Methods in Computer Science