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Statistical Methods in
Computer Science
Data 2:
Central Tendency & Variability
Ido Dagan
Frequency Distributions and Scales
Nominal
Ordinal
Numerical Numerical/Continuous
f Distrib.
Groupedf
Distrib.
Apparent/Real Limits
Relativef
F
(Accumulativef)
Percentile,
Per. Ranks
Empirical Methods in Computer Science
© 2006-now Gal Kaminka
2
Characteristics of Distributions
Shape, Central Tendency, Variability
Different Central
Tendency
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Different
Variability
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3
This Lesson

Examine measures of central tendency




Examine measures of variability (dispersion)



Mode (Nominal)
Median (Ordinal)
Mean (Numerical)
Entropy (Nominal)
Variance (Numerical), Standard Deviation
Standard scores (z-score)
Empirical Methods in Computer Science
© 2006-now Gal Kaminka
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Centrality/Variability Measures
and Scales
Nominal
Ordinal
Numerical Numerical/Continuous
Mode
Median
Mean
Entropy
Variance,
Std. Deviation
z-Scores
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The Mode (Mo)
‫השכיח‬

The mode of a variable is the value that is most frequent
Mo = argmax f(x)

For categorical variable: The category that appeared most

For grouped data: The midpoint of the most frequent interval
 Under the assumption that values are evenly distributed in the
interval
Empirical Methods in Computer Science
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Finding the Mode: Example 1
The collection of values that a variable X took during the
measurement
Student Grade
X1
60
X2
43
X3
57
X4
82
X5
75
X6
32
X7
82
X8
60
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System
Windows
Linux
BSD
MacOS
Algorithm Name
Round-Robin
Round-Robin
Prioritized Scheduling
Preemptive Scheduling
Trial Run-Time
#1 23.234
#2 15.471
#3 12.220
#4 23.100
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?
Depends on
Grouping
7
Finding the Mode: Example 2
The mode of a grouped frequency distribution depends on
grouping
86
Score
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
f
1
0
0
0
0
1
1
3
2
2
6
2
2
1
3
2
2
2
Score
78
77
76
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
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f
4
2
1
1
0
1
2
0
3
1
2
1
0
0
0
0
1
1
88
i=5
Score
96-100
91-95
86-90
81-85
76-80
71-75
66-70
61-65
N=
f
1
1
14
10
11
4
7
2
50
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87
i=5
Score
95-99
90-94
85-89
80-84
75-79
70-74
65-69
60-64
N=
f
1
2
15
10
10
6
4
2
50
8
The Median (Mdn)
‫החציון‬


The median of a variable is its 50th percentile, P50.
The point below which 50% of all measurements fall


Requires ordering: Only ordinal and the numerical scales
Examples:


0,8,8,11,15,16,20
==> Mdn = 11
12,14,15,18,19,20 ==> Mdn = 16.5 (halfway between 15 and 18).
Empirical Methods in Computer Science
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The Median (Mdn)
‫החציון‬


The median of a variable is its 50th percentile, P50.
The point below which 50% of all measurements fall


Requires ordering: Only ordinal and the numerical scales
Examples:



0,8,8,11,15,16,20
==> Mdn = 11
12,14,15,18,19,20 ==> Mdn = 16.5 (halfway between 15 and 18).
5,7,8,8,8,8
==> Mdn = ?
Empirical Methods in Computer Science
© 2006-now Gal Kaminka
10
The Median (Mdn)
‫החציון‬


The median of a variable is its 50th percentile, P50.
The point below which 50% of all measurements fall


Requires ordering: Only ordinal and the numerical scales
Examples:



0,8,8,11,15,16,20
==> Mdn = 11
12,14,15,18,19,20
==> Mdn = 16.5 (halfway between 15 and
18).
5,7,8,8,8,8
==> Mdn = ?
 One method: Halfway between first and second 8, Mdn = 8
 Another: Use linear interpolation as we did in intervals, Mdn =
7.75
 7.75 = 7.5 + (¼ * 1.0)
Empirical Methods in Computer Science
© 2006-now Gal Kaminka
11
The Median (Mdn)
‫החציון‬


The median of a variable is its 50th percentile, P50.
The point below which 50% of all measurements fall


Requires ordering: Only ordinal and the numerical scales
Examples:
0,8,8,11,15,16,20
==> Mdn = 11
 12,14,15,18,19,20
==> Mdn = 16.5 (halfway between 15 and
18).
 5,7,8,8,8,8
==> Mdn = ?
 One method: Halfway between first and second 8, Mdn = 8
 Another: Use linear interpolation as we did in intervals, Mdn =
7.75
 7.75
= 7.5
between
7 and
8 + (¼ * 1.0)

Empirical Methods in Computer Science
© 2006-now Gal Kaminka
12
The Median (Mdn)
‫החציון‬


The median of a variable is its 50th percentile, P50.
The point below which 50% of all measurements fall


Requires ordering: Only ordinal and the numerical scales
Examples:
0,8,8,11,15,16,20
==> Mdn = 11
 12,14,15,18,19,20
==> Mdn = 16.5 (halfway between 15 and
18).
 5,7,8,8,8,8
==> Mdn = ?
 One method: Halfway between first and second 8, Mdn = 8
 Another: Use linear interpolation as we did in intervals, Mdn =
7.75
 7.75 = 7.5 + (¼ * 1.0)
1 of four 8's

Empirical Methods in Computer Science
© 2006-now Gal Kaminka
13
The Median (Mdn)
‫החציון‬


The median of a variable is its 50th percentile, P50.
The point below which 50% of all measurements fall


Requires ordering: Only ordinal and the numerical scales
Examples:



0,8,8,11,15,16,20
==> Mdn = 11
12,14,15,18,19,20
==> Mdn = 16.5 (halfway between 15 and
18).
5,7,8,8,8,8
==> Mdn = ?
 One method: Halfway between first and second 8, Mdn = 8
 Another: Use linear interpolation as we did in intervals, Mdn =
7.75
 7.75 = 7.5 + (¼ * 1.0)
Width of interval containing 8's (real limits)
Empirical Methods in Computer Science
© 2006-now Gal Kaminka
14
The Arithmetic Mean
‫ממוצע חשבוני‬
Arithmetic mean (mean, for short)
Average is colloquial: Not precisely defined when used, so we avoid the
term.
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Properties of
Central Tendency Measures
Mo:
 Relatively unstable between samples
 Problematic in grouped distributions
 Can be more than one:


Distributions that have more than one sometimes called multi-modal
For uniform distributions, all values are possible modes
Typically used only on nominal data
Empirical Methods in Computer Science
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Properties of
Central Tendency Measures
Mean:
 Responsive to exact value of each score

Only interval and ratio scales

Takes total of scores into account: Does not ignore any value
Sum of deviations from mean is always zero:

Because of this: sensitive to outliers



Presence/absence of scores at extreme values
Stable between samples, and basis for many other statistical
measures
Empirical Methods in Computer Science
© 2006-now Gal Kaminka
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Properties of
Central Tendency Measures
Median:
 Robust to extreme values


Only cares about ordering, not magnitude of intervals
Often used with skewed distributions
Mo
Mdn
Mean
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Properties of
Central Tendency Measures
Contrasting Mode, Median, Mean
Mo
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Mdn
Mean
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Properties of
Central Tendency Measures
Contrasting Mode, Median, Mean
Mo
Mdn
Mean
Empirical Methods in Computer Science
© 2006-now Gal Kaminka
20