Post on 18-Dec-2015
© 2002 Thomson / South-Western Slide 3-2
Learning ObjectivesLearning Objectives
• Distinguish between measures of central tendency, measures of variability, and measures of shape
• Understand the meanings of mean, median, mode, quartile, percentile, and range
• Compute mean, median, mode, percentile, quartile, range, variance, standard deviation, and mean absolute deviation
© 2002 Thomson / South-Western Slide 3-3
Learning Objectives -- ContinuedLearning Objectives -- Continued
• Differentiate between sample and population variance and standard deviation
• Understand the meaning of standard deviation as it is applied by using the empirical rule
• Understand box and whisker plots, skewness, and kurtosis
© 2002 Thomson / South-Western Slide 3-4
Measures of Central TendencyMeasures of Central Tendency
• Measures of central tendency yield information about “particular places or locations in a group of numbers.”
• Common Measures of Location–Mode–Median–Mean–Percentiles–Quartiles
© 2002 Thomson / South-Western Slide 3-5
ModeMode
• The most frequently occurring value in a data set
• Applicable to all levels of data measurement (nominal, ordinal, interval, and ratio)
• Bimodal -- Data sets that have two modes
• Multimodal -- Data sets that contain more than two modes
© 2002 Thomson / South-Western Slide 3-6
• The mode is 44.• There are more 44s
than any other value.
35
37
37
39
40
40
41
41
43
43
43
43
44
44
44
44
44
45
45
46
46
46
46
48
Mode -- ExampleMode -- Example
© 2002 Thomson / South-Western Slide 3-7
Median (ΔΙΑΜΕΣΟΣ)Median (ΔΙΑΜΕΣΟΣ)
• Middle value in an ordered array of numbers.
• Applicable for ordinal, interval, and ratio data
• Not applicable for nominal data• Unaffected by extremely large and
extremely small values.
© 2002 Thomson / South-Western Slide 3-8
Median: Computational ProcedureMedian: Computational Procedure
• First Procedure– Arrange observations in an ordered array.– If number of terms is odd, the median is
the middle term of the ordered array.– If number of terms is even, the median is
the average of the middle two terms.
• Second Procedure– The median’s position in an ordered array
is given by (n+1)/2.
© 2002 Thomson / South-Western Slide 3-9
Median: Example with an Odd Number of TermsMedian: Example with
an Odd Number of TermsOrdered Array includes: 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22
• There are 17 terms in the ordered array.• Position of median = (n+1)/2 = (17+1)/2 = 9• The median is the 9th term, 15.• If the 22 is replaced by 100, the median
remains at 15.• If the 3 is replaced by -103, the median
remains at 15.
© 2002 Thomson / South-Western Slide 3-10
Mean (ΜΕΣΟΣ)Mean (ΜΕΣΟΣ)
• Is the average of a group of numbers• Applicable for interval and ratio data,
not applicable for nominal or ordinal data
• Affected by each value in the data set, including extreme values
• Computed by summing all values in the data set and dividing the sum by the number of values in the data set
© 2002 Thomson / South-Western Slide 3-11
Population MeanPopulation Mean
X
N NX X X X N1 2 3
24 13 19 26 11
593
518 6
...
.
© 2002 Thomson / South-Western Slide 3-12
Sample MeanSample Mean
XX
n nX X X X n
1 2 3
57 86 42 38 90 66
6379
663 167
...
.
© 2002 Thomson / South-Western Slide 3-13
QuartilesQuartiles
Measures of central tendency that divide a group of data into four subgroups
• Q1: 25% of the data set is below the first quartile
• Q2: 50% of the data set is below the second quartile
• Q3: 75% of the data set is below the third quartile
© 2002 Thomson / South-Western Slide 3-14
Quartiles, continued
• Q1 is equal to the 25th percentile
• Q2 is located at 50th percentile and equals the median
• Q3 is equal to the 75th percentile
Quartile values are not necessarily members of the data set
© 2002 Thomson / South-Western Slide 3-16
• Ordered array: 106, 109, 114, 116, 121, 122, 125, 129
• Q1:• Q2:• Q3:
Quartiles: ExampleQuartiles: Example
i Q
25
1008 2
109 114
2111 51( ) .
i Q
50
1008 4
116 121
2118 52( ) .
i Q
75
1008 6
122 125
2123 53( ) .
© 2002 Thomson / South-Western Slide 3-17
Measures of Variability Measures of Variability • Measures of variability describe the spread
or the dispersion of a set of data.• Common Measures of Variability
–Range–Interquartile Range–Mean Absolute Deviation–Variance–Standard Deviation– Z scores–Coefficient of Variation
© 2002 Thomson / South-Western Slide 3-18
VariabilityVariability
Mean
Mean
MeanNo Variability in Cash Flow
Variability in Cash Flow Mean
© 2002 Thomson / South-Western Slide 3-20
RangeRange• The difference between the largest and
the smallest values in a set of data• Simple to compute• Ignores all data points
except the two extremes
• Example: Range = Largest - Smallest
= 48 - 35 = 13
35
37
37
39
40
40
41
41
43
43
43
43
44
44
44
44
44
45
45
46
46
46
46
48
© 2002 Thomson / South-Western Slide 3-21
Interquartile RangeInterquartile Range
• Range of values between the first and third quartiles
• Range of the “middle half”• Less influenced by extremes
Interquartile Range Q Q 3 1
© 2002 Thomson / South-Western Slide 3-22
Deviation from the MeanDeviation from the Mean
• Data set: 5, 9, 16, 17, 18• Mean:
• Deviations from the mean: -8, -4, 3, 4, 5
X
N
65
513
0 5 10 15 20
-8 -4 +3 +4+5
© 2002 Thomson / South-Western Slide 3-23
Mean Absolute DeviationMean Absolute Deviation
• Average of the absolute deviations from the mean
59
161718
-8-4+3+4+5
0
+8+4+3+4+524
X X X M A D
X
N. . .
.
24
54 8
© 2002 Thomson / South-Western Slide 3-24
Population VariancePopulation Variance
• Average of the squared deviations from the arithmetic mean
59
161718
-8-4+3+4+5
0
6416
916
25130
X X 2
X 2
2
130
526 0
XN
.
© 2002 Thomson / South-Western Slide 3-25
Population Standard DeviationPopulation Standard Deviation• Square root of the
variance
2
2
2
130
526 0
26 0
51
XN
.
.
.
59
161718
-8-4+3+4+5
0
6416
916
25130
X X 2
X
© 2002 Thomson / South-Western Slide 3-26
Empirical RuleEmpirical Rule
• Data are normally distributed (or approximately normal)
1 2 3
95
99.7
68
Distance from the Mean
Percentage of ValuesFalling Within Distance
© 2002 Thomson / South-Western Slide 3-27
Sample VarianceSample Variance
• Average of the squared deviations from the arithmetic mean
2,3981,8441,5391,3117,092
62571
-234-462
0
390,6255,041
54,756213,444663,866
X X X 2
X X 2
2
1663 866
3221 288 67
SX Xn
,
, .
© 2002 Thomson / South-Western Slide 3-28
Sample Standard DeviationSample Standard Deviation
• Square root of the sample variance 2
2
2
1663 866
3221 288 67
221 288 67
470 41
SX X
S
n
S
,
, .
, .
.
2,3981,8441,5391,3117,092
62571
-234-462
0
390,6255,041
54,756213,444663,866
X X X 2
X X
© 2002 Thomson / South-Western Slide 3-29
Coefficient of VariationCoefficient of Variation
• Ratio of the standard deviation to the mean, expressed as a percentage
• Measurement of relative dispersion
C V. .
100
© 2002 Thomson / South-Western Slide 3-30
Coefficient of VariationCoefficient of Variation
1 29
4 6
100
4 6
29100
1586
1
11
1
.
.
.
. .CV
2 84
10
100
10
84100
1190
2
22
2
CV. .
.
© 2002 Thomson / South-Western Slide 3-31
Measures of ShapeMeasures of Shape
• Skewness– Absence of symmetry– Extreme values in one side of a
distribution• Kurtosis
– Peakedness of a distribution• Box and Whisker Plots
– Graphic display of a distribution– Reveals skewness
© 2002 Thomson / South-Western Slide 3-32
SkewnessSkewness
NegativelySkewed
PositivelySkewed
Symmetric(Not Skewed)
© 2002 Thomson / South-Western Slide 3-33
SkewnessSkewness
NegativelySkewed
Mode
Median
Mean
Symmetric(Not Skewed)
MeanMedianMode
PositivelySkewed
Mode
Median
Mean
© 2002 Thomson / South-Western Slide 3-34
Coefficient of SkewnessCoefficient of Skewness• Summary measure for skewness
• If S < 0, the distribution is negatively skewed (skewed to the left).
• If S = 0, the distribution is symmetric (not skewed).
• If S > 0, the distribution is positively skewed (skewed to the right).
S
M d
3
© 2002 Thomson / South-Western Slide 3-35
Coefficient of SkewnessCoefficient of Skewness
1
1
1
1
1 1
1
23
26
12 3
3
3 23 26
12 3073
M
SM
d
d
.
..
2
2
2
2
2 2
2
26
26
12 3
3
3 26 26
12 30
M
SM
d
d
.
.
3
3
3
3
3 3
3
29
26
12 3
3
3 29 26
12 3073
M
SM
d
d
.
..
© 2002 Thomson / South-Western Slide 3-36
KurtosisKurtosis• Peakedness of a distribution
– Leptokurtic: high and thin– Mesokurtic: normal in shape– Platykurtic: flat and spread out
Leptokurtic
MesokurticPlatykurtic
© 2002 Thomson / South-Western Slide 3-37
Box and Whisker PlotBox and Whisker Plot
• Five specific values are used:– Median, Q2
– First quartile, Q1
– Third quartile, Q3
– Minimum value in the data set– Maximum value in the data set
© 2002 Thomson / South-Western Slide 3-38
Box and Whisker Plot, continued
• Inner Fences– IQR = Q3 - Q1
– Lower inner fence = Q1 - 1.5 IQR– Upper inner fence = Q3 + 1.5 IQR
• Outer Fences– Lower outer fence = Q1 - 3.0 IQR– Upper outer fence = Q3 + 3.0 IQR
© 2002 Thomson / South-Western Slide 3-39
Box and Whisker PlotBox and Whisker Plot
Q1 Q3Q2Minimum Maximum