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Numerical Descriptive Measures
1
Motivation
• What is the “average consumer” exactly?
• Why is it that if an average yield on an investment (e.g. mutual
fund) is 28% that I’ve lost money?
• At Morning Brew people arrive on average 1 per min and it takes
typically 1 min to serve them, why is it that if I staff the register
with 1 person people complain that the lines are too long and
often leave before purchasing something?
• If I plan our payables based on an average daily receivables of
$7,000 why have I gone bankrupt?
• Why do students always want to know what the average on the
exam was?
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Summary Measures
Describing Data Numerically
Central Tendency
Quartiles
Variation
Arithmetic Mean
Range
Median
Interquartile Range
Shape
Skewness
Kurtosis
Mode
Standard Deviation
Coefficient of Variation
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Measures of Central
Tendency
Overview
Central Tendency
Arithmetic Mean
“Balance Point” of
data. Usually not
in data set.
Median
Midpoint of ranked
values. In an ordered
array, the median is
the “middle” number
(50% above, 50%
below). May by in
data set.
Mode
Most frequently
observed value
(multiple or may
not exist esp.
continuous data).
Always in data set.
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Which One to Use?
• Mean is generally used, unless extreme
values (outliers) exist
• Median is often used, since the median is
not sensitive to extreme values.
• Mode is rarely used because there may be
no mode, and there may be several modes
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 …. 500
Mean = 3
Mean = 58
Median = 3
Median = 3
Mode = 2, 4
Mode = 2, 4
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Quartiles
• Quartiles split the ranked data into 4 segments with
an equal number of values per segment
25%
25%
Q1
25th Percentile
25%
25%
Q2
Q3
50th percentile 75th percentile
• The first quartile, Q1, is the value for which 25% of the
observations are smaller and 75% are larger
• Q2 is the same as the median (50% are smaller, 50%
are larger)
• Only 25% of the observations are greater than the third
quartile
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Class Exercise:
Tendency & Histograms
mode
25
5
Histogram
BEER %Alcohol Content
median
20
Q1
4.4
Q2
4.9
Q3
5.1
Q4
6
Percent
15
4.9
10
5
0
mean
Place the mean, median, mode on the histogram. What do you see?
4.42029
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Example
• Median home prices usually are reported for a region – less sensitive to
outliers
• Example: Five houses on a hill by the beach
$2,000 K
$500 K
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Sum $3,000,000
$300 K
•
•
•
Mean:
($3,000,000/5)
= $600,000
Median: middle value of
ranked data
= $300,000
Mode: most frequent
value
= $100,000
$100 K
$100 K
Think about this: Which average best helps you decide what to offer for a house?
How about set selling price? What other considerations might there be?
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What’s The Difference?
Histogram
Histogram
45
45
40
40
35
35
30
Percent
Percent
30
25
20
25
20
15
15
10
10
5
5
0
0
Data
Data
Mean $600,000
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Measures of Variation
• Measures of variation give information on the spread or
variability of the data values.
Range
More variable
Less variable
Interquartile
Range
Variation
Standard
Deviation
Coefficient
of Variation
Same center,
different variation
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Range and Interquartile Range
Example:
X minimu
25%
m
12
Q1
25%
30
Median
(Q2)
Q3
25%
25%
45
57
X maximum
70
Range = 70 – 12 = 58
Range
 = Xmaximum– Xminimum
Interquartile range
= 57 – 30 = 27
Interquartile Range
 Simplest measure
 = Q3 – Q1
 Sensitive to outliers
 Measure Middle 50%
 Eliminate outliers Problem
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Disadvantages of the Range
• Range ignores the way in which data are distributed
6
7
8 9 10 11 12
Range = 13 - 6 = 7
13
IQR = 11 – 8 = 3

6
7
8 9 10 11 12
Range = 13 - 6 = 7
13
IQR = 11 – 10 = 1
Range is also sensitive to outliers
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5
Range = 5 - 1 = 4
IQR = 2 – 1 = 1
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,120
Range = 120 - 1 = 119
IQR = 2 – 1 = 1
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Standard Deviation
•
•
•
•
Most commonly used measure of variation
Each value in the data set is used in the calculation
Shows variation from the mean
Values far from the mean are given extra weight
(because deviations from the mean are squared)
• Has the same units as the original data
n
• Sample standard deviation:
SD 
2
(X

X
)
 i
i 1
n -1
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Coefficient of Variation
• Measures relative variation
• Always in percentage (%)
• Shows variation relative to mean
• Can be used to compare two or more sets of data
measured in different units
 SD 
 100%
CV  

 X 
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Comparing Variation
Standard Deviations
Data A
11
12
13
14
15
16
17
18
19
20 21
Mean = 15.5
SD = 3.338
20 21
Mean = 15.5
SD = 0.926
Data B
11
12
13
14
15
16
17
18
19
Data C
11
12
13
14
15
16
17
18
19
20 21
Mean = 15.5
SD = 4.567
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Comparing Variation
Coefficient of Variation
• Stock A:
– Average price last year = $50
– Standard deviation = $5
 S 
$5
CVA     100% 
 100%  10%
$50
X 
• Stock B:
– Average price last year = $100
– Standard deviation = $5
Both stocks
have the same
standard
deviation, but
stock B is less
variable relative
to its price
S
$5


CVB     100% 
 100%  5%
$100
X
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Shape of a Distribution
• Describes how data are distributed
• Measures of shape
– Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median
Mean = Median
Median < Mean
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Normal Exact
Normal Ok
Skewness = 0
Kortosis = 0
-1< SK <+1
-1< K <+1
Right Skewed
Left Skewed
SK >0
SK <0
Peaked
Flat
K >0
K <0
- 45
-30
Normal or Bell-shaped Curve
Mean and Standard Deviation define
what a normal curve looks like
Example:
Mean: μ =0
Standard Deviation: σ=15
IQR=20
Box-and-Whisker
-15
0
Approx. 50%
15
30
45
Approx. 68%
Approx. 95%
Almost 100%
K>0
SK>0
K<0
Mean, Mode, Median
Peaked
Flat
Right skewed
SK<0
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Left skewed
The Empirical Rule
• If the data distribution is approximately bell-shaped,
then the interval:
• μ  1σ contains about 68% of the values in
the population or the sample
68%
μ
μ  1σ
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The Empirical Rule
•
μ  2σ
contains about 95% of the values in the
population or the sample
• μ  3σ contains about 99.7% of the values in the
population or the sample
95%
99.7%
μ  2σ
μ  3σ
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Exploratory Data Analysis
• 5-number summary:
Minimum -- Q1 -- Median -- Q3 -- Maximum
• Box-and-Whisker Plot: A Graphical display of 5-number
summary. It shows both Central Tendency, Variation, and
Shape of the numerical variable.
25%
Minimum
Minimum
25%
1st
Quartile
1st
Quartile
25%
Median
Median
25%
3rd
Quartile
3rd
Quartile
Maximum
Maximum
Central Tendency
Variation
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Shape and B-n-W Plot
Left-Skewed
Q1
Q2 Q3
Symmetric
Q1 Q2 Q3
Right-Skewed
Q1 Q2 Q3
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Shape and B-n-W Plot Cont’d
Left-Skewed
Symmetric
Right-Skewed
Peaked
Flat
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