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1
Pre University-Mathematics for Business
Session 2
Numerical Descriptive Measures
2
Learning Objectives
• Explain Numerical Data Properties• Describe Summary Measures• Analyze Numerical Data Using Summary
Measures
3
Summary Measures
Arithmetic Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Coefficient of Variation
Range
Interquartile Range
Geometric Mean
Skewness
Central Tendency Variation ShapeQuartiles
4
Measures of Central Tendency
Central Tendency
Arithmetic Mean Median Mode Geometric Mean
n
XX
n
ii
1
n/1n21G )XXX(X
Overview
Midpoint of ranked values
Most frequently observed value
5
Arithmetic Mean
• The arithmetic mean (mean) is the most common measure of central tendency
– For a sample of size n:
Sample size
n
XXX
n
XX n21
n
1ii
Observed values
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Arithmetic Mean
• The most common measure of central tendency• Mean = sum of values divided by the number of values• Affected by extreme values (outliers)
(continued)
0 1 2 3 4 5 6 7 8 9 10
Mean = 3
0 1 2 3 4 5 6 7 8 9 10
Mean = 4
35
15
5
54321
4
5
20
5
104321
7
Median
• In an ordered array, the median is the “middle” number (50% above, 50% below)
• Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Median = 3
0 1 2 3 4 5 6 7 8 9 10
Median = 3
8
Finding the Median
• The location of the median:
– If the number of values is odd, the median is the middle number– If the number of values is even, the median is the average of the
two middle numbers
• Note that is not the value of the median, only the
position of the median in the ranked data
dataorderedtheinposition2
1npositionMedian
2
1n
9
Mode• A measure of central tendency• Value that occurs most often• Not affected by extreme values• Used for either numerical or categorical
(nominal) data• There may be no mode• There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
Chap 3-10
• Five houses on a hill by the beach
Review Example
$2,000 K
$500 K
$300 K
$100 K
$100 K
House Prices:
$2,000,000 500,000 300,000 100,000 100,000
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Review Example:Summary Statistics
• Mean: ($3,000,000/5)
= $600,000
• Median: middle value of ranked data
= $300,000
• Mode: most frequent value = $100,000
House Prices:
$2,000,000
500,000 300,000 100,000 100,000
Sum $3,000,000
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• Mean is generally used, unless extreme values (outliers) exist
• Then median is often used, since the median is not sensitive to extreme values.– Example: Median home prices may be
reported for a region – less sensitive to outliers
Which measure of location is the “best”?
13
Quartiles
• Quartiles split the ranked data into 4 segments with an equal number of values per segment
25% 25% 25% 25%
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
Q1 Q2 Q3
14
Quartile Formulas
Find a quartile by determining the value in the appropriate position in the ranked data, where
First quartile position: Q1 = (n+1)/4
Second quartile position: Q2 = (n+1)/2 (the median position)
Third quartile position: Q3 = 3(n+1)/4
where n is the number of observed values
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(n = 9)
Q1 is in the (9+1)/4 = 2.5 position of the ranked data
so use the value half way between the 2nd and 3rd values,
so Q1 = 12.5
Quartiles
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Example: Find the first quartile
Q1 and Q3 are measures of noncentral location
Q2 = median, a measure of central tendency
16
(n = 9)
Q1 is in the (9+1)/4 = 2.5 position of the ranked data,
so Q1 = 12.5
Q2 is in the (9+1)/2 = 5th position of the ranked data,
so Q2 = median = 16
Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data,
so Q3 = 19.5
Quartiles
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Example:(continued)
17
Geometric Mean• Geometric mean
– Used to measure the rate of change of a variable over time
• Geometric mean rate of return– Measures the status of an investment over time
– Where Ri is the rate of return in time period i
n/1n21G )XXX(X
1)]R1()R1()R1[(R n/1n21G
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Example
An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two:
000,100$X000,50$X000,100$X 321
50% decrease 100% increase
The overall two-year return is zero, since it started and ended at the same level.
19
Example
Use the 1-year returns to compute the arithmetic mean and the geometric mean:
%0111)]2()50[(.
1%))]100(1(%))50(1[(
1)]R1()R1()R1[(R
2/12/1
2/1
n/1n21G
%252
%)100(%)50(X
Arithmetic mean rate of return:
Geometric mean rate of return:
Misleading result
More accurate result
(continued)
20Same center,
different variation
Measures of Variation
Variation
Variance Standard Deviation
Coefficient of Variation
Range Interquartile Range
Measures of variation give information on the spread or variability of the data values.
21
Range• Simplest measure of variation• Difference between the largest and the
smallest values in a set of data:
Range = Xlargest – Xsmallest
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 14 - 1 = 13
Example:
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• Ignores the way in which data are distributed
• Sensitive to outliers
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
Disadvantages of the Range
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 5 - 1 = 4
Range = 120 - 1 = 119
23
Interquartile Range• Can eliminate some outlier problems by
using the interquartile range
• Eliminate some high- and low-valued observations and calculate the range from the remaining values
• Interquartile range = 3rd quartile – 1st quartile
= Q3 – Q1
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Interquartile Range
Median
(Q2)
XmaximumX
minimum Q1 Q3
Example:
25% 25% 25% 25%
12 30 45 57 70
Interquartile range
= 57 – 30 = 27
25
• Average (approximately) of squared deviations of values from the mean
– Sample variance:
Variance
1-n
)X(XS
n
1i
2i
2
Where = mean
n = sample size
Xi = ith value of the variable X
X
26
Standard Deviation
• Most commonly used measure of variation• Shows variation about the mean• Is the square root of the variance• Has the same units as the original data
– Sample standard deviation:
1-n
)X(XS
n
1i
2i
27
Calculation Example:Sample Standard Deviation
Sample Data (Xi) : 10 12 14 15 17 18 18 24
n = 8 Mean = X = 16
4.30957
130
18
16)(2416)(1416)(1216)(10
1n
)X(24)X(14)X(12)X(10S
2222
2222
A measure of the “average” scatter around the mean
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Measuring variation
Small standard deviation
Large standard deviation
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Comparing Standard Deviations
Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5 S = 0.926
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5 S = 4.567
Data C
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Advantages of Variance and Standard Deviation
• Each value in the data set is used in the calculation
• Values far from the mean are given extra weight (because deviations from the mean are squared)
31
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
100%X
SCV
32
Comparing Coefficient of Variation
• Stock A:– Average price last year = $50– Standard deviation = $5
• 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
10%100%$50
$5100%
X
SCVA
5%100%$100
$5100%
X
SCVB
33
Shape of a Distribution
• Describes how data are distributed• Measures of shape
– Symmetric or skewed
Mean = Median Mean < Median Median < Mean
Right-SkewedLeft-Skewed Symmetric
34
Numerical Measures for a Population
• Population summary measures are called parameters
• The population mean is the sum of the values in the
population divided by the population size, N
N
XXX
N
XN21
N
1ii
μ = population mean
N = population size
Xi = ith value of the variable X
Where
35
• Average of squared deviations of values from the mean
– Population variance:
Population Variance
N
μ)(Xσ
N
1i
2i
2
Where μ = population mean
N = population size
Xi = ith value of the variable X
36
Population Standard Deviation• Most commonly used measure of variation• Shows variation about the mean• Is the square root of the population
variance• Has the same units as the original data
– Population standard deviation:
N
μ)(Xσ
N
1i
2i
37
Summary of Variation Measures
Measure Formula Description
Range X largest – X smallest Total Spread
Standard Deviation(Sample)
X X
ni
2
1
Dispersion aboutSample Mean
Standard Deviation(Population)
X
N
i X 2 Dispersion about
Population Mean
Variance(Sample)
(Xi X )2
n – 1Squared Dispersionabout Sample Mean
38
• If the data distribution is approximately bell-shaped, then the interval:
• contains about 68% of the values in the population or the sample
The Empirical Rule
1σμ
μ
68%
1σμ
39
• contains about 95% of the values in the population or the sample
• contains about 99.7% of the values in the population or the sample
The Empirical Rule
2σμ
3σμ
3σμ
99.7%95%
2σμ
40
Interpreting Standard Deviation
41
• Regardless of how the data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1)
– Examples:
(1 - 1/12) x 100% = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22) x 100% = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32) x 100% = 89% ………. k=3 (μ ± 3σ)
Chebyshev Rule
withinAt least
42
Interpreting Standard Deviation: Chebyshev’s Theorem
• Applies to any shape data set
No useful information about the fraction of data in the interval x – s to x + s
At least 3/4 of the data lies in the interval x – 2s to x + 2s
At least 8/9 of the data lies in the interval x – 3s to x + 3s
In general, for k > 1, at least 1 – 1/k2 of the data lies in the interval x – ks to x + ks
43
Interpreting Standard Deviation: Chebyshev’s Theorem
sx 3 sx 3sx 2 sx 2sx xsx
No useful information
At least 3/4 of the data
At least 8/9 of the data
44
Thinking Challenge
• You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11.
• What are the variance and standard deviation of the stock prices?
45
Variation Solution*
Sample Variance
Raw Data: 17 16 21 18 13 16 1211
S
X X
nX
X
n
S
ii
n
ii
n
2
2
1 1
2
2 2 21
15 5
17 15 5 16 15 5 11 15 5
8 11114
( )
( ) ( ) ( )where .
. . .
.
…
46
Variation Solution*
Sample Standard Deviation
S S
X X
n
ii
n
2
2
1
11114 3 34
( ). .
47
Chebyshev’s Theorem Example
• Previously we found the mean closing stock price of new stock issues is 15.5 and the standard deviation is 3.34.
• Use this information to form an interval that will contain at least 75% of the closing stock prices of new stock issues.
48
Chebyshev’s Theorem Example
At least 75% of the closing stock prices of new stock issues will lie within 2 standard deviations of the mean.
x = 15.5 s = 3.34
(x – 2s, x + 2s) = (15.5 – 2∙3.34, 15.5 + 2∙3.34)
= (8.82, 22.18)
49
Ethical Considerations
Numerical descriptive measures:
• Should document both good and bad results
• Should be presented in a fair, objective and neutral manner
• Should not use inappropriate summary measures to distort facts
50
Chapter Summary
• Described measures of central tendency– Mean, median, mode, geometric mean
• Discussed quartiles• Described measures of variation
– Range, interquartile range, variance and standard deviation, coefficient of variation, Z-scores
• Illustrated shape of distribution– Symmetric, skewed, box-and-whisker plots
