Transcript Basic Business Statistics, 8th Edition
Statistics for Managers 4th Edition
Chapter 3 Numerical Descriptive Measures
Chap 3-1
Chapter Topics
Measures of central tendency Mean, median, mode, geometric mean, Quartile Measure of variation Range, interquartile range, average deviation, variance and standard deviation, coefficient of variation, standard units, Sharpe ratio, Sortino ratio Shape Chap 3-2
Chapter Topics
(continued) Ethical considerations Chap 3-3
Summary Measures
Summary Measures Central Tendency Mean Median Mode Quartile Range Variance Variation Coefficient of Variation Geometric Mean Standard Deviation
Chap 3-4
Measures of Central Tendency
Central Tendency Average
X
i n
1
X i n
i N
1
X i N
Median Mode
X G
Geometric Mean
X
1
X
2
X n
1/
n
Chap 3-5
Mean (Arithmetic Mean)
Mean (arithmetic mean) of data values Sample mean Sample Size
X
i n
1
X i
X
1
X
2
n X n
Population Size
i N
1
X i N
X
1
X
2
X N N
Chap 3-6
Mean (Arithmetic Mean)
(continued) The most common measure of central tendency Affected by extreme values (outliers)
0 1 2 3 4 5 6 7 8 9 10 Mean = 5 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6
Chap 3-7
Mean of Grouped Data
Class 10 but under 20 20 but under 30 30 but under 40 40 but under 50 50 but under 60 X = (F) Frequency 3 6 5 4 2 20 S M
•
n F) = 660 20 = 33 (M) Mid-Point 15 25 35 45 55 M
•
F 45 150 175 180 110 660 Chap 3-8
Median
Robust measure of central tendency Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5
In an ordered array, the median is the “middle” number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers Chap 3-9
Median of Group Data
Class 10 but under 20 20 but under 30 30 but under 40 40 but under 50 50 but under 60 (F) Frequency 3 6 5 4 2 20 Step 1: Locate Median Term
Median Class
M T = n 2 = 20 2 Step 2: Assign a Value to the Median Term M D = L+ (M T S F P ) F MD
•
(i)= 30+ (10 - 9) 5
•
10 = 32 = 10 Chap 3-10
Mode
A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may 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 3-11
Geometric Mean
Useful in the measure of rate of change of a variable over time
X G
X
1
X
2
X n
1/
n
Geometric mean rate of return Measures the status of an investment over time
R G
1
R
1
R
2
R n
1/
n
1 Chap 3-12
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:
X
1 $100, 000
X
2 $50, 000
X
3 Average rate of return: $100, 000
X
25% 2 Geometric rate of return:
R G
1
50%
0.50
1/ 2
100%
1/ 2 1/ 2 1 Chap 3-13
Quartiles
Split Ordered Data into 4 Quarters
25% 25% 25% 25%
i
1
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of
Q
1
Q Q
1 2
Q
3 4 2.5
Q
1 2 12.5
= Median, A Measure of Central Tendency Chap 3-14
Measures of Variation
Variation Range Variance Population Variance Sample Variance Interquartile Range Standard Deviation Population Standard Deviation Sample Standard Deviation Coefficient of Variation
Chap 3-15
Range
Measure of variation Difference between the largest and the smallest observations:
Range
X
Largest
X
Smallest Ignores the way in which data are distributed
Range = 12 - 7 = 5 Range = 12 - 7 = 5 7 8 9 10 11 12 7 8 9 10 11 12
Chap 3-16
Interquartile Range
Measure of variation Also known as midspread Spread in the middle 50% Difference between the first and third quartiles
Data in Ordered Array: 11 12 13 16 16 17 17 18 21
5 Interquartile Range
Q
3
Q
1 Not affected by extreme values Chap 3-17
9 6 3 6
X
1
25 (X-X)
- 4 - 2 + 1 + 4 + 1
0
Average Deviation
X-X
4 2 1 4 1
12
X = S X n = 25 5 = 5 AD = S
X-X
n = 12 5 = 2.4
Chap 3-18
Variance
Important measure of variation Shows variation about the mean Sample variance:
S
2
i n
1
X i n
1
X
2 Population variance: 2
i N
1
X i
2
N
Chap 3-19
Standard Deviation
Most important measure of variation Shows variation about the mean Has the same units as the original data Sample standard deviation:
S
i n
1
X i n
1
X
2 Population standard deviation:
i N
1
X i N
2 Chap 3-20
Comparing Standard Deviations
Data A 11 12 13 14 15 16 17 18 19 20 21 Data B 11 12 13 14 15 16 17 18 19 20 21 Data C 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5
s = 3.338
Mean = 15.5
s = .9258
Mean = 15.5
s = 4.57
Chap 3-21
Coefficient of Variation
Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units
CV
X
100% Chap 3-22
Comparing Coefficient of Variation
Stock A: Average price last year = $50 Standard deviation = $5 Stock B: Average price last year = $100 Standard deviation = $5 Coefficient of variation: Stock A:
CV
S X
100% Stock B:
S CV
100%
X
$5 $50 $5 $100 100% 5% Chap 3-23
Using
z scores
to evaluate performance (Example
)
The industry in which sales rep Bill works has average annual sales of $2,500,000 with a standard deviation of $500,000. The industry in which sales rep Paula works has average annual sales of $4,800,000 with a standard deviation of $600,000. Last year Rep Bill’s sales were $4,000,000 and Rep Paula’s sales were $6,000,000. Which of the representatives would you hire if you had one sales position to fill?
Chap 3-24
Standard Units
Sales person Bill B = $2,500,000 B = $500,000 X B = $4,000,000 Z Z B = P = X X B B P P P B = = Sales person Paula P =$4,800,000 P = $600,000 X P = $6,000,000 4,000,000 – 2,500,000 = 500,000 +3 6,000,000 – 4,800,000 = 600,000 +2 Chap 3-25
SHARPE RATIO
Sharpe ratio = (Prr – RFrr)/Srr Where: Prr = Annualized average return on the portfolio RFrr = Annualized average return on risk free proxy Srr = Annualized standard deviation of average returns Sharpe R = (10.5 – 2.5)/ 3.5 = 2.29
Generally, the higher the better.
Chap 3-26
SORTINO RATIO
Sortino Ratio = (Prr – RFrr)/Srr(downside) Where: Prr = Annualized rate of return on portfolio RFrr= Annualized risk free annualized rate of return on portfolio Srr(downside) = downside semi-standard deviation Sortino = (10.5-2.5)/ 2.5 = 3.20
Doesn’t penalize for positive upside returns which the Sharpe ratio does Chap 3-27
Shape of a Distribution
Describes how data is distributed Measures of shape Symmetric or skewed
Left-Skewed Mean < Median < Mode Symmetric Mean = Median = Mode Right-Skewed Mode < Median < Mean
Chap 3-28
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 Chap 3-29
Chapter Summary
Described measures of central tendency Mean, median, mode, geometric mean Discussed quartile Described measure of variation Range, interquartile range, average deviation, variance, and standard deviation, coefficient of variation, standard units, Sharp ratio, Sortino ratio Illustrated shape of distribution Symmetric, skewed Chap 3-30