Download Descriptive Statistics - San Francisco State University

Descriptive Statistics
A.A. Elimam
College of Business
San Francisco State University
Statistics
The Science of collecting,
organizing, analyzing,
interpreting and presenting data
Topics
• Descriptive Statistics
• Frequency Distributions and Histograms
Relative / Cumulative Frequency
• Measures of Central Tendency
Mean, Median, Mode, Midrange
Topics
• Measures of Dispersion (Variation)
Range, Standard Deviation,
Variance and Coefficient of variation
• Shape
Symmetric, Skewed, using Box-andWhisker Plots
• Quartile
• Statistical Relationships
Correlation , Covariance
Descriptive Statistics
A collection of quantitative measures and
ways of describing data. This includes:
Frequency distributions & histograms,
measures of central tendency
and
measures of dispersion
Descriptive Statistics
•Collect Data
e.g. Survey
•Present Data
e.g. Tables and Graphs
•Characterize Data
e.g. Mean
 xi
n
A Characteristic of a:
Population is a Parameter
Sample is a Statistic.
Summary Measures
Summary Measures
Central Tendency
Mean
Quartile
Mode
Median
Range
Midrange
Variation
Coefficient of
Variation
Variance
Standard Deviation
Measures of Central Tendency
Central Tendency
Mean
Median
Mode
n
xi
i 1
n
Midrange
The Mean (Arithmetic Average)
•It is the Arithmetic Average of data values:
x 
Sample Mean
n
 xi
i 1
n
xi  x2      xn

n
•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
The Median
•Important Measure of Central Tendency
•In an ordered array, the median is the
“middle” number.
•If n is odd, the median is the middle number.
•If n is even, the median is the average of the 2
middle numbers.
•Not Affected by Extreme Values
0 1 2 3 4 5 6 7 8 9 10
Median = 5
0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
The Mode
•A Measure of Central Tendency
•Value that Occurs Most Often
•Not Affected by Extreme Values
•There May Not be a Mode
•There May be Several Modes
•Used for Either Numerical or Categorical Data
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
Midrange
•A Measure of Central Tendency
•Average of Smallest and Largest
Observation:
Midrange

x l arg est  x smallest
2
•Affected by Extreme Value
0 1 2 3 4 5 6 7 8 9 10
Midrange = 5
0 1 2 3 4 5 6 7 8 9 10
Midrange = 5
Quartiles
•
•
Not a Measure of Central Tendency
Split Ordered Data into 4 Quarters
25%
25%
Q1
•
25%
Q2
Position of i-th Quartile:
25%
Q3
position of point
Qi 
i(n+1)
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of Q1 =
1•(9 + 1)
4
= 2.50
Q1 =12.5
Quartiles
•
•
Not a Measure of Central Tendency
Split Ordered Data into 4 Quarters
25%
25%
Q1
•
25%
Q2
Position of i-th Quartile:
25%
Q3
position of point
Qi 
i(n+1)
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of Q3 =
3•(9 + 1)
4
= 7.50
Q3 =19.5
Summary Measures
Summary Measures
Central Tendency
Mean
Median
n
xi
i 1
n
Mode
Midrange
Quartile
Range
Variance
 x i  x 
s 
n 1
2
2
Variation
Coefficient of
Variation
Standard Deviation
Measures of Dispersion (Variation)
Variation
Variance
Range
Population
Variance
Sample
Variance
Standard Deviation
Population
Standard
Deviation
Sample
Standard
Deviation
Coefficient of
Variation
S
CV  
X

  100%

Understanding Variation
• The more Spread out or dispersed data
the larger the measures of variation
• The more concentrated or homogenous the data
the smaller the measures of variation
• If all observations are equal
measures of variation = Zero
• All measures of variation are Nonnegative
The Range
• Measure of Variation
• Difference Between Largest & Smallest
Observations:
Range =
x La rgest  x Smallest
• Ignores How Data Are Distributed:
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
9
10
11
12
7
8
9
10
11
12
Variance
•Important Measure of Variation
•Shows Variation About the Mean:
2
2 Xi   
•For the Population:  
N
•For the Sample:
 X i  X 
s 
n1
2
2
For the Population: use N in the
denominator.
For the Sample : use n - 1
in the denominator.
Standard Deviation
•Most Important Measure of Variation
•Shows Variation About the Mean:
•For the Population:
•For the Sample:
s 
For the Population: use N in the
denominator.

2


X


 i
N
 X i
 X
n 1
2
For the Sample : use n - 1
in the denominator.
Sample Standard Deviation
 X i  X 
n1
2
s
Data:

Xi :
10
12
n=8
s=
For the Sample : use n - 1
in the denominator.
14
15
17 18 18 24
Mean =16
(10  16)2  (12  16)2  (14  16)2  (15  16)2  (17  16)2  (18  16)2  (24  16)2
81
= 4.2426
Comparing Standard Deviations
Data :
X i : 10
N= 8
12
14
15 17 18 18 24
Mean =16
s =
 X i  X 
n 1
 
 X i   
N
2
=
4.2426
=
3.9686
2
Value for the Standard Deviation is larger for data considered as a Sample.
Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 3.338
Data B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .9258
Data C
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Coefficient of Variation
•Measure of Relative Variation
•Always a %
•Shows Variation Relative to Mean
•Used to Compare 2 or More Groups
•Formula ( for Sample):
S 
CV     100%
X 
Comparing Coefficient of Variation
Stock A: Average Price last year = $50
Standard Deviation = $5
Stock B: Average Price last year = $100
Standard Deviation = $5
S 
CV     100%
X 
Coefficient of Variation:
Stock A: CV = 10%
Stock B: CV = 5%
Shape
•
•
Describes How Data Are Distributed
Measures of Shape:
Symmetric or skewed
Shape
•
•
Describes How Data Are Distributed
Measures of Shape:
Symmetric or skewed
-0.5 <0 < 0.5
Symmetric
Mean = Median = Mode
Shape
•
•
Describes How Data Are Distributed
Measures of Shape:
Symmetric or skewed
< -1
-0.5 <0 < 0.5
Left-Skewed
Symmetric
Mean Median Mod
e
Mean = Median = Mode
Shape
•
•
Describes How Data Are Distributed
Measures of Shape:
Symmetric or skewed
< -1
-0.5 <0 < 0.5
Left-Skewed
Symmetric
Mean Median Mod
e
Mean = Median = Mode
>1
Right-Skewed
Mode Median Mean
Box-and-Whisker Plot
Graphical Display of Data Using
5-Number Summary
X smallest Q1 Median Q3
4
6
8
10
Xlargest
12
Distribution Shape &
Box-and-Whisker Plots
Left-Skewed
Q1 Median Q3
Symmetric
Q1
Median Q3
Right-Skewed
Q1 Median Q3
Correlation
A measure of the strength of linear
relationship between two variables X and
Y , and is measured by the (population)
correlation coefficient:
cov  X , Y 


xy
 
x
y
The numerator is the covariance
Covariance
The average of the products of the deviations of
each observation from its respective mean:
 x    y  
N
cov  X , Y  
i 1
i
x
N
i
y

Sample Correlation Coefficient



 xi  x   y i 



i 1 
r
 n  1 s x s y
n

y

Correlation Coefficient ranges from –1 to +1
+1 perfect positive correlation
0 no linear correlation
-1 perfect negative correlation
Summary
• Discussed Measures of Central Tendency
Mean, Median, Mode, Midrange
• Quartiles
• Addressed Measures of Variation
The Range, Interquartile Range, Variance,
Standard Deviation, Coefficient of Variation
• Determined Shape of Distributions
Symmetric, Skewed, Box-and-Whisker Plot
Mean Median Mode
Mean = Median = Mode
Mode Median Mean