Download Descriptive Statistics

LSSG Green Belt
Training
Descriptive Statistics
Descriptive Statistics
1
Describing Data: Summary Measures
Measures of Central Location
Mean, Median, Mode
Measures of Variation
Range, Variance and Standard Deviation
Descriptive Statistics
2
Mean
•It is the Arithmetic Average of data values:
x=
Sample Mean
n
 xi
i =1
n
xi + x 2 +    + 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
Descriptive Statistics
Mean = 6
3
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 Median
4 5 6 7 =85 9 10
0 1 2 Median
3 4 5 6 =7 58 9 10 12 14
Descriptive Statistics
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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
Mode = 5
Descriptive Statistics
No
0 1Mode
2 3 4
5 6
5
Measures Of Variability
Range and Inter Quartile Range
Variance and Standard Deviation
Coefficient of Variation
Descriptive Statistics
6
Range
• Measure of Variation
• Difference Between Largest & Smallest
Observations:
Range = Highest Value – Lowest Value
• Ignores How Data Are Distributed:
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
9
10
11
12
7
Descriptive Statistics
8
9
10
11
12
7
Inter Quartile Range

Difference between the 75th percentile (3rd Quartile) and
the 25th percentile (1st Quartile)

Eliminates Effects of Outliers

Captures how data are distributed around the median
(2nd Quartile)
Q1
Q2
Q3
IQR
Descriptive Statistics
8
Variance
•Important Measure of Variation
•Shows Variation About the Mean
2
(
)
m
2
 Xi
2
s = 
m)
s = N
•For the Population:
•For the Sample:
S2=
2
2
(
)
 Xi
X
(X-i - )
2
s = n
Descriptive Statistics
1
n -1
9
Standard Deviation
•Most Important Measure of Variation
•Shows Variation About the Mean
2
(
)
m

Xi
•For the Population: s =
N
•For the Sample:
s =
 (X i
For the Population: use N in
the denominator.
- X
n -1
)2
For the Sample : use n - 1 in the
denominator.
Descriptive Statistics
10
Sample Standard Deviation
s =
 (X i
- X
n -1
X
i :
Data:
10
12
n=8
)2
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
Sample Standard Deviation= 4.24
Descriptive Statistics
11
Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 3.3
Data B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .92
Data C
11 12 13 14 15 16 17 18 19 20 21
Descriptive Statistics
Mean = 15.5
s = 4.57
12
Coefficient of Variation
Relative
Variation (adjusted for the mean)
Measured
Adjusts
as a %
for differences in magnitude of data
Comparison
of variation across groups
S 
CV =    100%
X 
Descriptive Statistics
13
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%
Descriptive Statistics
14
Shape of Distribution



Describes How Data Are Distributed
Measures of Shape:
Symmetric or skewed
Left-Skewed
Mean Median Mode
Symmetric
Mean = Median = Mode
Descriptive Statistics
Right-Skewed
Mode Median Mean
15
BOX PLOTS
Captures Many Statistics in One Chart
Mean
Max
Min
Q1
Median
Q3
Descriptive Statistics
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