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Lesson03_static11
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Lesson. 3 - 1
Statistics
for Management
Summarizing and Describing Numerical Data
Lesson. 3 - 2
Lesson Topics•Measures of Central Tendency Mean, Median, Mode, Midrange, Midhinge•Quartile
•Measures of Variation The Range, Interquartile Range, Variance and Standard Deviation, Coefficient of variation•Shape Symmetric, Skewed, using Box-and-Whisker Plots
Lesson. 3 - 3
Summary Measures
Central Tendency
MeanMedian
Mode
Midrange
Quartile
Midhinge
Summary Measures
Variation
Variance
Standard Deviation
Coefficient of Variation
Range
Lesson. 3 - 4
Measures of Central Tendency
Central Tendency
Mean Median Mode
Midrange
Midhinge
n
xn
ii
1
Lesson. 3 - 5
The Mean (Arithmetic Average)
•It is the Arithmetic Average of data values:
•The Most Common Measure of Central Tendency
•Affected by Extreme Values (Outliers)
n
xn
1ii
n
xxx n2i
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5 Mean = 6
xSample Mean
Lesson. 3 - 6
The Median
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
•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
Lesson. 3 - 7
The Mode
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
•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
No Mode
Lesson. 3 - 8
Midrange
•A Measure of Central Tendency
•Average of Smallest and Largest
Observation:
•Affected by Extreme Value
2
xx smallestestl arg
Midrange
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Midrange = 5 Midrange = 5
Lesson. 3 - 9
Quartiles
• Not a Measure of Central Tendency• Split Ordered Data into 4 Quarters
• Position of i-th Quartile: position of point
25% 25% 25% 25%
Q1 Q2 Q3
Q i(n+1)i 4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of Q1 = 2.50 Q1 =12.5= 1•(9 + 1)4
Lesson. 3 - 10
Midhinge
• A Measure of Central Tendency
• The Middle point of 1st and 3rd Quarters
• Not Affected by Extreme Values
Midhinge = 2
31 QQ
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Midhinge = 162
519512
231
Lesson. 3 - 11
Summary Measures
Central Tendency
MeanMedian
Mode
Midrange
Quartile
Midhinge
n
xn
ii
1
Summary Measures
Variation
Variance
Standard Deviation
Coefficient of Variation
Range
1n
xxs
2i2
Lesson. 3 - 12
Measures of Variation
Variation
Variance Standard Deviation Coefficient of Variation
PopulationVariance
Sample
Variance
PopulationStandardDeviationSample
Standard
Deviation
Range
Interquartile Range
100%
X
SCV
Lesson. 3 - 13
• Measure of Variation
• Difference Between Largest & Smallest Observations:
Range =
• Ignores How Data Are Distributed:
The Range
SmallestrgestLa xx
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
Lesson. 3 - 14
• Measure of Variation
• Also Known as Midspread: Spread in the Middle 50%
• Difference Between Third & First
Quartiles: Interquartile Range =
• Not Affected by Extreme Values
Interquartile Range
13 QQ Data in Ordered Array: 11 12 13 16 16 17 17 18 21
13 QQ = 17.5 - 12.5 = 5
Lesson. 3 - 15
•Important Measure of Variation
•Shows Variation About the Mean:
•For the Population:
•For the Sample:
Variance
N
X i
22
1
22
n
XXs i
For the Population: use N in the denominator.
For the Sample : use n - 1 in the denominator.
Lesson. 3 - 16
•Most Important Measure of Variation
•Shows Variation About the Mean:
•For the Population:
•For the Sample:
Standard Deviation
N
X i
2
1
2
n
XXs i
For the Population: use N in the denominator.
For the Sample : use n - 1 in the denominator.
Lesson. 3 - 17
Sample Standard Deviation
1
2
n
XX i For the Sample : use n - 1 in the denominator.
Data: 10 12 14 15 17 18 18 24
s =
n = 8 Mean =16
18
1624161816171615161416121610 2222222
)()()()()()()(
= 4.2426
s
:X i
Lesson. 3 - 18
Comparing Standard Deviations
1
2
n
XX is =
= 4.2426
N
X i
2 = 3.9686
Value for the Standard Deviation is larger for data considered as a Sample.
Data : 10 12 14 15 17 18 18 24:X i
N= 8 Mean =16
Lesson. 3 - 19
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 = .9258
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5 s = 4.57
Data C
Lesson. 3 - 20
Coefficient of Variation
•Measure of Relative Variation
•Always a %
•Shows Variation Relative to Mean
•Used to Compare 2 or More Groups
•Formula ( for Sample):
100%
X
SCV
Lesson. 3 - 21
Comparing Coefficient of Variation
Stock A: Average Price last year = $50
Standard Deviation = $5
Stock B: Average Price last year = $100
Standard Deviation = $5
100%
X
SCV
Coefficient of Variation:
Stock A: CV = 10%
Stock B: CV = 5%
Lesson. 3 - 22
Shape
• Describes How Data Are Distributed
• Measures of Shape: Symmetric or skewed
Right-SkewedLeft-Skewed SymmetricMean = Median = ModeMean Median Mode Median MeanMode
Lesson. 3 - 23
Box-and-Whisker Plot
Graphical Display of Data Using50%-Number Summary
Median
4 6 8 10 12
Q3Q1 XlargestXsmallest
Lesson. 3 - 24
Distribution Shape & Box-and-Whisker Plots
Right-SkewedLeft-Skewed Symmetric
Q1 Median Q3Q1 Median Q3 Q1
Median Q3
Lesson. 3 - 25
Lesson Summary• Discussed Measures of Central Tendency Mean, Median, Mode, Midrange, Midhinge
• 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 = ModeMean Median Mode Mode Median Mean