Lecture 3. NUMERICAL SUMMARY Lecture 3. NUMERICAL SUMMARY
Data Measurements
Locations Variability Measures Shape
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[1] Chapter 3, pp. 99 - 162 [3] Chapter 2
Comparison Comparison
Profit of Project A (million)
30%
Profit of two project A & B
20%
20%
15%
10%
5%
1
2
3
4
5
6
Profit of Project B (million)
30%
20%
20%
15%
10%
5%
1
2
3
4
5
6
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Comparison Comparison
Profit of Project C (million)
30%
20%
20%
15%
8%
5%
2%
0%
0%
1
2
3
4
5
6
7
8
9
Profit of Project D (million)
30%
20%
20%
15%
8%
5%
2%
0%
0%
1
2
3
4
5
6
7
8
9
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Comparison Comparison
Profit of Project E (million)
20%
20%
15%
15%
10%
10%
5%
5%
-1
0
1
2
3
4
5
6
Profit of Project F (million)
40%
40%
10%
10%
0%
0%
0%
0%
-1
0
1
2
3
4
5
6
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Data Measurements Data Measurements
Location:
Minimum, Maximum Central Tendency: Mean, Median, Mode Quantile: Quartile, Percentile
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Variability: Range Variance (Var) Standard Deviation (SD) Coefficient of Variation (CV) Interquartile Range (IQR)
3.1. Mean (arithmetic mean) 3.1. Mean (arithmetic mean)
Apply for scale variable only
=
Population Data: { , ,… , } Sample Data: { , ,… , }
= =
+ + ⋯ + + + ⋯ +
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Have the same unit as the original data
Weighted mean Weighted mean
Price ($) in Quarter 1, 2, 3, 4 are 10, 12, 18, 14,
respectively.
= =
10 + 12 + 18 + 14 4
Any difference if the volume of sales in Quarter 1, 2, 3,
4 are 70, 90, 110, 130?
Value xi
Q1
Q2
Q3
Q4
Price
10
12
18
14
Weight wi
Volume
70
90
110
130
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Weighted Mean Weighted Mean
In general, for grouped data:
= =
+ + ⋯ + + + ⋯ + ∑ ∑
For Example of Price:
̅ = =
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70 ∗ 10 + 90 ∗ 12 + 110 ∗ 18 + 130 ∗ 14 70 + 90 + 110 + 130
Mean of Grouped data Mean of Grouped data
Frequency, Proportion, Percent table
Wage ($) 7 8 9
4 10 6
0.2 0.5 0.3
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Number of worker (Frequency) Proportion (Relative frequency) Percent 20% 50% 30%
Compare the Mean Compare the Mean
Compare the mean of following data: Data 1: {10, 10, 11, 12, 12} Data 2: {5, 5, 6, 6, 100}
The mean is easily affected by the extreme or outlier
value
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May lead to biased comparison Use the other measures
3.2. Median 3.2. Median
Median, denoted by me, is the midpoint of ordered
list of values
Median could be applied for ordinal variable
Ex. Data: { 5, 6, 9, 5, 6 }
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Ordered data: { 5, 5, 6, 6, 9 } : Median = Ordered Data {6, 6, 7, 8, 9, 11} : Median = Data: {XXS, XS, S, S, S, M, L, XL, XXL}: Median =
MedianMedian
Median is the ‘cutoff point’ of lower 50% - upper 50%
Discrete vs Continous
Lower 50%
Upper 50%
Discrete
Continuous
Median
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parts
3.3. Mode 3.3. Mode
Mode, denoted by m0, is the value that occurs most
often, frequency of (X = m0) is the largest. There may be no mode or several modes. Mode could be applied for nominal variable
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Example What are the modes? Data 1: { 5, 6, 6, 7, 7, 7, 9 } Data 2: { 5, 6, 7, 8, 9 } Data 3: { 5, 6, 9, 5, 6 } Data 4: { Yellow, Yellow, Red, Blue, Green}
Mean, Median, Mode Mean, Median, Mode
No Mode
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Mean = 3
Median = 3
Mean = 4
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Mean = 4.8
Mode: 7
Mean = Median = Mode = 5
Median = 5.5
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Mean, Median, Mode Mean, Median, Mode
Left skewed
Symmetric
Right skewed
Mean Median Mode
Mean < Median < Mode
Mode < Median < Mean
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Grouped data Grouped data
Customer’s waiting time
Waiting time Frequency
0 – 5 15
5 – 10 20
10 – 15 8
15 – 20 5
20 + 2
Median is in group of [5 – 10) Modal group: Mean: using middle value
Waiting time
2.5
7.5
12.5
17.5
22.5
Frequency
15
20
8
5
2
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3.4. Quartile 3.4. Quartile
Divide data into 4 equal-parts by 3 cutoff points: 3
quartile , ,
25% 25% 25% 25%
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2nd quartile: =
Quantile Quantile
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Divide into 5 equal-parts by 4 cutoff point: 4 Quintile Divide into 10 equal-parts by 9 cutoff point: 9 Decile 100 equal-parts: 99 percentile 10th percentile = 1st decile 20th percentile = 2nd decile = 1st quintile 25th percentile = 1st quartile 50th percentile = 2nd quartile = median
Micrsoft Excel Function Micrsoft Excel Function
Measures
Command / Function
Mean
= average(data)
Median
= median(data)
Mode
= mode(data)
Quartile k (k = 1,2,3)
= quartile(data, k)
Percentile k (k = 1,2,…,99) = percentile(data, k)
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Variability Variability
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
Central Tendency may not provide efficient information of the data.
0 1 2 3 4 5 6 7 8 9
Mean = Median = 5
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Data have the same Mean, Median, but differ in variability (dispersion, spread).
3.5. Range 3.5. Range
Range
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Range = 7
Range = 6
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= largest value – smallest value = xmax – xmin Simplest, but poorest information.
3.6. Variance & Standard Deviation 3.6. Variance & Standard Deviation
Sample Data: , ,… , the mean ̅ Deviation: − ̅ : (+) or (–) or zero Sum of Squares: = ∑ − ̅ Variance:
∑ −
= =
− −
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Unit of Variance is squared unit of
Standard Deviation Standard Deviation
Standard Deviation is square root of Variance
then:
=
>
Standard Deviation has the same unit as Variance & S.D measure the “absolute” variability If
is more variability, dispersed, widespread,
fluctuated than
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is more stable, concentrated than
Population and Sample Population and Sample
Difference between Population and Sample
Population Sample
Data { , ,… , } { , ,… , }
Mean = =
∑ ∑
= ∑ − = ∑ − ̅
SS Variance
= =
−
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Std. Dev. = =
Compare variability Compare variability
Compare 3 samples Firm A: Profit ($ mil.): ( 5, 6, 7, 8, 9 ) Firm B: Profit ($ mil.): ( 51, 53, 55, 57, 59 ) Firm C: Price ($): ( 15, 16, 17, 18, 19 )
Mean S2 S CV SS
A 7 ($m) 2.5 ($m)2 1.58 ($m) 22.6 % 10
B 55 ($m) 10 ($m)2 3.16 ($m) 5.7 % 40
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C 17 ($) 2.5 ($)2 1.58 ($) 9.3 % 10
3.7. Coefficient of Variation 3.7. Coefficient of Variation
= × 100%
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CV has unit of %, independent to unit of the data. CV measures “relative” variation
3.8. Interquartile Range 3.8. Interquartile Range
Interquartile Range is range between 3rd quartile
and 1st quartile
= 3 − 1 = − IQR is the width of 50% middle value of data
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25% 25% 25% 25%
Outlier Outlier
There are Lower Limit and Upper Limit for the data Observations smaller than LL or greater than UL are
Outlier
By Quartiles:
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Lower Limit is − 1.5 Upper Limit is + 1.5
Key-point and Boxplot Key-point and Boxplot
Find 5 key-point and Outliers
Salary No. of Worker 10 11 12 13 14 15 16 17 18 1 10 16 30 19 14 10 0 0
Boxplot
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1.5 1.5
Table, Histogram, Boxplot Table, Histogram, Boxplot
Value Freq.
Salary
35
30
25
20
15
10
5
0
10
11
12
13
14
15
16
17
18
10 11 12 13.5 18
10 11 12 13 14 15 16 17 18
10 16 30 19 14 10 0 0 1
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Boxplot : Key values and Whiskers Boxplot : Key values and Whiskers
A
B
C
D
E
F
6
Max 6
7
9
6
4
5
4
6
6
4
3
Q3
4.5 2.5 5.5 4.5 2.5 2.5
Q2
3
2
4
4
1
2
Q1
Min
1
1
1
3
-1
1
̅
4.2 2.8 5.16 4.84 2.5 2.5
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Boxplot Boxplot
2014
2015
2016
2017
Max
Q3
Q2
Q1
Min
Mean
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3.9. Skewness (Sk) 3.9. Skewness (Sk)
Sk = 0.3 Right short tail Sk = 0 Two-tail Sk = – 0.3 Left short tail
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Sk = – 1.3 Left long tail Sk = 1.3 Right long tail
3.10. Covariance & Correlation 3.10. Covariance & Correlation
Covariance: combined variability of , , in sample:
Positive covariance
Negative covariance
Y f o n a e M
Y f o n a e M
Mean of X
Mean of X
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, = = ∑ ( − ̅)( − ) − 1
Correlation Coefficient Correlation Coefficient
∑ ( − ̅)( − )
= =
− ̅
( , ) ∑ ∑ −
−1 ≤ ≤ 1, no unit measures linear relationship between and
: linear negative
: no correlated
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= −1 −1 < < 0 : negatively correlated = 0 0 < < 1 : positively correlated = 1 : linear positive
Correlation Correlation
r = 0.5
Positively
Week
r = 0.8
Strong
r = 0
Negatively
No correlated
r = – 0.5
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Graph and Correlation Coefficient ( )
Correlation Coefficient Correlation Coefficient
− ̅
−
− ̅
−
− ̅ ∗ −
X: Advertising; Y: sales
5 10 Jan
6 8 15 10 Feb Mar
9 12 18 32 Apr May
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Sum Mean
3.11. Standardized value 3.11. Standardized value
Z-score of one value in data, have no unit
=
− .
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Ex. Compare score of Microeconomics and Macroeconomics of one student in one class if: Micro score = 7.5; Marcro score = 9 Mean of Micro in class = 6; Mean of Macro = 7 S.D of Micro = 1; S.D of Macro = 2
Excel: Statistic Functions Excel: Statistic Functions
Statistic
Function
= SUM(array)
= AVERAGE(array)
= MEDIAN(array)
= QUARTILE(array, k)
= VAR(array)
Sum Mean X Median Kth Quartile (Q1,Q2,Q3) Sample variance (S2) Sample S.D (S)
= STDEV(array)
= COVAR(array1, array2)
= CORREL(array1, array2) = NORMDIST(b, µ, σ, 1)
Covariance Cov(X,Y) Correlation rXY X ~ N(µ,σ2); P(X < b)
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Exercise Exercise
[1] Chapter 3: (p110) 2, 3, 6, 7, 11, 13, (p120) 26, 27, 29, 33, (p133) 49, 50, 52, (p143) 56, 58, 59, (p152) 62, 63, 70,
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Case Problem 1, 4
