Lecture Probability & statistics: Chapter 3 - Bùi Dương Hải

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Lecture "Probability & statistics - Chapter 3: Numerical summary" has contents: Data measurements, locations, variability measures, shape, arithmetic mean, compare the mean, mode, quartile, range, variance & standard deviation.... Invite you to consult the content.

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Nội dung Text: Lecture Probability & statistics: Chapter 3 - Bùi Dương Hải

Lecture 3. NUMERICAL SUMMARY  Data Measurements  Locations  Variability Measures  Shape  [1] Chapter 3, pp. 99 - 162  [3] Chapter 2 PROBABILITY & STATISTICS– Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 1
Comparison Profit of Project A (million)  Profit of two 30% 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 PROBABILITY & STATISTICS– Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 2
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 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 3
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 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 4
Data Measurements  Location:  Minimum, Maximum  Central Tendency: Mean, Median, Mode  Quantile: Quartile, Percentile  Variability:  Range  Variance (Var)  Standard Deviation (SD)  Coefficient of Variation (CV)  Interquartile Range (IQR) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 5
3.1. Mean (arithmetic mean)  Apply for scale variable only  = Population Sample Data: { , , … , } Data: { , ,…, } + + ⋯+ + + ⋯+ = =  Have the same unit as the original data PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 6
Weighted mean  Price ($) in Quarter 1, 2, 3, 4 are 10, 12, 18, 14, respectively. 10 + 12 + 18 + 14 = = 4  Any difference if the volume of sales in Quarter 1, 2, 3, 4 are 70, 90, 110, 130? Q1 Q2 Q3 Q4 Value xi Price 10 12 18 14 Volume 70 90 110 130 Weight wi PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 7
Weighted Mean  In general, for grouped data: + + ⋯+ ∑ = = + + ⋯+ ∑  For Example of Price: 70 ∗ 10 + 90 ∗ 12 + 110 ∗ 18 + 130 ∗ 14 ̅= = 70 + 90 + 110 + 130 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 8
Mean of Grouped data  Frequency, Proportion, Percent table Wage ($) 7 8 9 Number of worker 4 10 6 (Frequency) Proportion 0.2 0.5 0.3 (Relative frequency) Percent 20% 50% 30% PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 9
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  May lead to biased comparison   Use the other measures PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 10
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 } 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 = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 11
Median  Median is the ‘cutoff point’ of lower 50% - upper 50% parts Discrete vs Continous Lower 50% Upper 50% Discrete Continuous Median PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 12
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  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} PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 13
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 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 14
Mean, Median, Mode Left skewed Symmetric Right skewed Mean Median Mean < Median < Mode Mode Mode < Median < Mean PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 15
Grouped data  Customer’s waiting time Waiting time 0–5 5 – 10 10 – 15 15 – 20 20 + Frequency 15 20 8 5 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 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 16
3.4. Quartile  Divide data into 4 equal-parts by 3 cutoff points: 3 quartile , , 25% 25% 25% 25%  2nd quartile: = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 17
Quantile  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 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 18
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) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 19
Variability  Central Tendency may not provide efficient 0 1 2 3 4 5 6 7 8 9 information of the data.  Data have the same 0 1 2 3 4 5 6 7 8 9 Mean, Median, but differ in variability 0 1 2 3 4 5 6 7 8 9 (dispersion, spread). 0 1 2 3 4 5 6 7 8 9 Mean = Median = 5 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 20