8. Biostatistics standard deviation and coefficient of variation for grouped data
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Biological data shows variations in the qualitative and quantitative attributes. Each individual value of observation scatters around the central tendency. measurement of the dispersion helps to understand the distribution of individual values around the central tendency. this ppt explains measures of dispersion namely range, mean deviation, standard deviation, and coefficient of deviation.
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1
FACULTY OF SCIENCE AND ENGINEERING
SCHOOL OF COMPUTING, MATHEMATICS & DIGITAL MEDIA
REASSESSMENT COURSEWORK 2013/14
UNIT CODE:
6G6Z3005
UNIT DESC:
APPLIED REGRESSION AND MULTIVARIATE ANALYSIS
ASSESSMENT ID:
1CWK30
ASSESSMENT NAME:
Courswork 30%
WEIGHT
FACTOR: 30%
See below.
NAME OF STAFF SETTING ASSIGNMENT: Dr B L Shea
0
MANCHESTER METROPOLITAN UNIVERSITY
FACULTY OF SCIENCE AND ENGINEERING
SCHOOL OF COMPUTING, MATHEMATICS & DIGITAL TECHNOLOGY
ACADEMIC YEAR 2013-2014:
REFERRED COURSEWORK
BSC(HONS) FINANCIAL MATHEMATICS
BSC(HONS) MATHEMATICS
YEAR/STAGE THREE
UNIT 6G6Z3005 : APPLIED REGRESSION AND MULTIVARIATE ANALYSIS
Answer ALL questions.
The pass mark is 40% which corresponds to a minimum of 72
marks out of a possible 180 marks.
The deadline is 8th August 2014.
SECTION A
1. (a) Three measurementsx1, x2 andx3 have the following sample covariance matrix.
∑̂ =
9 2 0
2 4 1
0 1 4
(i) Verify that the corresponding sample correlation matrix C, is given by
C =
1 13 0
1
3 1
1
4
0 14 1
[2]
(ii) Given that one of the eigenvalues of C is equal to one, calculate the other two
eigenvalues and determine the proportion of the variation in the data explained
by the first principal component.
[6]
(iii) Using the sample correlation matrix C, calculate the first principal component.
[6]
(b) A Principal Components Analysis of the prices of food items in 23 cities was carried
out with a view to forming a measure of the Consumer Price Index(CPI). A Minitab
analysis of this data is attached.
(i) Explain why Principal Components Analysis was performedon the correlation
matrix instead of the covariance matrix.
[2]
(ii) If the first Principal Component is taken as a measure of the CPI calculate, to
one decimal place, the value of the index for Atlanta.
[2]
(iii) Which is the most expensive city and which is the least expensive city?
[2]
(Question 1 continued overleaf)
1
(Question 1 continued)
Minitab output for Question 1
Descriptive Statistics: bread, burger, milk, oranges, tomatoes
Variable N Mean Median TrMean StDev SE Mean
bread 23 25.291 25.300 25.267 2.507 0.523
burger 23 91.86 91.00 91.63 7.55 1.58
milk 23 62.30 62.50 61.96 6.95 1.45
oranges 23 102.99 105.90 102.90 14.24 2.97
tomatoes 23 48.77 46.80 48.74 7.60 1.59
Principal Component Analysis: bread, burger, milk, oranges, tomatoes
Eigenanalysis of the Correlation Matrix
Eigenvalue 2.4225 1.1047 0.7385 0.4936 0.2408
Proportion 0.484 0.221 0.148 0.099 0.048
Cumulative 0.484 0.705 0.853 0.952 1.000
Variable PC1 PC2 PC3 PC4 PC5
bread 0.496 -0.309 0.386 -0.509 -0.500
burger 0.576 -0.044 0.262 0.028 0.773
milk 0.340 -0.431 -0.835 -0.049 0.008
oranges 0.225 0.797 -0.292 -0.479 -0.006
tomatoes 0.506 0.287 0.012 0.713 -0.391
(Question 1 continued overleaf)
2
(Question 1 continued)
Data Display
Row c ...
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1 FACULTY OF SCIENCE AND ENGINEERING SCHOOL OF COMPUT.docx
1
FACULTY OF SCIENCE AND ENGINEERING
SCHOOL OF COMPUTING, MATHEMATICS & DIGITAL MEDIA
REASSESSMENT COURSEWORK 2013/14
UNIT CODE:
6G6Z3005
UNIT DESC:
APPLIED REGRESSION AND MULTIVARIATE ANALYSIS
ASSESSMENT ID:
1CWK30
ASSESSMENT NAME:
Courswork 30%
WEIGHT
FACTOR: 30%
See below.
NAME OF STAFF SETTING ASSIGNMENT: Dr B L Shea
0
MANCHESTER METROPOLITAN UNIVERSITY
FACULTY OF SCIENCE AND ENGINEERING
SCHOOL OF COMPUTING, MATHEMATICS & DIGITAL TECHNOLOGY
ACADEMIC YEAR 2013-2014:
REFERRED COURSEWORK
BSC(HONS) FINANCIAL MATHEMATICS
BSC(HONS) MATHEMATICS
YEAR/STAGE THREE
UNIT 6G6Z3005 : APPLIED REGRESSION AND MULTIVARIATE ANALYSIS
Answer ALL questions.
The pass mark is 40% which corresponds to a minimum of 72
marks out of a possible 180 marks.
The deadline is 8th August 2014.
SECTION A
1. (a) Three measurementsx1, x2 andx3 have the following sample covariance matrix.
∑̂ =
9 2 0
2 4 1
0 1 4
(i) Verify that the corresponding sample correlation matrix C, is given by
C =
1 13 0
1
3 1
1
4
0 14 1
[2]
(ii) Given that one of the eigenvalues of C is equal to one, calculate the other two
eigenvalues and determine the proportion of the variation in the data explained
by the first principal component.
[6]
(iii) Using the sample correlation matrix C, calculate the first principal component.
[6]
(b) A Principal Components Analysis of the prices of food items in 23 cities was carried
out with a view to forming a measure of the Consumer Price Index(CPI). A Minitab
analysis of this data is attached.
(i) Explain why Principal Components Analysis was performedon the correlation
matrix instead of the covariance matrix.
[2]
(ii) If the first Principal Component is taken as a measure of the CPI calculate, to
one decimal place, the value of the index for Atlanta.
[2]
(iii) Which is the most expensive city and which is the least expensive city?
[2]
(Question 1 continued overleaf)
1
(Question 1 continued)
Minitab output for Question 1
Descriptive Statistics: bread, burger, milk, oranges, tomatoes
Variable N Mean Median TrMean StDev SE Mean
bread 23 25.291 25.300 25.267 2.507 0.523
burger 23 91.86 91.00 91.63 7.55 1.58
milk 23 62.30 62.50 61.96 6.95 1.45
oranges 23 102.99 105.90 102.90 14.24 2.97
tomatoes 23 48.77 46.80 48.74 7.60 1.59
Principal Component Analysis: bread, burger, milk, oranges, tomatoes
Eigenanalysis of the Correlation Matrix
Eigenvalue 2.4225 1.1047 0.7385 0.4936 0.2408
Proportion 0.484 0.221 0.148 0.099 0.048
Cumulative 0.484 0.705 0.853 0.952 1.000
Variable PC1 PC2 PC3 PC4 PC5
bread 0.496 -0.309 0.386 -0.509 -0.500
burger 0.576 -0.044 0.262 0.028 0.773
milk 0.340 -0.431 -0.835 -0.049 0.008
oranges 0.225 0.797 -0.292 -0.479 -0.006
tomatoes 0.506 0.287 0.012 0.713 -0.391
(Question 1 continued overleaf)
2
(Question 1 continued)
Data Display
Row c ...
12 ch ken black solution
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- 1. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany skhot1976@gmail.com Paper XIV Unit. 2. Biostatistics 2.8 Measures of Dispersion Standard Deviation and Coefficient of Variation (CV) (for grouped data) Dr. Sudhakar Sambhaji Khot M.Sc., Ph.D., SET Assistant Professor in Botany Y. C. Warana Mahavidyalaya, Warananagar
- 2. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany Dr. S. S. KHOT 2 B.Sc. Part- III Botany Paper- XIV DSE –F26 Bioinformatics, Biostatistics and Economic Botany Unit 2: Biostatistics (11) 2.1 Introduction, definition, terminology. 2.2 Collection and presentation of data: Types of data, techniques of data collection- Census method, sampling method- simple random, stratified and systematic sampling. Classification, tabulation, graphical representation- Histogram and polygon. 2.3 Measures of central tendency and Dispersion: Arithmetic mean, Mode, Median, Range, Deviation, Mean deviation, Standard Deviation, Coefficient of Variation. 2.4 Statistical methods for testing the hypothesis’ i) Students’ T-test ii) Chi-square test
- 3. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany Dr. S. S. KHOT 3 Story of SIX SIGMA (6σ) SIX SIGMA (6σ): defined and disciplined business methodology to increase customer satisfaction and profitability by streamlining operations, improving quality and eliminating defects in any organization-wide process. Originated at Motorola in 1980s 6σ = 99.99966 % Chance of defect = 0.00034 % = 3.4 defects /10,00,000
- 4. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany Number of seeds per pod (x) No. of pods (f) fx Deviation (dx= x-x̄ ) Deviation square (dx2 ) fdx2 2 2 4 -3.5 12.25 24.5 3 2 6 -2.5 6.25 12.5 4 3 12 -1.5 2.25 6.75 5 8 40 -0.5 0.25 2 6 7 42 0.5 0.25 1.75 7 4 28 1.5 2.25 9 8 3 24 2.5 6.25 18.75 9 1 9 3.5 12.25 12.25 ∑f=30 ∑fx=165 ∑dx= 0 ∑dx2 = 42 ∑fdx2 =87.5 Dr. S. S. KHOT 4 Example 2: The data on number of seeds per pod in Albizia lebbeck is as below. Calculate mean and standard deviation and CV. Solution: This is a grouped data. Number of seeds (x) 2 3 4 5 6 7 8 9 Number of pods (f) 2 2 3 8 7 4 3 1 Mean (x̄ ) = ∑fx/ ∑f = 165/30 = 5.5
- 5. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany ∑f = 30 ∑fx = 165 ∑dx= 0 ∑dx2 = 42 ∑fdx2 = 87.5 Dr. S. S. KHOT 5 Example 2: The data on number of seeds per pod in Albizia lebbeck is as below. Calculate mean and standard deviation. Solution: This is a grouped data. Number of seeds (x) 2 3 4 5 6 7 8 9 Number of pods (f) 2 2 3 8 7 4 3 1 Mean (x bar) = ∑fx/ ∑f = 165/30 = 5.5 Computation of S.D. for grouped data 𝝈 = ∑𝒇𝑿 ∑𝒇 𝟐 = ∑𝒇𝒅𝒙 ∑𝒇 𝟐 𝝈 = 𝟖𝟕. 𝟓/𝟑𝟎 𝝈 = 𝟐. 𝟗𝟐 = 1.71 seeds per pod Answer: Mean= 5.5 seeds per pod and SD (σ) = 1.71 seeds per pod CV = 31.09 % Computation of CV 𝑪𝑽 = 𝑺𝑫 𝒙 𝒙 𝟏𝟎𝟎 CV= 1.71/5.5 x 100 CV= 31.09 %
- 6. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany Dr. S. S. KHOT Questions, if any ? Thank You