1504 basic statistics
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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
![7
Saad Haj Bakry
MEAN : AVERAGE : EXPECTATIONMEAN : AVERAGE : EXPECTATION
Statistical Distributions
Definition Arithmetic mean
Raw Data Given values: x[1], x[2], …. x[N[
Mean m=
Raw Data
Given ranges: y[1], y[2], ….y[n[
Frequencies: f[1], f[2], …. f[n[
Weighted
Mean
m= :
Problem Write and test computer functions /
Give Illustrations
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Basic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-7-320.jpg)
![8
Saad Haj Bakry
MODE / MEDIANMODE / MEDIAN
Statistical Distributions
Median Middle value
Mode Value with highest frequency
Raw Data Given values: x[1], x[2], …. x[N]
Median
For ODD N: m = x[(N+1)/2]
EVEN: m = (1/2) {x[N/2]+x[(N/2)+1]}
Mode
Find frequency distribution:
m = x[k] : f[k] highest frequency
Problem Write and test computer functions /
Give Illustrations
Basic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-8-320.jpg)
![9
Saad Haj Bakry
DEVIATION / VARIANCEDEVIATION / VARIANCE
Statistical Distributions
Deviation
Deviation from the “mean”
d[i] = |x[i] – m|
Mean
Deviation
Variance /
Standard
Deviation
Standard
Score
Standardized Variables: z[i] = d[i] / s
Problem Write and test computer functions /
Give Illustrations
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Basic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-9-320.jpg)
![10
Saad Haj Bakry
UNIFORM DISTRIBUTIONUNIFORM DISTRIBUTION
Statistical Distributions
Features
Range
“min”: Minimum number
“max”: Maximum number
Principle All numbers “x[i] : x” are equally likely
Probability Density p(x,min,max) = 1 / (max-min(
Mean m = (max + min) / 2
Variance v = (max - min)2
/ 12
Problem Write and test computer functions /
Give Illustrations
Basic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-10-320.jpg)
![15
Saad Haj Bakry
t DISTRIBUTIONt DISTRIBUTION
Statistical Distributions
Features
Range - (infinity) < t < + (infinity)
m mean / median / mode at “zero”
f Degree of freedom: 0 < f < + (infinity(
Principle Used for estimation: small sample
t Distribution variable: variance unknown
Probability Density
Gamma Function
Problem Write and test computer functions /
Give Illustrations
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Basic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-15-320.jpg)
![19
Saad Haj Bakry
UNIFORM RNG: U (0,1)UNIFORM RNG: U (0,1)
Features
m
Modulus factor: large (st) prime number within
memory cell size (for wide repeated sequence cycle(
m = 231
- 1 = 2,147,483,674 (for 32 bit cell(
a Multiplier: a = 314,159,269
b Increment: b = 453,806,245
X[0[ Starting value: X[0] = 577,215,665 (the seed)
X[i-1[ “)ith-1)” value: seed for X[i[
Uniform: X (0,m( X[i] = {a . X[i-1] + b} MOD m
Uniform: U (0,1( U[i] = X[i] / m
Problem Write and test computer functions /
Give Illustrations
Generation of Random NumbersBasic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-19-320.jpg)
![20
Saad Haj Bakry
UNIFORM RNG: U (min , max)UNIFORM RNG: U (min , max)
Integer
Range
min Required minimum integer value
max Required maximum integer value
U (min, max( min + TRUNC [(max – min + 1) . U (0, 1([
Problem Write and test computer functions /
Give Illustrations
Test
100,000“runs”: Test
• Frequency Distribution
• Mean
• Variance & Standard Deviation
)Relative to theoretical expectations(
Note Every new set of runs should start with
a different seed: X[0]
Generation of Random NumbersBasic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-20-320.jpg)
![22
Saad Haj Bakry
NORMAL RNG: N (m, s)NORMAL RNG: N (m, s)
Features
m Required “mean” of the normal RNG
s Required “standard deviation”
STEPS
1
V[1] = 2 . { U(0,1)[1] } – 1
V[2] = 2 . { U(0,1)[2] } – 1
2 SUM = V2
[1] + V2
[2[
3 IF SUM >= 1 GO TO STEP 1
4
5 N (m,s) = m + s . Y
Standard Normal N (0,1) = Y
Problem Write and test computer functions /
Give Illustrations
SUMSUMLnVY /)](.2[].2[ −=
Generation of Random NumbersBasic Statistics](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-22-320.jpg)
![24
Saad Haj Bakry
EVALUATION ALGORITHM: GeneralEVALUATION ALGORITHM: General
Confidence IntervalsBasic Statistics
INPUT: Measurements: N & x[1], x[2],….,x[N]
Required “confidence level” : L (%)
INPUT: Measurements: N & x[1], x[2],….,x[N]
Required “confidence level” : L (%)
MEAN: |MEAN: | ∑
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AREA UNDER CURVE: | a = 1 – (L/100)AREA UNDER CURVE: | a = 1 – (L/100)](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/1504-basicstatistics-161031144015/85/1504-basic-statistics-24-320.jpg)
![27
Saad Haj Bakry
EVALUATION ALGORITHM: N < 30EVALUATION ALGORITHM: N < 30
Confidence IntervalsBasic Statistics
| m – (T (a/2) . S) < sample < m – (T (a/2) . S)| m – (T (a/2) . S) < sample < m – (T (a/2) . S)
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1504 basic statistics
- 1. 1 SAAD HAJ BAKRY, PhD, CEng, FIEESAAD HAJ BAKRY, PhD, CEng, FIEE BASIC STATISTICS FOR SYSTEM STUDIES BASIC STATISTICS FOR SYSTEM STUDIES
- 2. 2 OBJECTIVES / CONTENTSOBJECTIVES / CONTENTS Saad Haj Bakry STATISTICAL DISTRIBUTIONSSTATISTICAL DISTRIBUTIONS GENERATION OF RANDOM NUMBERSGENERATION OF RANDOM NUMBERS CONFIDENCE INTERVALSCONFIDENCE INTERVALS Basic Statistics REQUIRED DEVELOPMENTREQUIRED DEVELOPMENT
- 3. 3 FREQUENCY DISTRIBUTION: 1/3FREQUENCY DISTRIBUTION: 1/3 Saad Haj Bakry Statistical Distributions INPUT: raw data “a set of (N) values”INPUT: raw data “a set of (N) values” ORGANIZE: values in ascending / descending order ORGANIZE: values in ascending / descending order DETERMINE: the range of raw data / valuesDETERMINE: the range of raw data / values DIVIDE: the range into sub-rangesDIVIDE: the range into sub-ranges Basic Statistics
- 4. 4 FREQUENCY DISTRIBUTION: 2/3FREQUENCY DISTRIBUTION: 2/3 Saad Haj Bakry Statistical Distributions FIND: number of values per sub-range “frequency”FIND: number of values per sub-range “frequency” RESULT: frequency distributionRESULT: frequency distribution DIVIDE: frequency of each sub-range by (N)DIVIDE: frequency of each sub-range by (N) RESULT: relative frequency distribution “probability density” RESULT: relative frequency distribution “probability density” Basic Statistics
- 5. 5 FREQUENCY DISTRIBUTION: 3/3FREQUENCY DISTRIBUTION: 3/3 Saad Haj Bakry Statistical Distributions ADD: frequencies sub-range by sub-rangeADD: frequencies sub-range by sub-range RESULT: cumulative frequency distributionRESULT: cumulative frequency distribution ADD: relative frequencies sub-range by sub-rangeADD: relative frequencies sub-range by sub-range RESULT: relative cumulative frequency distribution “cumulative probability” RESULT: relative cumulative frequency distribution “cumulative probability” Basic Statistics
- 6. 6 FREQUENCY DISTRIBUTION: ProblemFREQUENCY DISTRIBUTION: Problem Saad Haj Bakry Statistical Distributions Given: “N values” raw data (N is very large for probability considerations( Find (Graphs to Illustrate( Frequency Distribution Relative Frequency Distribution: (Probability Density( Cumulative Frequency Distribution Relative Cumulative Frequency Distribution (Cumulative Probability( Basic Statistics
- 7. 7 Saad Haj Bakry MEAN : AVERAGE : EXPECTATIONMEAN : AVERAGE : EXPECTATION Statistical Distributions Definition Arithmetic mean Raw Data Given values: x[1], x[2], …. x[N[ Mean m= Raw Data Given ranges: y[1], y[2], ….y[n[ Frequencies: f[1], f[2], …. f[n[ Weighted Mean m= : Problem Write and test computer functions / Give Illustrations ][].[ 1 1 ∑ = = nj j jyjf N ∑ = = Ni i ix N 1 ][ 1 ∑ = = = nj j Njf 1 ][ Basic Statistics
- 8. 8 Saad Haj Bakry MODE / MEDIANMODE / MEDIAN Statistical Distributions Median Middle value Mode Value with highest frequency Raw Data Given values: x[1], x[2], …. x[N] Median For ODD N: m = x[(N+1)/2] EVEN: m = (1/2) {x[N/2]+x[(N/2)+1]} Mode Find frequency distribution: m = x[k] : f[k] highest frequency Problem Write and test computer functions / Give Illustrations Basic Statistics
- 9. 9 Saad Haj Bakry DEVIATION / VARIANCEDEVIATION / VARIANCE Statistical Distributions Deviation Deviation from the “mean” d[i] = |x[i] – m| Mean Deviation Variance / Standard Deviation Standard Score Standardized Variables: z[i] = d[i] / s Problem Write and test computer functions / Give Illustrations ∑ = = = Ni i id N d 1 ][ 1 ∑= = n j jdjf n d 1 ][].[ 1 ∑ = = = nj j Njf 1 ][ ∑ = = = Ni i idN v 1 2 ][ 1 vs = Basic Statistics
- 10. 10 Saad Haj Bakry UNIFORM DISTRIBUTIONUNIFORM DISTRIBUTION Statistical Distributions Features Range “min”: Minimum number “max”: Maximum number Principle All numbers “x[i] : x” are equally likely Probability Density p(x,min,max) = 1 / (max-min( Mean m = (max + min) / 2 Variance v = (max - min)2 / 12 Problem Write and test computer functions / Give Illustrations Basic Statistics
- 11. 11 Saad Haj Bakry BINOMIAL DISTRIBUTIONBINOMIAL DISTRIBUTION Statistical Distributions Features N Number of trials p Probability of success q Probability of failure: q = 1 - p x Number of successful trials in N Probability Density Mean m = N . p Variance v = N . p . q Problem Write and test computer functions / Give Illustrations qpC xNxN x pNxp − =(,,( Basic Statistics
- 12. 12 Saad Haj Bakry POISSON DISTRIBUTIONPOISSON DISTRIBUTION Statistical Distributions Features r Rate of arrivals: mean t Time interval, may be “t=1 time unit” x Possible number of arrivals during “t” Probability Density Mean m = r Variance v = r Problem Write and test computer functions / Give Illustrations rt x e x rt trxp − = ! (( (,,( Basic Statistics
- 13. 13 Saad Haj Bakry EXPONENTIAL DISTRIBUTIONEXPONENTIAL DISTRIBUTION Statistical Distributions Features r Rate of arrivals w Waiting time for next arrival: Inter-arrival Principle Distribution for the value of “w” Probability Density Mean m = 1 / r (Poisson inter-arrival mean( Variance v = 1 / r2 Problem Write and test computer functions / Give Illustrations rw errwp − = .(,( Basic Statistics
- 14. 14 Saad Haj Bakry NORMAL DISTRIBUTIONNORMAL DISTRIBUTION Statistical Distributions Features Range - (infinity) < x < + (infinity) m Average value: mean / median / mode v Variance: v = s2 x Possible value Principle Usually a measurement process Probability Density Standard form: Problem Write and test computer functions / Give Illustrations 2 2 (( 2 1 2 1 (,,( mx s e s smxp −− = π 2 2 1 2 1 (1,0,( z ezp − = π Basic Statistics
- 15. 15 Saad Haj Bakry t DISTRIBUTIONt DISTRIBUTION Statistical Distributions Features Range - (infinity) < t < + (infinity) m mean / median / mode at “zero” f Degree of freedom: 0 < f < + (infinity( Principle Used for estimation: small sample t Distribution variable: variance unknown Probability Density Gamma Function Problem Write and test computer functions / Give Illustrations 2/(1( 2 (1( (2/( ]2/(1[( (,( +− + Γ +Γ = f f t ff f ftp π (!1((( 0 1 −==Γ − ∞ − ∫ fdxexf xf Basic Statistics
- 16. 16 Saad Haj Bakry COMPUTATION TIPS: 1/2COMPUTATION TIPS: 1/2 Statistical Distributions Integer Factorial 0 ! = 1 (i+1) ! = i! (i+1( Real Factorial Sterling Formula: Gamma Function Useful for its computation Problem Write and test computer functions / Give Illustrations ii eiii − ≈ π2! Basic Statistics
- 17. 17 Saad Haj Bakry Statistical Distributions i iN CC N i N i )1( 1 +− =+ Combination Poisson Integration Trapezoid Rule: Summation in small steps Problem Write and test computer functions / Give Illustrations 1 !0 0 = A 10 =N C )1(!)!1( 1 + = + + i A i A i A ii COMPUTATION TIPS: 2/2COMPUTATION TIPS: 2/2 Basic Statistics
- 18. 18 RNGS: WHYRNGS: WHY Saad Haj Bakry SYSTEM MODELING / SIMULATION SYSTEM MODELING / SIMULATION SAMPLINGSAMPLING NUMERICAL ANALYSISNUMERICAL ANALYSISRANDOM PROCESSES RANDOM PROCESSES TESTING COMPUTER ALGORITHMS TESTING COMPUTER ALGORITHMS DECISION MAKING DECISION MAKING OTHER REASONSOTHER REASONS Generation of Random NumbersBasic Statistics
- 19. 19 Saad Haj Bakry UNIFORM RNG: U (0,1)UNIFORM RNG: U (0,1) Features m Modulus factor: large (st) prime number within memory cell size (for wide repeated sequence cycle( m = 231 - 1 = 2,147,483,674 (for 32 bit cell( a Multiplier: a = 314,159,269 b Increment: b = 453,806,245 X[0[ Starting value: X[0] = 577,215,665 (the seed) X[i-1[ “)ith-1)” value: seed for X[i[ Uniform: X (0,m( X[i] = {a . X[i-1] + b} MOD m Uniform: U (0,1( U[i] = X[i] / m Problem Write and test computer functions / Give Illustrations Generation of Random NumbersBasic Statistics
- 20. 20 Saad Haj Bakry UNIFORM RNG: U (min , max)UNIFORM RNG: U (min , max) Integer Range min Required minimum integer value max Required maximum integer value U (min, max( min + TRUNC [(max – min + 1) . U (0, 1([ Problem Write and test computer functions / Give Illustrations Test 100,000“runs”: Test • Frequency Distribution • Mean • Variance & Standard Deviation )Relative to theoretical expectations( Note Every new set of runs should start with a different seed: X[0] Generation of Random NumbersBasic Statistics
- 21. 21 Saad Haj Bakry EXPONENTIAL RNG: E (m)EXPONENTIAL RNG: E (m) Required “mean”: “inter-event” Poisson m E (mean( - )m) . Ln [U(0, 1([ Problem Write and test computer functions / Give Illustrations Test 100,000“runs”: Test • Frequency Distribution • Mean • Variance & Standard Deviation )Relative to theoretical expectations( Generation of Random NumbersBasic Statistics
- 22. 22 Saad Haj Bakry NORMAL RNG: N (m, s)NORMAL RNG: N (m, s) Features m Required “mean” of the normal RNG s Required “standard deviation” STEPS 1 V[1] = 2 . { U(0,1)[1] } – 1 V[2] = 2 . { U(0,1)[2] } – 1 2 SUM = V2 [1] + V2 [2[ 3 IF SUM >= 1 GO TO STEP 1 4 5 N (m,s) = m + s . Y Standard Normal N (0,1) = Y Problem Write and test computer functions / Give Illustrations SUMSUMLnVY /)](.2[].2[ −= Generation of Random NumbersBasic Statistics
- 23. 23 Saad Haj Bakry ConfidenceBasic Statistics MEASUREMENTS & ESTIMATIONSMEASUREMENTS & ESTIMATIONS MEASUREMENTS: Experiments on real systems / models / Simulation MEASUREMENTS: Experiments on real systems / models / Simulation Confidence for “Mean” ESTIMATION THEORY ESTIMATION THEORY SET OF VALUES: Sample of results (N)SET OF VALUES: Sample of results (N) Confidence for “Sample” Large Sample: N >= 30 Small Sample: N < 30
- 24. 24 Saad Haj Bakry EVALUATION ALGORITHM: GeneralEVALUATION ALGORITHM: General Confidence IntervalsBasic Statistics INPUT: Measurements: N & x[1], x[2],….,x[N] Required “confidence level” : L (%) INPUT: Measurements: N & x[1], x[2],….,x[N] Required “confidence level” : L (%) MEAN: |MEAN: | ∑ = = = Ni i ix N m 1 ][ 1 STANDARD DEVIATION:|STANDARD DEVIATION:| ∑ = = − − = ni i mix N s 1 2 }][{ 1 1 AREA UNDER CURVE: | a = 1 – (L/100)AREA UNDER CURVE: | a = 1 – (L/100)
- 25. 25 Saad Haj Bakry NORMAL DISTRIBUTIONNORMAL DISTRIBUTION Confidence IntervalsBasic Statistics 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 3.9 -3 -2 -1 -0.5 0 0.5 1 2 3 3.9 Probability Density Mean Standard Deviations 68.27 % 95.45 % - z (a/2) + z (a/2) Standard Deviations
- 26. 26 Saad Haj Bakry EVALUATION ALGORITHM: N >=30EVALUATION ALGORITHM: N >=30 Confidence IntervalsBasic Statistics | m – (Z (a/2) . S) < sample < m – (Z (a/2) . S)| m – (Z (a/2) . S) < sample < m – (Z (a/2) . S) CONFIDENCE COEFFICIENT: Z (a/2) | CONFIDENCE COEFFICIENT: Z (a/2) | ∫ − = − )2/( 2 0 2 2 2 2 1 aZ z dze a π || N S Zmmean N S Zm aa )2/()2/( +<<−
- 27. 27 Saad Haj Bakry EVALUATION ALGORITHM: N < 30EVALUATION ALGORITHM: N < 30 Confidence IntervalsBasic Statistics | m – (T (a/2) . S) < sample < m – (T (a/2) . S)| m – (T (a/2) . S) < sample < m – (T (a/2) . S) || ∫ − − + −−Γ Γ = − )2/( 0 )2/( 2 ) 1 1( )1(]2/)1[( )2/( 2 1 aT N dt N t NN Na π || N S Tmmean N S Tm aa )2/()2/( +<<− || )!1(.)( 0 1 −==Γ ∫ ∞ −− fdxexf xf
- 28. 28 SOFTWARE FUNCTIONS: 1/4SOFTWARE FUNCTIONS: 1/4 Saad Haj Bakry Basic Statistics Required Development SUBJECT FUNCTION INPUT OUTPUT Frequency Distribution (Raw Data: Empirical Distributions( “N: Integer”: Number of values “Array of N values: Real Array” “n: Integer”: Number of ranges “Array of n ranges: Real Array” Array of n frequencies: Integer Array” Central Measures Mean “N: Integer”: Number of values “Array of N values: Real Array” )Can also be considered with frequency ranges( “Mean value: Real” Median “Median: Real” Mode “Mode: Real” Dispersion Measures Mean Deviation As above )May be with Mean: Real( )It can also be: Variance for Standard Deviation and Vice Versa( “Average Deviation: Real” Variance “Variance: Real” Standard Deviation “Standard Deviation: Real”
- 29. 29 SOFTWARE FUNCTIONS: 2/4SOFTWARE FUNCTIONS: 2/4 Saad Haj Bakry Basic Statistics Required Development SUBJECT FUNCTION INPUT OUTPUT Binomial Distribution Combination “N: Integer”: Number of values (objects / independent trails) “X: Integer”: Number of selected values (success) “Value of Combination: Real” Probability Density As above plus: “p: Real”: probability of success “Probability of X successes: Real” Cumulative Probability “N, p”: As above. “X1-X2: integers”: Range values “Sum of probabilities (range): Real” Poisson Distribution Probability Density “X: Integer”: Number of (arrivals( “m: Real”: Mean number of (arrivals( “Probability of X arrivals: Real” Cumulative Probability “m”: As above. “X1-X2: Integers”: Range values “Sum of probabilities (range): Real”
- 30. 30 SOFTWARE FUNCTIONS: 3/4SOFTWARE FUNCTIONS: 3/4 Saad Haj Bakry Basic Statistics Required Development SUBJECT FUNCTION INPUT OUTPUT Standard Normal Distribution Probability Density “z: Real”: Random variable (measurement( “Probability of value z: Real” Cumulative Probability “z1-z2: Real”: Range of values “Sum of probabilities (range): Real” Gamma Function “f: Integer”: Degree of freedom “Value of gamma function: Real” “f: Real”: Degree of freedom “f: Integer or Real”: Degree of freedom T- Distribution Probability Density “f: Integer or Real” “t: Real”: Random variable “Probability of value t: Real” Cumulative Probability “f1-f2: Real”: Range of values “Sum of probabilities (range): Real”
- 31. 31 SOFTWARE FUNCTIONS: 4/4SOFTWARE FUNCTIONS: 4/4 Saad Haj Bakry Basic Statistics Required Development SUBJECT FUNCTION INPUT OUTPUT Confidence Co-efficient For Large Sample “L: Real”: Level of Confidence (%) Z(a/2): Real For Small Sample “L”: As above. “N: Integer”: Sample size T(a/2): Real Random Number Generators Uniform “Seed: Real / Integer” : According to requirements “Uniform random value (0-1): Real” U (min, max( “Min, Max: Integers”: Range “Uniform random value in the range: Integer” E (m( “m: Real”: Mean value (duration( “Exponential random value of mean m: Real” N (m, s( “m: Real”: Mean value (measure( “s: Real”: Standard deviation “Normal random value of mean m, and standard deviation s: Real”
- 32. 32 REFERENCESREFERENCES Saad Haj Bakry Seq. Authors / References Title Publication 1 Murray R Spiegel Statistics Schaum’s Outline Series, McGraw-Hill, 1972 2 Ronald E. Walpole Raymond H. Myers Probability and Statistics for Engineers & Scientists Collier Macmillan, 1972 Donald E. Knuth The Art of Computer Programming, Vol.2 Addison-Wesley, 1969 4 Saad Haj Bakry and Mustafa Shatila Pascal Functions for the Generation of Random Numbers Journal of Computer, Mathematics & Applications, Vol. 15, No. 11, pp. 969-973, 1988 Pergamon Press, UK 5 Saad Haj Bakry and Mustafa Shatila A Computer Algorithm for Comp the Confidence Limits of Measured Factors Journal of Engineering Science, KSU, Vol. 2, 1990, pp. 195-200 6 Averill M. Law W. David Kelton Simulation Modeling and Analysis McGraw-Hill, 2000 Basic Statistics