This document provides an overview of key concepts in applied statistics and design of experiments (DOE). It defines common measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation, coefficient of variation). It also describes hypothesis testing using z-tests, t-tests, F-tests and ANOVA. Key concepts in regression, correlation and different experimental designs like factorial, fractional factorial and response surface methodology are summarized.

1. Applied Statistics and DOEMayank

2. Applied StatisticsMeasures of central tendency (central position of data)µMeanPopulation :Sample:MedianModeMeasures of dispersion (spread of data)Varianceσ2s2Population :Sample:Standard deviationσsPopulation :Sample:Coefficient of variation

3. Measures of Central tendencyData: 34, 43, 81, 106, 106 and 115MeanAverage Σx/n =80.83ModeHighest frequency =106MedianMiddle score (81+106)/2 =93.5

4. Measures of dispersionVariance:Standard deviation:xSSSS/(n-1)MSsd√MSMost of the data lies between 44.5±4,57 = 39to 49

5. Measures of dispersionCoefficient of VarianceCV = s/ *100% 4.57/44.5*100% = 10.28%Standard deviation is 10.28% of the mean

6. Measures of dispersionNormal DistributionExample: IQ Score

7. Measures of dispersionNormal DistributionIQ ScoreCountScore<55115130145100857055145<

8. Measures of dispersionNormal Distribution34.13%34.13%Probability13.59%13.59 %Score2.14%2.14%0.13%0.13%0.0031%0.0031%0.000028%0.000028%Sd from-6σ-5σ-4σ-2σ-1σ1σ2σ3σ-3σ5σ6σ4σμ68.2689%95.4499%99.7300%99.9936%99.999942669%99.999999802%

9. Measures of dispersionNormal DistributionSix SigmaDPMODPHOLSLUSLSd from-6σ-5σ-4σ-2σ-1σ1σ2σ3σ-3σ5σ6σ4σμ99.999999802%

10. Measures of dispersionNormal DistributionLSLLSLUSLUSL

11. Measures of dispersionNormal Distribution1.5 σLSLUSL3.4 DMPO-6σ-5σ-4σ-2σ-1σ1σ2σ3σ4σ-3σ5σ6σμ

12. Statistical significance testsSignificance tests Z- test t- test F- test ANOVA

13. Statistical significance testsZ - test Z-value :How many standard deviations away from mean?+ve z: values are above the mean, -ve z: values are below the meanPopulationSampleGroup compared to population1 point compared to population

14. Statistical significance testsZ - test Sample :BMIMean ( ) = 26.20Standard deviation (s) = 6.57What is the probability that of a person having BMI 19.2 sdbelow the mean19.2 sd above the meanA person with a BMI of 19.2 has a z score of:So this person has a BMI 1.07 standard deviations below the mean

15. Statistical significance testsZ - test Sample :Probability<19.6>19.6Sd16 %84 %-1σμStandard deviationZ score0-1

16. Statistical significance testsZ - test Population :Test group : Employee having two wheelerTest : Commuting time from home to BioconClaim : Average commuting time is less than 24 minAt 0.01 level of significance (α=0.01):Is there enough evidence to support the research claim???Samples : 3018 16 23 19 25 48 13 17 20 2316 21 18 16 29 15 8 19 20 715 16 24 15 6 11 14 23 18 12

17. Statistical significance testsZ - test Population :Assumption: Population is normally distributed ProbabilityScore24MeanX

18. Statistical significance testsZ - test Population :Hypothesis testingTest vs PopulationComparison of means:Null hypothesis : H0No difference (Claim not true)H0 : x ≥ µµ = 24Alternate hypothesis : H1It is different (Claim is true)H1 : x < µ

19. Statistical significance testsZ - test Population :ProbabilityProbability24MeanXZ valueScoreLevel of significanceα = 0.01CriticalvalueZ0-2.33

20. Statistical significance testsZ - test Population :Ztest< ZcriticalZtest>ZcriticalRejection regionAcceptance region-2.33Z = 18.2s = 7.7Z = - 4.13µ = 24n = 30

21. Statistical significance testsZ - test Population :Rejection region-2.33- 4.13ZSo is test value is significantly different (lower) than the mean Yes: There are significant evidence to reject the null hypothesisH0 : s ≥ 24Rejectedand therefore accept the claimH1 : s < 24Significantly supported

22. Statistical significance testst - testComparison of means between two groupsH0: H1: Null hypothesis will be rejectedttest > tcriticalNull hypothesis will not be rejectedttest < tcritical

23. Statistical significance testst - testComparison of means between two groupsSignalDifference between group meanst = =NoiseVariability of groups

24. Statistical significance testst - testEffect of fertilizer on plant heightCase 1Fertilizerw/o Fertilizer27.15 – 17.9t test = = 2.4t critical with 38 df at 0.05 significance level= 2.03Plant heightdf = 2n-2ttest > tcriticalSo is significantly different from H0: RejectedH1: s2

25. Statistical significance testst - testCase 2Fertilizerw/o Fertilizert critical =2.031.3t test = Plant heightttest < tcriticalSo is not significantly different from H0: Not rejectedRejectedH1: s2

26. Statistical significance testst - testOverview

27. Statistical significance testsF - testComparison of variances where and are the sample variancesF =The F hypothesis test is defined as:H0: =RejectedHa: <>≠If Ftest > Fcritical (at significant level)

28. Statistical significance testsANOVAANalysisOf VArianceOne way : Effect of one factor (variable)Two way : Effect of two factors (variables)

29. Effect of interactionStatistical significance testsOne way ANOVAStrategy:Compare variability within group MSwg to between groups MSbgMSbgF = MSwgGroup 1Group 1Group 2Group 2Between groupsWithin groups

30. Statistical significance testsOne way ANOVAIs there any impact of exam room temperature on student performance?Factor ( Independent Variable): Temperature (cold, optimum, hot)Effect ( Dependent Variable): Score (marks obtained)Null hypothesis (H0) : No effect (µ1= µ2 = µ3)Alternate hypothesis (H1) : There is an effect (µ1 ≠ µ2 ≠ µ3)

31. Statistical significance testsOne way ANOVACOHColdOptHotNumber of AttendeesSS= X̄

32. Statistical significance testsOne way ANOVAMSbg==F = 6.40MSwgFcriticalfor Numerator degrees of freedom : 2Denominator degrees of freedom : 33 At significance level (α) : 0.05=4.17Ftest > FcriticalSo there are enough evidence to reject null hypothesisH0: All means are same (no effect of Temperature)RejectedAt 95% confidence level we can say:That the variation between means is not just by chanceExamination Room temperature matters significantly

33. Statistical significance testsTwo way ANOVAFactors ( Independent Variable): 1) Gender:Man Woman2) Type of sport Indoor OutdoorEffect ( Dependent Variable): 1) Number of participantsRelative impact of gender or type of sprot?Any interaction between gender and type of sport?Null hypothesis (H0a) : No effect of gender Null hypothesis (H0b) : No effect of type of sportNull hypothesis (H0c) : No interaction Alternate hypothesis (H1) : There is an effect

34. Statistical significance testsTwo way ANOVAMan Woman s↓g->IndoorOutdoor

35. Statistical significance testsTwo way ANOVAIndoor OutdoorNull hypothesis (H0a) : No effect of genderRejectedRejectedNull hypothesis (H0b) : No effect of type of sportsRejectedNull hypothesis (H0c) : No interaction

36. Statistical significance testsTwo way ANOVAFactors ( Independent Variable): 1) Temperature:30 352) pH 5 7Effect ( Dependent Variable): 1) Total product (g)pH 7pH 530o C 35o C

37. Regression and correlationRegression analysis:Investigation of relationship between variables

38. Regression and correlationRegression analysis:Investigation of relationship between variablesy = -0.951x + 50.49y = ax +bR² = 0.955One independent variableSimple linear regression

39. Regression and correlationRegression analysis:Simple linear regressiony = ax + bNon linearMultiple linear regressiony = a1x1+ a2x2+ a11 x2 + a12 x1x2+by = a1x1+ a2x2+ a3x3+ bLinearNon Linear

40. Regression and correlationCorrelation analysis:To find how well (or badly) a line fits the observationWhat is the strength of this relationship- r2 (coefficient of determination) or adjusted r2Is the relationship we have described statistically significant?-Significant tests

41. Regression and correlationCorrelation analysis:ŷ = ax + binterceptslopeε= ŷ, predicted value= y i, true valueε =residual error =y - ŷA and b values are calculated that minimize Sum of Squares (SS) of residuals =Σ (y – ŷ)2 : minimum

42. Regression and correlationCorrelation analysis:r2 : Coefficient of determinationErrorTotal(yi – y)2(y – ŷ)2Always between 0 and 1Increase with number of predictorSSError= 1-r2SSTotalIt can be negative alsoSSError/(n-p-1)Adjusted r2= 1- True representative of relationship strengthSSTotal/(n-1)n= total observationp= Number of predictor

43. MSbgMSModel==FFMSwgMSErrorGroup 1Group 1Group 2Group 2Regression and correlationCorrelation analysis:Statistical significance of relationshipErrorModel

44. Design of experimentTraditional methodOne factor at time (OFAT)Statistical methodMultiple factor at time (MFAT)

45. Design of experiment

46. Design of experimentHow to select a design?

47. Design of experiment- terminologyIndependent variable/sFactorsContinuousNumeric: any value between lower and upper valueeg. Temperature, pH, concentrationCategoricalNumeric/non-numeric : only characters or levelseg. Gender, operator, type, temperatureLevels-1(lower)+1(higher)0(middle)Range of a factor/sEffectsDependent variable/s: ResponseMain effect/sEffect/s due to individual factor/sInteraction effect/sEffect/s due to interaction of multiple factorsConfounding/AliasingWhen two or more effects can not be distinguishedeg. Main effect is confounded with interaction effects Main effects and interaction effects are aliased

48. Design of experimentResolution of a designPower of a designHigher order interaction are less significant than lower order interaction

49. Design of experimentFactorial designFactorLfFull factorial:Level

50. Design of experimentFactorial design22ba4 experiments

51. Design of experimentFactorial design23cba8 experiments

52. Design of experimentFactorial design32ba9 experiments

53. Design of experimentFactorial design33cb27 experiments

54. Design of experimentFractional Factorial design23-1238 experiments4 experiments

55. Design of experimentResponse surface methodology

56. Design of experimentGeometry of some important response surface designsBox - Behnkeneg. 3 factor 3 level12 experiments

57. Design of experimentGeometry of some important response surface designsCentral composite designeg. 2 factor 2level+=

58. Design of experimentGeometry of some important response surface designsTaguchi designSignalMedia, pH, feed rateInner array:Controllable variables during productionOuter array:Uncontrollable variables during productionNoiseTemp, DO,