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Researchmethods2012

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Short course on research methods used in Agronomy (Montpellier, France) Vinifera Master

Short course on research methods used in Agronomy (Montpellier, France) Vinifera Master

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  • 1. Experimental methodology and statistics 42 5 1 0011 0010 1010 1101 0001 0100 1011 Pedro Aguiar Pinto papinto@isa.utl.pt January 2012 Instituto Superior de Agronomia Universidade Técnica de Lisboa Portugal
  • 2. Research What for? 42 5 1 0011 0010 1010 1101 0001 0100 1011 What for? What are we talking about? How to?
  • 3. Bernard de Clairvaux (1090-1153) • There are five stimulii that push man towards Science: – There are men that want to know for the simple pleasure of knowing • It is low curiosity 42 5 1 0011 0010 1010 1101 0001 0100 1011 • It is low curiosity – There are other that want to know to be known: • It is vanity – Others want to possess science in order to sell it and make profit and get honours • It is a selfish motivation – But there are some who want to know in order to edify – and this is charity • Others to be edified – and this is wisdom
  • 4. What is all about? It is a matter of knowing 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 5. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 6. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 7. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 8. Observation and experiment • Experimental method – Experimental “pathway” – Trial and error 42 5 1 0011 0010 1010 1101 0001 0100 1011 – Trial and error – Logical deduction (deductive method) – Test
  • 9. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 10. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 11. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 12. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 13. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 14. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 15. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 16. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 17. 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 18. 42 5 1 0011 0010 1010 1101 0001 0100 1011 ISO 3591, Sensory analysis - Wine tasting glass,
  • 19. Sensorial analysis 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 20. Environment / reality • Scenary where the activity takes place • The environment is a reality – that is external to the activity, but has a marked 42 5 1 0011 0010 1010 1101 0001 0100 1011 – that is external to the activity, but has a marked effect on it – that can only be partially modified and in a limited manner
  • 21. Environmental characterization • This reality, strange to the will or action of man is given, is there, it is not made by him. 42 5 1 0011 0010 1010 1101 0001 0100 1011 him. • Environmental characterization deals with knowing reality as such, as it is given. – Deals with knowing its characteristics (most remarkable details)
  • 22. Observation • One gets to know reality (environment) by observation • Observation must prevent the observer’s 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Observation must prevent the observer’s influence as well as the influence of observation tools, otherwise, what is being observed differs from what is given.
  • 23. Quantitative observation • The characteristics of reality that matter to our purpose are known by measurement (observation) of physical dimensions or 42 5 1 0011 0010 1010 1101 0001 0100 1011 (observation) of physical dimensions or quantities • Observation error / measurement error
  • 24. Variability • Measuring the same characeristic results in diferente values as a function of: – observer (observation error) 42 5 1 0011 0010 1010 1101 0001 0100 1011 – observer (observation error) – measurement tool (instrumental error) – location (space and time) of the observation • Environmental variability
  • 25. Example: Soil characterization • Soil physical characteristics have a variability that is mainly spatial – Visible in soil maps – Giving meaning to Precision Agriculture 42 5 1 0011 0010 1010 1101 0001 0100 1011 Precision Agriculture • Time variability (%H2O, %OM,…)
  • 26. Climate characterization I • Aerial environment characterization departs from observations that are, by nature, instantaneous 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Weather characteristics (temperature, atmospheric pressure, radiation,…) are in each instant the focus of observation
  • 27. Climate characterization II • The nature of the factors that determine weather (radiation, general atmosphere circulation and rainfall mechanics) introduces a large temporal variability that 42 5 1 0011 0010 1010 1101 0001 0100 1011 introduces a large temporal variability that adds up to spatial variability
  • 28. Climate characterization III • Note that, – soil characterization is attained by a set of observations (in different locations and along soil profile) that are not repeated in time 42 5 1 0011 0010 1010 1101 0001 0100 1011 – weather characterization demands observation time series that incorporate time variation
  • 29. Climate • Results from the aggregation of a series of observations of instantaneous weather measurements – for example, daily average temperature results 42 5 1 0011 0010 1010 1101 0001 0100 1011 – for example, daily average temperature results from the arithmetic average between maximum and minimun daily temperatures - 2 instantaneous observations used as estimates of the daily “thermal climate”
  • 30. Climate • The concept we call climateclimate results from a greater aggregation of climate data already aggregated , 42 5 1 0011 0010 1010 1101 0001 0100 1011 (averages and arithmetic sums over monthly periods of observation), integrated in indices that allow the differentiation of spatial and geographical units – Climate classification
  • 31. Climate • Each CLIMATE correspondes to a set of climatological normals. – Instantaneous observations – Daily sums or averages 42 5 1 0011 0010 1010 1101 0001 0100 1011 – Daily sums or averages – Monthly sums or averages – Averages of monthly averages or averages of monthly sums for a standard period (30 years)
  • 32. Annual rainfall variability 625 700 775 850 925 30 year average = 617 mm Annual rainfall variability Mora, Portugal 30 years is a long time,but not long enough to find a pattern 42 5 1 0011 0010 1010 1101 0001 0100 1011 325 400 475 550 1955 1960 1965 1970 1975 1980 1985 617 mm
  • 33. Climate normality • The successive data agreggation that leads to what one may call a “normal climate”s useful to several purposes, but has the cost of cancelling the natural variability of instantaneous observations 42 5 1 0011 0010 1010 1101 0001 0100 1011 observations • As a consequence, “the normal climate” is not data anymore, but rather some sort of data manipulation. – Therefore, it is only by coincidence that ond can run into a “normal year”
  • 34. Uncertainty • Variability, and in a more evident and sensible fashion, time variability, illustrates the question of uncertainty in the knowledge of reality 42 5 1 0011 0010 1010 1101 0001 0100 1011 • In any case, it is with this “uncertain” knowledge that we depart to make decisions • By the way, one needs to decide, to make a decision, when he/she is not sure about the outcome
  • 35. Uncertainty • In many cases it is useful to know the degree of uncertainty linked to a decision; and some times one can use predictive models to make predictions/forecasts • As opposed to observations, predictions are not 42 5 1 0011 0010 1010 1101 0001 0100 1011 data: they did not happen… • Assumption: prediction supposes that the pattern that was verified in the past holds in the future (or changes in a given hypothetical manner).
  • 36. Induction and deduction • Data, observation, data analysis, descriptive statistics, polls, samples deductive method 42 5 1 0011 0010 1010 1101 0001 0100 1011 deductive method • Experiments, results, observations, conclusions, generalization, statistical inference inductive method
  • 37. Deductive reasoning • Given some general principle what happens in a specific set of conditions: – Given the formula for the area of a circle, what is the area of a circle whose raius is 5? 42 5 1 0011 0010 1010 1101 0001 0100 1011 whose raius is 5? – Given a key and description of herbaceous species in Southern France, to what species does a certain plant belong? – Give a coin whose probability of coming up heads when tossed is ½, what will happen when the coin is tossed 10 times?
  • 38. Inductive reasoning • Given some specific cases, arrive to some general principles that will apply to all: – Given the areas and radii of several circles, what general formula can we give to express the relation between the areas and the radii? 42 5 1 0011 0010 1010 1101 0001 0100 1011 give to express the relation between the areas and the radii? – Given several specimens of an undescribed weed species, how would you describe the species as a whole and express its relation to other species in a key?? – Given the results of tossing a coin10 times what conclusions can we draw regarding the bias or lack of bias of the coin?
  • 39. Prediction / induction • What happened 1, 2, …, k times can be generalized in the next …. - future … (until n) times 42 5 1 0011 0010 1010 1101 0001 0100 1011 n) times • These implies a statistical description of “what happened”
  • 40. Three questions • What is a sample? • What is the meaning of random? 42 5 1 0011 0010 1010 1101 0001 0100 1011 • What is the meaning of random? • What is a variance?
  • 41. Statistics 42 5 1 0011 0010 1010 1101 0001 0100 1011 Population census Demographics Taxes
  • 42. Statistical data analysys • A data series (for ex. average February temperature for the years 1960-90) might be sinthetically described as – Measures of central tendency • Average, median, mode 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Average, median, mode – Measures of dispersion the way values are distributed around a central tendency • variance • amplitude
  • 43. Sample standard deviation 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 44. Calculation in Excel 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 45. Raíz nr. % sucrose Raíz nr. % sucrose Raíz nr. % sucrose Raíz nr. % sucrose 1 11,8 26 13,5 51 10,1 76 9,0 2 13,1 27 11,9 52 12,4 77 14,0 3 9,2 28 16,7 53 10,8 78 13,2 4 8,7 29 9,6 54 11,3 79 15,0 5 12,9 30 15,1 55 6,3 80 13,8 6 13,7 31 14,6 56 15,7 81 15,1 7 9,6 32 10,4 57 14,3 82 14,9 8 13,7 33 13,4 58 15,0 83 12,6 9 8,5 34 14,6 59 12,5 84 14,1 10 15,7 35 10,5 60 11,8 85 11,4 11 14,1 36 8,6 61 11,6 86 9,4 12 11,9 37 15,2 62 11,2 87 12,4 42 5 1 0011 0010 1010 1101 0001 0100 1011 13 16,7 38 11,1 63 7,5 88 15,0 14 7,4 39 14,5 64 13,4 89 9,4 15 10,0 40 12,1 65 14,7 90 12,9 16 4,4 41 14,9 66 14,2 91 13,4 17 13,2 42 15,0 67 14,0 92 10,6 18 13,8 43 12,1 68 15,1 93 6,5 19 9,1 44 12,6 69 6,5 94 11,0 20 11,9 45 13,0 70 8,7 95 11,9 21 12,8 46 14,1 71 11,0 96 11,8 22 15,3 47 14,4 72 13,0 97 12,6 23 12,6 48 13,1 73 9,2 98 9,5 24 16,1 49 13,3 74 7,0 99 12,2 25 17,2 50 15,0 75 13,2 100 8,2
  • 46. X1 X2 X3 … a1 a2 a3 … ak b1 b2 b3 … sa, a 42 5 1 0011 0010 1010 1101 0001 0100 1011 Xn … bm c1 c2 c3 … cp σ, µsb, b sc, c
  • 47. A two-way table RowsRowsRowsRows (i)(i)(i)(i) ColumnsColumnsColumnsColumns 1111 (j)(j)(j)(j) 2222 …………………… rrrr TotalsTotalsTotalsTotals YiYiYiYi.... MeansMeansMeansMeans 1 Y11 Y12 Y1r Y1. Ÿ1. 2 Y21 Y22 Y2r Y2. Ÿ2. 42 5 1 0011 0010 1010 1101 0001 0100 1011 2 Y21 Y22 Y2r Y2. Ÿ2. … Yij n Yn1 Ynr Totals Y.j Y.1 Y.2 Y.r Y.. Means Ÿ.1 Ÿ.2 Ÿ..
  • 48. Frequency distribution • Maximum and minimum values • Amplitude • Number of classes: 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Number of classes: Sturges’ rule: k= 1 + 3.3 log N • Class interval: amplitude/k
  • 49. Distributions • The way a series of data is distributed as a function of its relative frequency (frequency curve or polygon) • Normal distribution as interesting and useful 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Normal distribution as interesting and useful properties – simmetry – average, mode and median coincide – we can know the probability or ocurrence of any value
  • 50. Frequency polygon 16 20 24 Frequency polygon 42 5 1 0011 0010 1010 1101 0001 0100 1011 0 4 8 12 4,80 6,30 7,80 9,31 10,81 12,31 13,81 15,31 16,81 % de Sucrose Histogram
  • 51. Normal distribution 2,0% 2,5% 3,0% 60% 70% 80% 90% 100% 42 5 1 0011 0010 1010 1101 0001 0100 1011 0,0% 0,5% 1,0% 1,5% 0 20 40 60 80 100 0% 10% 20% 30% 40% 50%
  • 52. Standard deviations 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 53. Normal distribution and scales 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 54. Different normal distributions Differences in position Differences in dispersion 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 55. Normal deviations 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 56. Normal distribution table 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 57. • Student'sStudent'sStudent'sStudent's tttt----distributiondistributiondistributiondistribution (or simply the tttt---- distributiondistributiondistributiondistribution) is a continuous probability distribution that arises in the problem of t-Student’s distribution 42 5 1 0011 0010 1010 1101 0001 0100 1011 distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small
  • 58. t distribution df=1 df=30 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 59. t-Student’s distribution 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 60. Chi-square distribution 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 61. Finite vs. infinite • The observation of reality is always finite – 20 years vs. all years…. • The data I have are a sample of all data 42 5 1 0011 0010 1010 1101 0001 0100 1011 • The data I have are a sample of all data possible about this subject • Frequency distribution is only an approximation (estimate) of the true distribution
  • 62. Sample dimensions • The larger the sample size, the closer the frequency distribution is to the “theoretical distribution”. • When sample size tends to infinity, the 42 5 1 0011 0010 1010 1101 0001 0100 1011 • When sample size tends to infinity, the distribution tends to be well represented by the normal distribution
  • 63. Finite samples • Is the distribution normal? • Chi- Square test χ2 χ2 a quocient of variances 42 5 1 0011 0010 1010 1101 0001 0100 1011 χ2 a quocient of variances
  • 64. Hipothesis testing • Chi-square test as well as other statistic tests (t-Student, Fischer’s F, etc.) are tests that use the instruments of 42 5 1 0011 0010 1010 1101 0001 0100 1011 are tests that use the instruments of classical logic • Null hypothesis: Ho (no dif.) • Alternative hypothesis: H1 (sign. dif.)
  • 65. Null Hypothesis • Represents the attitude of observer’s independence, i. e., the real attitude that accepts reality as given, as data, as 42 5 1 0011 0010 1010 1101 0001 0100 1011 accepts reality as given, as data, as opposed to manipulating it as a result of a prejudice – the idea we make of it.
  • 66. Probability and significance • When the test value is larger than a table value for the same degrees of freedom and a chosen probability level (the power of the 42 5 1 0011 0010 1010 1101 0001 0100 1011 a chosen probability level (the power of the test) the null hypothesis is refused.
  • 67. type I error • In 100 cases, I fail 5 times if the power of the test is 95% (5% significance) 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 68. Analysys of variance involves • Partition of the sum of squares by origins of variation • Estimation of variance for each origin of 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Estimation of variance for each origin of variation • Comparison of variances by F tests
  • 69. ANOVA VarVarVarVar.... ReplicationsReplicationsReplicationsReplications Y1.Y1.Y1.Y1. Y1Y1Y1Y1.(.(.(.(averaveraveraver)))) Yields of 2 wheat varieties from plots to which the varieties (A and B) where randomly assigned (values in 100 kg) 42 5 1 0011 0010 1010 1101 0001 0100 1011 VarVarVarVar.... ReplicationsReplicationsReplicationsReplications Y1.Y1.Y1.Y1. Y1Y1Y1Y1.(.(.(.(averaveraveraver)))) A 19 14 15 17 20 85 17 Y1. (aver.) B 23 19 19 21 18 100 20 Y2. (aver.) 100 kg is an old unit of mass: quintal or centner in English, quintal in French. It is equivalente in the pound system to the unit hundredweight http://en.wikipedia.org/wiki/Quintal
  • 70. ANOVA – Analysis of variance OriginOriginOriginOrigin ofofofof variationvariationvariationvariation DegreesDegreesDegreesDegrees ofofofof freedomfreedomfreedomfreedom SumSumSumSum ofofofof SquaresSquaresSquaresSquares MeanMeanMeanMean SumSumSumSum ofofofof SquaresSquaresSquaresSquares Total kr-1 SS MS 42 5 1 0011 0010 1010 1101 0001 0100 1011 Total kr-1 SS MS Treatments k-1 SST MST Within treatments (Experimental Error) k(r-1) SSError MSE
  • 71. ANOVA (Wheat yield varieties) • Step 1 – Outline the ANOVA table and list the sources of variation and degrees of freedom • Two sources of variation: 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Between treatments (Varieties) • Within treatments (replications)
  • 72. • Anova table for the wheat example OriginOriginOriginOrigin ofofofof variationvariationvariationvariation DegreesDegreesDegreesDegrees ofofofof freedomfreedomfreedomfreedom SumSumSumSum ofofofof SquaresSquaresSquaresSquares MeanMeanMeanMean SquaresSquaresSquaresSquares Total kr-1 (2x5-1) 9 SST MST ANOVA (wheat example) (cont.) 42 5 1 0011 0010 1010 1101 0001 0100 1011 Total kr-1 (2x5-1) 9 SST MST Treatments k-1 (2-1)1 SStreatments MSTreatments Within treatments (Experimental Error) k(r-1) (2x(5-1)) 8 SSError MSE
  • 73. ANOVA (wheat example) (cont.) • Step 2 – Calculate the total sum of squares • SS = Σ (Yij – overall mean)2 64,5 • Step 3 – Calculate the sum of squares for treatments • SST = Σ (Yi. – overall mean)2 4,5 • Step 4 – Calculate the sum of squares for error 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Step 4 – Calculate the sum of squares for error • SSE= SS-SST 64,5 – 4,5 = 60 • Step 5 – Calculate the mean squares • MST = SST/(k-1) 4,5 MSE = SSE/k(r-1) 60 /8 = 7,5 • Step 6 – Calculate the F value • F = MST / MSE 4,5 / 7,5 = 0,6
  • 74. OriginOriginOriginOrigin ofofofof variationvariationvariationvariation dfdfdfdf SumSumSumSum ofofofof SquaresSquaresSquaresSquares MeanMeanMeanMean SquaresSquaresSquaresSquares FFFF valuesvaluesvaluesvalues Total 9 64,5 ANOVA (wheat example) (cont.) 42 5 1 0011 0010 1010 1101 0001 0100 1011 Total 9 64,5 Treatments 1 4,5 4,5 Within treatments (Experimental Error) 8 60,0 7,5 MST/MSE =F* =4,5/7,5 =0,6
  • 75. F- table 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 76. F-distribution (5;20) 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 77. Partitioning of the sum of squares SS = ∑ ∑ (Yij- Y..) 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 78. Glossary so far • Reality • Data • Sample • Chance – Randomness • Frequency • Frequency polygon • Distribution functions • Median • Mode • Deviation 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Frequency – Relative – Absolute • Average • Mean • Frequency classes • Deviation • Variance • Standard deviation • Coefficient of variation • Hypothesis testing • Confidence intervals • ANOVA
  • 79. Research, scientific method and the experiment • Research – A systematic inquiry into a subject to discover new facts or principles. The procedure for 42 5 1 0011 0010 1010 1101 0001 0100 1011 new facts or principles. The procedure for research is generally known as the scientific method
  • 80. Scientific method 1. Formulation of an hypothesis 2. Planning an experiment to test the hypothesis 42 5 1 0011 0010 1010 1101 0001 0100 1011 hypothesis 3. Careful observation and collection of data from the experiment 4. Interpretation of the experimental results
  • 81. Characteristics of a well planned experiment • Simplicity • Degree of precision – Appropriate design and sufficient replication • Absence of systematic error 42 5 1 0011 0010 1010 1101 0001 0100 1011 • Absence of systematic error – No bias • Range of validity of conclusions – Replication on time and space • Calculation of the degree of uncertainty – Probability of obtaining the observed results by chance alone
  • 82. Steps in experimentation 1. Definition of the problem Clearly and concisely; if you can’t define there is little chance you can solve it 2. Statement of objectives 42 5 1 0011 0010 1010 1101 0001 0100 1011 2. Statement of objectives Write down in precise terms; hierarchy 3. Selection of treatments 4. Selection of experimental material Material used should be representative of the population
  • 83. Steps in experimentation 5. Selection of experimental design Parcimony – the simplest possible 6. Selection of the unit for observation and the number of replications 42 5 1 0011 0010 1010 1101 0001 0100 1011 the number of replications 7. Control of the “border effect” 8. Consideration of data to be collected 9. Outlinig statistical analysis and summarization of results Sources of variation in ANOVA What means to compare?
  • 84. Steps in experimentation 10. Conducting the experiment Procedures free from personal biases (fatigue, double- checking, careful note-taking) 11. Analysing data and interpreting results 42 5 1 0011 0010 1010 1101 0001 0100 1011 Dont’t jump into conclusions even if statistically significant 12. Preparation of a complete, readable and correct report of the research There is no such thing as a negative result
  • 85. The three R’s of experimentation I. Replicate 42 5 1 0011 0010 1010 1101 0001 0100 1011 II. Randomize III. Request help
  • 86. Linear correlation and regression • The idea – The more, the merrier – The bigger they are, the harder they fall – Easy come, easy go 42 5 1 0011 0010 1010 1101 0001 0100 1011 – Much haste, little speed – The best gifts come in small packages • 2 variables: dependent, independent • Direct or inverse correlation; • Measuring correlation: – correlation coefficient ( r )
  • 87. Regression • The amount of change in one variable associated with a unit change in the other variable 42 5 1 0011 0010 1010 1101 0001 0100 1011 variable • Correlation – refers to the fact that two variables are related and to the closeness of the relationship • Regression – refers to the nature of the relationship
  • 88. Regression examples • A penny saved is a penny earned • A bird in hand is worth two in the bush 42 5 1 0011 0010 1010 1101 0001 0100 1011 • A stitch in time saves nine • One picture is worth a thousand words
  • 89. Sayings in math terms IndependentIndependentIndependentIndependent varvarvarvar. X. X. X. X DependentDependentDependentDependent varvarvarvar. Y. Y. Y. Y RegressionRegressionRegressionRegression eqeqeqeq.... RegressionRegressionRegressionRegression coeffcoeffcoeffcoeff.... Pennies saved Pennies earned Y=X 1 Hand birds Bush birds Y=2X 2 Stitches in time Stitches saved Y=9X 9 42 5 1 0011 0010 1010 1101 0001 0100 1011 Stitches in time Stitches saved Y=9X 9 Pictures Words Y=1000X 1000 Y = mx + b
  • 90. Y = mx+b 42 5 1 0011 0010 1010 1101 0001 0100 1011
  • 91. y = 2,856x - 104908 R² = 0,9633 y = 0,972x - 35146 R² = 0,7851 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Regression in Excel 42 5 1 0011 0010 1010 1101 0001 0100 1011 y = -1,884x + 69762 R² = 0,9846 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 Abr-01 Ago-01 Dez-01 Abr-02 Ago-02 Dez-02 Abr-03 Ago-03 Dez-03 Abr-04 Jul-04 Nov-04 Mar-05 Jul-05 Nov-05 Mar-06 Inscrições Desistências e inactivações Membros
  • 92. y = 0,246x - 9540,3 R² = 0,1352800 1000 1200 42 5 1 0011 0010 1010 1101 0001 0100 1011 0 200 400 600 14-Nov 22-Fev 1-Jun 9-Set 18-Dez 28-Mar 6-Jul 14-Out 22-Jan 2-Mai 10-Ago
  • 93. Linear regression and Excel • On-line tutorial – http://phoenix.phys.clemson.edu/tutorials/exc el/regression.html 42 5 1 0011 0010 1010 1101 0001 0100 1011 el/regression.html