Part I, Chapters 4 & 5

1. Part I, Chapters 4 & 5 Data Tables and Data Analysis Statistics and Figures

2. Descriptive Statistics 1 • Are data points clumped? • (order variable / exp. variable) • Concentrated around one value? • Concentrated in several areas? • Do data point pairs show a pattern? • (exp. variable #1 / exp. variable #2) • Straight line? • Parabola? • Sin function? This is the first decision you need to make!

3. Scatter Graphs • A scatterplot is a useful summary of a set of bivariate data (two variables), usually drawn before working out a linear correlation coefficient or fitting a regression line. Each unit contributes one point to the scatterplot, on which points are plotted but not joined. The resulting pattern indicates the type and strength of the relationship between the two variables. • Gives a good visual picture of the relationship between the variables • Aids the interpretation of the correlation coefficient or regression model http://www.stats.gla.ac.uk/steps/glossary/presenting_data.html

4. Bar and Column Graphs • A bar / column graph is a way of summarising a set of categorical data. It is often used in exploratory data analysis to illustrate the major features of the distribution of the data in a convenient form. It displays the data using a number of rectangles, of the same width, each of which represents a particular category. The length (and hence area) of each rectangle is proportional to the number of cases in the category it represents, for example, age group, religious affiliation. • Summarize nominal or ordinal data • Displayed horizontally (bars) or vertically (column) • Drawn with a gap between the bars (rectangles) http://www.stats.gla.ac.uk/steps/glossary/presenting_data.html

5. Frequency Analysis: Histograms • A histogram is a way of summarizing data that are measured on an interval scale (either discrete or continuous). It is often used in exploratory data analysis to illustrate the major features of the distribution of the data in a convenient form. It divides up the range of possible values in a data set into classes or groups. For each group, a rectangle is constructed with a base length equal to the range of values in that specific group, and an area proportional to the number of observations falling into that group. This means that the rectangles might be drawn of non-uniform height. • Variables are numerical • Variables are measured on an interval scale • Used with large data sets (>100 observations) • Detect unusual observations (outliers, gaps) http://www.stats.gla.ac.uk/steps/glossary/presenting_data.html

6. Descriptive Statistics 2 Size – Total number of Data Points, N -- Number of reactions performed -- Numbers of points measured (i.e., for a spectrum) Range – Distance between smallest and largest data value -- Min., Max., and Range = Max. – Min. -- Average ± Difference/2 Middle – There are many types of averages -- Average = Mean = Arithmetic Mean: Sum(Data) / N -- Geometric Mean: [ Product(Data) ]1/N -- Mode: Most frequent data value -- Median: N/2 data are below and above. Spread – Frequency of Significant Deviation -- Standard Deviation -- Central 50% Does the Study Matter? Does the Parameter Matter? How Does the Parameter Matter? How Frequent Are Deviations?

7. Standard Deviation

8. Standard Deviation Real Work Sample: Data Histogram Mathematical World: Inferential statistics maps the data by a formula that describes the population.

9. Normal or Gaussian DistributionBell Curve A normal distribution in a variante x with mean a and variance s 2is a statistic distribution with probability density function If Gaussian, then: Mean = Median = Mode Mean and Standard Deviation determine the distribution http://mathworld.wolfram.com/NormalDistribution.html

10. Significance Test & p-Valuep-value = alpha level = significance level For significance tests there is a hypothesized condition (called null hypothesis, H0) that one is testing to see if it is true.  For a test of fit the hypothesized condition is that the selected distribution (i.e., a normal distribution) generated the data. The p-value is the probability that the data have been generated under the hypothesized condition. A p-value of 0.05 indicates that the chance of the observed data is low, 1 in 20, due to variation alone. This is good evidence that the data was not generated under the hypothesized condition.  The hypothesized condition is rejected if the p-value is 0.05 or below.  A p-value of 0.05 provides 95% confidence the hypothesized condition is not true (i.e., no normal distribution). The confidence level is calculated from the p-value as 100*(1 – p). 

11. Cramer von Mises Test Andersen-Darling Test A “quadratic EDF statistics” EDF = empirical dist. function AD value –{Statistics}-> p-value

12. Normal Distribution, 1D-Gaussian

13. Scatter Graph In Electronics Research

14. Gaussian Dist. In Polymer Research

15. Gaussian Dist. In Food Chemistry

16. Histogram In Food Chemistry A log-normal distribution: variable whose logarithm is normally distributed.

17. Histogram In Nano Chemistry PRD = Particle Radius Distribution A log-normal distribution: variable whose logarithm is normally distributed.

18. Histogram In Polymer Research

19. Histogram In Solid State Research

20. How to Compute Descriptive Statistics,Present Scatter Graphs & Histograms, and Plot Functions using Excel Example in Lecture: Test Scores Assign. #3: Handout & online.

21. Create a Bar Graph 1

22. Create a Bar Graph 2

23. Create a Bar Graph 3

24. Create a Column Graph 1

25. Create a Column Graph 2

26. Create a Scatter Graph 1

27. Create a Scatter Graph 2

28. Stats 1

29. Stats 2

30. Histogram: Load Analysis 1

31. Histogram: Load Analysis 2

32. Histogram: Load Analysis 3

33. Histogram: Load Analysis 4

34. Histogram: Load Analysis 5

35. Histogram: Load Analysis 6

40. Distribution Functions Gaussian, Normal Distribution Poisson Distribution Binominal Distribution Pascal Distribution (Neg. Binom. Dist.)

41. Normal or Gaussian DistributionBell Curve A normal distribution in a variante x with mean a and variance s 2is a statistic distribution with probability density function If Gaussian, then: Mean = Median = Mode Mean and Standard Deviation determine the distribution http://mathworld.wolfram.com/NormalDistribution.html

42. Poisson Distribution Frequency Outcome Bins http://mathworld.wolfram.com/PoissonDistribution.html

43. Poisson Distribution http://demonstrations.wolfram.com/PoissonDistribution/

44. Poisson Dist. in Single Photon Detection The Poisson distribution can be used to describe how rare, random events are distributed in time or space and applies to measurements in which the number of observations is large but the number of events is very small. In chemical measurements Poisson statistics apply to the extraordinarily small signals commonly found in ultrasensitive analytical instrumentation: for example, single-photon counting fluorescence detection. Harbron, E. J.; Barbara, P. F. J. Chem. Educ. 2002, 79, 211-213.

45. Binominal Distribution Frequency The binomial distribution gives the discrete probability distribution P(n|N) of obtaining exactly n successes out of N Bernoulli trials where the result of each Bernoulli trial is true with probability pand false with probability q. Outcome Bins http://mathworld.wolfram.com/BinomialDistribution.html

46. Binominal Dist.: N Coin Flips p = q = 0.5 N = 10, 20, and 40.

47. Binom. Dist.: N People There are N people at Shakespeare’s and on average 75% are adults. We are sitting outside and watch the door as all people exit. What are the probabilities that n adolescents exit the place any given time? We watch the door many, many times and then construct a histogram. N = 10, 20, and 40. p = 0.25 for not being adult

48. Binom. Dist. In Nanochemistry Au25(SCH2CH2Ph)18 + (HSC6H5 or HSPh)  Au25(SCH2CH2Ph)18-m(SR)m + HSCH2CH2Ph Large excess of added thiol. Probabilities for ligand exchange are different for each type of thiol. For each thiol, the probability for exchange does not depend on environment in cluster. Use mass spectrometry to analyze the exchange reaction.

49. Binominal Dist. In Nanochemistry

50. Poisson DistributionBinominal Distribution for Large N Binominal Frequency Poisson Outcome Bins The N is gone!! We can determineN based on the measured distribution. http://mathworld.wolfram.com/PoissonDistribution.html