BIOL 570 1nd Edition Lecture 3 Outline of Last Lecture I. Displaying dataa. Bar charts, histograms, cumulative frequency distributions, contingency tables, grouped bar charts, multiple histograms, scatterplotsII. Descriptive statisticsa. Measures of location: mean, median, modeb. Measures of spread: range, variance, standard deviationOutline of Current Lecture I. Numerical data a. empirical ruleb. coefficient of variationc. interquartile range, box plots, outliersII. Categorical dataa. proportionsIII. Brief summary of variables, samples, and populationsIV. Properties of descriptive statisticsCurrent LectureVariance (s2): Standard deviation (SD, s): * has to be positive (can be zero), CANNOT be negativeThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.Empirical Rule: If data are roughly bell shaped (normal distribution), then…Ȳ ± 1SD – will include 67-68% of the dataȲ ± 2SD – will include 95% of the data10.45 + .818 = 11.268 cm10.45 - .818 = 9.632 cmSee p. 80 problem 14Coefficient of variation *no unitsLeft hand bar Right hand bar Ȳ = 2 Ȳ = 3.5s = .25 s = .4(.25/2) * 100= 12.5% (.4/3.5) * 100= 11.4%Interquartile Range or lengthIQR = Q3 – Q117 38 23 14 18 34 16 23 - raw data14 16 17 18 23 23 34 38 – orderedMedian (Q2) = (18+23)/2= 20.514 16 17 18 23 23 34 38 ↓ ↓ ↓ 16.5 20.5 28.5 Q1 Q2 Q3IQR = Q3 – Q1 = 28.5 – 16.5 = 12*represents 50% of dataData14 16 17 18 23 23 34 Q1 Q2 Q3Box plot- Q1 to Q3 with a line for Q2 , whiskers from min to max 25% 25% 25% 25% 0 10 16 20 30 40A graph of this data will be skewed-More consistent with…A) B) If the data were a normal distribution- *25% of the data is between (16-18), the amount of data in thisnarrow range results in a “hump”Bell shaped curveData points (in order) 1) No outliers14 16 17 18 23 23 34 Q1 Q2 Q32) Outliers 3) 2.2 2.7…. 5.6 5.8 7.5 10.3Min= 2.2Q1 =3.6 Q2 = 4.3 Q3 =4.85 Outliers: 7.5, 10.3Max =10.3IQR =1.25 center line (median)0 1 2 3 4 5 6 7 8 9 10 11 12Outlier- data that either…1) Exceeds Q3 + 1.5 (IQR)*outliers are represented as dots on the box plot4.85 + 1.5 (1.25) = 6.375 → 7.5, 10.32) Is smaller than Q1 - 1.5 (IQR)3.6 – 1.5 (1.25) = 1.725Categorical data- sum by proportionsVinyl Latex Did not leak 89 223 312Leaked 151 17 168240 240 480Proportion: 151/240 = .629 17/240 = .0708Mean of a combined data setMean of 2 samples = (m(x-bar) + n(Ȳ))/ m+nm= sample size of x-barn= sample size of ȲSummary Table Graph Descriptive statisticsSingle variableNumerical Frequency table HistogramBox plotCumulative frequency distribution(2 variables: scatterplot)Location: mean, median, modeSpread: standard deviation, variance, range Categorical Frequency table Bar graph chartProportions Sample (statistic) Population (parameter)Mean Ȳ (Y-bar) µVariance s2σ2Standard deviation s (SD) σProportion p-hat P1) If a constant c is added to all measurements then…Ȳ’ = Ȳ + cȲ’ = mean of new data setȲ = original meanc = 31Feb 2 31 + 2 = 33Feb 4 31 + 4 = 35Feb 6 31 + 6 = 37x-bar = 352) If a constant c is added to all measurements the....new SD = old SDnew variance = old variance* spread does NOT change3) If each observation is multiplied by a constant c, then… the mean of the new data set = the mean of the original data set multiplied by