Descriptive StatisticsMeasures of LocationArithmetic MeanWeighted Arithmetic MeanWeighted Sample Mean: ExampleGeometric MeanGeometric Mean: ExampleMedianMedian: ExampleMode (Mo)PercentilesSpecial PercentilesSkewnessMeasures of Variability (or Dispersion)Slide 15Slide 16Sample RangeVarianceSlide 19Variance: ExampleStandard Deviation (s.d.)Coefficient of VariationCoefficient of Variation: Example 1Tabular SummariesFrequency TableGraphical Methods1Descriptive StatisticsMathematical•Measures of Location•Measures of VariabilityTabularGraphical2Measures of LocationFind the center of the dataEstimate the center of the population•Arithmetic mean: average value•Weighted arithmetic mean•Geometric mean: constant growth rate•Median: middle value•Mode: most frequent value•Percentiles, Quantiles, Quartiles3Arithmetic Mean : population mean sample meanWhen  is unknown, estimates Excel functions•SUM(<data>)•AVERAGE(<data>)nxxxnxxnnii211:xx4Weighted Arithmetic MeanWeighted w or Excel: Use SUM function to create formula.niiniiiwwxwx115Weighted Sample Mean: ExampleCompute average grade point, G.P.A.GPA =27.31334)4(1)4(3)4(3)2(4wxCourse 1 2 3 4 Grade C A A A Credit Hr (wi) 4 3 3 1 Grade Point (xi) 2 4 4 46Geometric MeanGeometric mean of a1, a2,…, an =•I = initial investment•Ri = rate of return in period i•Wealth after n periods isI · (1+R1) · (1+R2) ··· (1+Rn)•Rg = constant growth rate that results in same wealth = GeoMean(1+R1, 1+R2 , … , 1+Rn )  1Excel function: GEOMEAN(<data>)nnaaa 217Geometric Mean: ExampleI = $100, R1 = 50%, R2 = 20%, R3 = –70%Wealth after period 1 = 100(1+0.5) = 150Wealth after period 2 = 150(1+0.2) = 180Wealth after period 3 = 180(1–0.7) = 54Wealth = 100(1–0.185674715)3 = 54Wealth dropped 18.57% per year, on average.185674715.0154.01)7.01)(2.01)(5.01(33gR8Median population median•50% of the values above and 50% of the values below sample median•For an odd number of observations, is the middle observation in an ordered set of observations•For an even number of observations, is the average of the two middle observationsWhen is unknown, estimatesExcel function: MEDIAN(<data>))(21~nxx2/)(~)1()(22nnxxx:~x:~x~x~x~~~9Median: ExampleCalculate the sample median and sample meana) –2, 0, 3, 4, 7, 7, 9 b) –2, 0, 3, 4, 7, 7, 9, 10  ,c) –2, 0, 3, 4, 7, 7, 9, 30  ,For symmetrical distributionsMedian is not sensitive to extreme valuesMean is sensitive to extreme values4~x5.5274~x~472879774302x75.4x5.5~x 25.7x10Mode (Mo)For individual observations, the mode is the most frequently occurring valueFor symmetrical distributions where Mo is the peak in the distributionA mode may not existAn existing mode may not be uniqueExcel function: MODE(<data>)oM~11Percentiles For a population distribution:•If z is the 20th percentile, then20% of the values are  z21~nxx2/)(~122nnxxxx~x~~~12Special PercentilesFirst/Lower Decile = 10th percentileNinth/Upper Decile = 90th percentileQuartiles•Q1 = First/Lower Quartile = 25th percentile•Q2 = Second/Middle Quartile = 50th percentile•Q3 = Third/Upper Quartile = 75th percentileMedian = Q2 = 50th percentileExcel function: PERCENTILE(<data>, f ) where 0 < f < 113SkewnessIf a distribution is not symmetric, then it is skewedmode < median < meantendency for outliers to rightRight-skewedmean < median < modetendency for outliers to left Left-skewedArea0.50Area0.5014Measures of Variability(or Dispersion)Represent the variability of the dataEstimate population variability•Range of the data values•Variance•Standard Deviation•Coefficient of Variation15Grouped data with equal means and different dispersions16Distributions with different variability about the mean17Sample RangeThe sample range is the difference between the maximum and minimum valuesR = xmax  xmin = x(n)  x(1)Excel functions•MAX(<data>)•MIN(<data>)IQR = Interquartile Range = Q3 – Q118Variance population variance sample variance•Average of squared deviations from the sample mean:2:2sniixxns122)(1119VarianceBy simple algebraic manipulation, the above formulas can be expressed as shown belowExcel function: VAR(<data>)][/11222)(  nxxnsii20Variance: ExampleData: –2, 0, 3, 4, 7, 7, 9  = (–2 )2 + 02 + 32 + 42 + 72 + 72 + 92 = 2082ix4,28 xxi1617)49()40()42(2222s16177)28(20822s21Standard Deviation (s.d.) : population standard deviations : sample standard deviation•The standard deviation is simply the square root of the variance•It has the same unit as the original variable unit and so is easier to interpret than varianceExcel function: STDEV(<data>) = SQRT(VAR(<data>))22Coefficient of VariationC.V. = orC.V. is unitlessC.V. is useful for comparing variations between samples (or populations) which have markedly different magnitudes or different units of measurementsxs23Coefficient of Variation: Example 1Weights in a sample of men have = 154 lb and s = 26 lb; weights in a sample of kindergarten children have = 56 lb and s = 20 lb. Which group has greater variation in weight ?Solution: Use C.V. not s.d.C.V. (men) = C.V. (children) =Therefore children have greater variation in weight17.015426lblb21.05620lblbxx24Tabular SummariesGroup data into distinct classes•Determine # of classes (Avoid < 5 or > 12)•Determine class range (C.R.)•Round off class limits - increments in multiples of 10’s, 5’s, etc. are bestclassesof#dataofRangeC.R. 25Frequency TableTally data by classCount the frequency of values in each classRelative Frequency = Cumulative Relative Frequency = See Example on handout: Oil Tanker datanCountnclasseslower and class in thisCount26Graphical MethodsHistogram: Bar chart of frequency vs. classEmpirical Cumulative Distribution: Bar chart of cum. relative freq. vs. classOgive: Line chart of cumulative relative frequency vs. classBox and Whisker plot•Draw box around 1st and 3rd