175 200 225 250 275 300 325 350 375036912tonnageFrequency2812 123102175 200 225 250 275 300 325 350 375020406080100tonnageCumulative Percent5.025.055.085.092.595.0 95.0100.0Descriptive Statistics The following data set refers to the tonnage (in thousands of tons) for a sample of large oil tankers. These numbers were taken from the 1976 International Petroleum Encyclopedia: 229 232 239 232 259 361 220 260 231 229 249 254 257 214 237 253 274 230 223 253 195 269 231 268 189 290 218 313 220 270 277 374 222 290 231 258 227 269 220 224n = COUNT = 40 Measures of Location: Sample Mean = AVERAGE = 249.775 Geometric Mean = GEOMEAN = 247.2717 Sample Median = MEDIAN = 238 Sample Mode = MODE = 220 Σ xi = SUM = 9991 Measures of Variability: Sample Range = MAX−MIN = 374−189 = 185 Sample Variance = VAR = 1428.999 Sample Standard Deviation = STDEV = 37.802 Coefficient of Variation = xs/ = 0.151345Percentiles: 21st Percentile = PERCENTILE(tonnage,0.21) = 223.19 First Decile = 10th Percentile = PERCENTILE(tonnage,0.10) = 219.8 Q1 = First Quartile = 25th Percentile = PERCENTILE(tonnage,0.25) = 226.25 Q2 = Second Quartile = 50th Percentile = Sample Median = PERCENTILE(tonnage,0.50) = 238 Q3 = Third Quartile = 75th Percentile = PERCENTILE(tonnage,0.75) = 268.25 Tabular Summary: Class Interval Tally Count or Frequency Relative Frequency Cumulative Rel Freq 175 ≤ x < 200 2 0.050 0.050 200 ≤ x < 225 8 0.200 0.250 225 ≤ x < 250 12 0.300 0.550 250 ≤ x < 275 12 0.300 0.850 275 ≤ x < 300 3 0.075 0.925 300 ≤ x < 325 1 0.025 0.950 325 ≤ x < 350 0 0.000 0.950 350 ≤ x < 375 2 0.050 1.000 Histogram (Bar Chart): Empirical Cumulative Distribution Function:Pie Chart: Ogive (Line Chart): Boxplot: Q1 = 226.25 Q2 = 238 Q3 = 268.25 IQR = Q3 – Q1 = 42. Lower Whisker: Draw whisker down to smallest value > Q1 – 1.5*IQR Q1 – 1.5*IQR = 163.25 → draw down to obsn value 189 (smallest). Upper Whisker: Draw whisker up to largest value < Q3 + 1.5*IQR Q3 + 1.5*IQR = 331.25 → draw up to obsn value 313 (third largest). Outliers: 361 (obsn #6), 374 (obsn #32) Software for boxplots: SPSS (example shown) Minitab Statistica S-PLUS Matlab Statistics Toolbox [250, 275)30%[275, 300)8%[300, 325)3%[325, 350)0%[350, 375)5%[175, 200)5%[200, 225)20%[225,