8. Measures of relative location and detecting outliers The Bureau of Transportation Statistics (BTS) collects, analyzes, and disseminates information on U.S. transportation systems, including data on airline on-time performance. Consider departure-time data for all flights of an unspecified major airline out of New York City's John F. Kennedy Airport from December 1 through 7, 2007. These data, obtained from BTS, can be viewed in the following DataView tool.Data Set Departure Departure Delay in minutes Variable Departure Delay Location Full Data Quartiles Full Data Departure Delay Minimum -10 Q1 -3.5 Quantitative Median 0.0 Median 0.0 Observations = 248 Mean 23.7 Q3 23.5 Missing = 0 Maximum 397 Values = 248 Variability & Shape Deciles Statistics Values 248 10% -6.0 Range 407 20% -5.0 Interquartile Range 27.0 30% .2.0 Histogram > Standard Deviation 54.6 40% -1.0 Coefficient of Variation 230% Median 0.0 Skewness 3.38 60% 5.0 Box Plots 70% 17.0 Filter: Include observations between... 80% 34.0 Minimum 90% 93.0 Variable Maximum Go Correlation CorrelationFor each ight in the sample, data was collected on two variables: its date and its departure delay (in minutes), which is computed as the difference between the actual and scheduled departure times. A negative value for the departure delay means that the ight departed early. Use the DataView tool to obtain the mean and standard deviation of the departure delays. The mean departure delay is 23.7 V minutes. The standard deviation of the departure delays is 54.6 7 minutes. (Hint: Click one of the Variable sliding panels on the left side of the tool screen, and select the variable named Departure Delay. Then click on the Statistics button. You will see a screen showing different statistics calculated for the variable.) Observation 140 in the data set shows a flight that was scheduled to leave at 6:45 PM but was delayed. The z-score for its departure delay is 3.6 V , which means that the departure delay is V standard deviations away from the mean. The departure delay for this observation can be considered an outlier, because it is V than 3 standard deviations away from the mean. Observation (with Departure Times) Date Departure Delay minutes V 139 (Scheduled: 0/:50; Actual: 0/:43) ZUV/-12-V4 - / 140 (Scheduled: 18:45; Actual: 22:23) 2007-12-04 218 141 (Scheduled: 08:50; Actual: 09:17) 2007-12-05 27 142 (Scheduled: 12:00; Actual: 11:56) 2007-12-05 143 (Scheduled: 11:00; Actual: 10:53) 2007-12-05 144 (Scheduled: 19:55; Actual: 20:19) 2007-12-05 BABYA 145 (Scheduled: 11:00; Actual: 10:56) 2007-12-05 146 (Scheduled: 19:00; Actual: 19:29) 2007-12-05 147 (Scheduled: 07:45; Actual: 07:40) 2007-12-05 -5 148 (Scheduled: 07:00; Actual: 06:57) 2007-12-05 149 (Scheduled: 15:30; Actual: 17:37) 2007-12-05 127 2007-12-05 174For any set of data, Chebyshev's theorem tells you that at least V of the data values must lie within 1.70 standard deviations of the mean. Use the DataView tool to determine the proportion of data values within 1.70 standard deviations of the mean. (Hint: In the tool, click the Variable panel for Departure Delay. Under the heading lter: Include observations between ..., enter the appropriate values for the fields Minimum and Maximum to lter out observations that are more than 1.70 standard deviations from the mean. Remember to erase these values when you want to use the tool with unfiltered data.) For the departure delays in this data set, V of the values lie within 1.70 standard deviations of the mean. This is V the proportion specied by Chebyshev's theorem. When a data set has a symmetrical mound-shaped or bell-shaped distribution, the Empirical Rule tells you that V of the data values will be within 1 standard deviation of the mean and V will be within 2 standard deviations of the mean. Use the DataView tool to examine the shape of the distribution of departure delays. The distribution of the departure delays is V . Therefore, the Empirical Rule V hold for this distribution. \f54.6 52.3 54.6 50.0 56.25.2 3.6 6.8 3.2\fDeparture Delay in minutes Location Filtered Full Data Quartiles Filtered Full Data Minimum 2 -10 Q1 12.0 -3.5 Median 27.0 0.0 Median 27.0 0.0 Mean 55.9 23.7 Q3 82.0 23.5 Maximum 397 397 Variability & Shape Deciles Values 113 248 10% 5.0 -6.0 Range 395 407 20% 9.0 -5.0 Interquartile Range 70.0 27.0 30% 14.0 -2.0 Standard Deviation 68.2 54.6 40% 19.0 -1.0 Coefficient of Variation 122% 230% Median 27.0 0.0 Skewness 2.40 3.38 60% 37.0 5.0 70% 68.0 17.0 Filter: Include observations between... 80% 99.0 34.0 Minimum 90% 126.0 93.0 Maximum Go Filter: Departure Delay > 2\fStandard deviations Ifof approximately 68% at least 95% he S approximately 95% at least 68% -S EData Set Departure Frequency Variable Departure Delay 120 Departure Delay 100 Quantitative 80 Observations = 248 Missing = 0 60 Values = 248 40 Statistics 20 Histogram 0 0 to 49 50 to 99 -50 to -1 350 to 399 300 to 349 150 to 199 100 to 149 200 to 249 250 to 299 Box Plots > Variable Correlation Departure Delay in minutes Correlation