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Notes 6B – Measures of Variation Range – the difference between its highest and lowest data values: Range = highest value (max)– lowest value (min) Example 1 Misleading Range Consider the following two sets of quiz scores for nine students. Which set has the greater range? Would you also say that the scores in this set are more varied? Quiz 1: 1 10 10 10 10 10 10 10 10 Quiz 2: 2 3 4 5 6 7 8 9 10 Now try Exercises 15-18, part a Lower Quartile (or first quartile) – divides the lowest fourth of a data set from the upper three-fourths. It is the median of the data values in the lower half of a data set. Middle Quartile (or second quartile) – the overall median Upper Quartile (or third quartile) - divides the lowest three-fourths of a data set from the upper fourth. It is the median of the data values in the upper half of a data set. Example 2 Consider the following two sets of twenty 100-meter running times (in seconds): Set 1: 9.92 9.97 9.99 10.01 10.06 10.7 10.08 10.10 10.13 10.13 10.14 10.15 10.17 10.17 10.18 10.21 10.24 10.26 10.31 10.38 Set 2: 9.89 9.90 9.98 10.05 10.35 10.41 10.54 10.76 10.93 10.98 11.05 11.21 11.30 11.46 11.55 11.76 11.81 11.85 11.87 12.00 Compare the variation in the two data sets with five-number summaries and boxplots. First Set of Times: Second Set of Times: Low = Lower quartile = Median = Upper quartile = High = Low = Lower quartile = Median = Upper quartile = High = Now try Exercises 15-18, part b Standard deviation – measure of how far data values are spread around the mean of a data set. Example 3 Calculating Standard Deviation Calculate the standard deviations for the following set of data 4, 9, 11, 12, 17, 5, 8, 12, 14. x xx x x 2 Now try Exercises 15-18, part c Example 4 Using the Range Rule of Thumb Use the range rule of thumb to estimate the standard deviations for 4, 9, 11, 12, 17, 5, 8, 12, 14. Compare the estimates to the actual standard deviations found in example 3. Now try Exercises 15-18, part d Example 5 Estimating a Range Studies of the gas mileage of a Prius under varying driving conditions show that it gets a mean of 45 miles per gallon with a standard deviation for 4 miles per gallon. Estimate the minimum and maximum gas mileage that you can expect under ordinary driving conditions. Now try Exercises 15-18, part e Graphing Calculator Imagine customers waiting in line for tellers at two different banks. Customers at Big Bank can enter any one of three different lines leading to three different tellers. Best Bank also has three tellers, but all customers wait in a single line and are called to the next available teller. The following values are waiting times, in minutes, for eleven customers at each bank. The times are arranges in ascending order. Big Bank (three lines): 4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0 Best Bank (one line): 6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8 Compare the variation in the two data sets with five-number summaries and boxplots using the graphing calculator. Stat 1: Edit List data values under L1 (for Big Bank) and L2 for Best Bank) 2nd Y= 1: Plot 1 Window Graph Turn On xmin= Trace Big Bank Use Type xmax= and ymin= ymax= to move over the five-number summaries Min = Q1 = Med = Q3 = Max = Best Bank Min = Q1 = Med = Q3 = Max = Calculate the standard deviations for the waiting times at Big Bank and Best Bank using the graphing calculator. Put data values in L1 and L2 as you did for the boxplot Stat Calc1: 1-Var Stats Enter Enter Stat Calc1: 1-Var Stats Enter L2 Enter x SD Best Bank x SD Big Bank Big Bank Best Bank Mean x Sum of the data values x x x Standard Deviation Sx Number of data values n Sx n Use the rule of thumb to estimate the standard deviations for the waiting times at Big Bank and Best Bank.