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The local amusement park was interested in the average wait time at their most popular roller coaster at the peak park time (2 p.m.). they selected 13 patrons and had them get in line between 2 and 3 p.m. each was given a stopwatch to record the time they spent in line. the times recorded were as follows (in minutes; mean = 114.15): 118, 124, 108, 116, 99, 120, 148, 118, 119, 121, 45, 130, 118. what is the variance?

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Kalahira
515.05    

The variance of a set of numbers is simply the average of the squares of the difference of the mean of those same numbers. It's an important value in calculating the standard deviation of the same set of numbers. In fact, the standard deviation is nothing more than the square root of the variance. So let's calculate the variance:    

(118-114.15)^2 = 3.85^2 = 14.8225  

(124-114.15)^2 = 9.85^2 = 97.0225  

(108-114.15)^2 = -6.15^2 = 37.8225  

(116-114.15)^2 = 1.85^2 = 3.4225  

(99-114.15)^2 = -15.15^2 = 229.5225  

(120-114.15)^2 = 5.85^2 = 34.2225  

(148-114.15)^2 = 33.85^2 = 1145.8225  

(118-114.15)^2 = 3.85^2 = 14.8225  

(119-114.15)^2 = 4.85^2 = 23.5225  

(121-114.15)^2 = 6.85^2 = 46.9225  

(45-114.15)^2 = -69.15^2 = 4781.7225  

(130-114.15)^2 = 15.85^2 = 251.2225  

(118-114.15)^2 = 3.85^2 = 14.8225    

Now that we have all the squares of the difference from the mean, let's calculate the average which is simply the sum of all those squares divided by the number of squares we have which is 13. So:  

(14.8225+97.0225+37.8225+3.4225+229.5225+34.2225+1145.8225+14.8225+23.5225+46.9225+4781.7225+251.2225+14.8225)/13  

= 6695.6925/13  

= 515.05333    

Rounding to 2 decimal places gives 515.05

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