Ron and Ara recorded the distances they covered while biking for 4 hours. The miles they biked on randomly selected days are as follows. Who has a more consistent biking record?
Ara: 39,35,49,32,45,39,33,38,40
Ron: 37,46,50,49,36,38,35,39,41
To determine who has a more consistent biking record, we need to look at the range of the distances they covered.
For Ara:
Range = highest value - lowest value = 49 - 32 = 17
For Ron:
Range = highest value - lowest value = 50 - 35 = 15
Therefore, Ara has a less consistent biking record compared to Ron.
To determine who has a more consistent biking record, we can calculate the standard deviation of the distances biked by Ara and Ron.
Standard deviation measures the amount of variation or dispersion in a set of values. Lower standard deviation indicates less variation and higher consistency.
For Ara:
1. Calculate the mean (average) of the distances:
Mean = (39 + 35 + 49 + 32 + 45 + 39 + 33 + 38 + 40) / 9 = 360 / 9 = 40
2. Calculate the deviations from the mean for each distance:
Deviations for Ar: (39-40), (35-40), (49-40), (32-40), (45-40), (39-40), (33-40), (38-40), (40-40) = -1,-5,9,-8,5,-1,-7,-2,0
3. Square each deviation:
Squares: (-1)^2, (-5)^2, 9^2, (-8)^2, 5^2, (-1)^2, (-7)^2, (-2)^2, 0^2 = 1,25,81,64,25,1,49,4,0
4. Calculate the average of the squared deviations (variance):
Variance = (1+25+81+64+25+1+49+4+0) / 9 = 250 / 9 = 27.78
5. Calculate the square root of the variance to get the standard deviation:
Standard Deviation = √(27.78) = 5.27
For Ron:
1. Calculate the mean (average) of the distances:
Mean = (37 + 46 + 50 + 49 + 36 + 38 + 35 + 39 + 41) / 9 = 361 / 9 = 40.11
2. Calculate the deviations from the mean for each distance:
Deviations for Ron: (37-40.11), (46-40.11), (50-40.11), (49-40.11), (36-40.11), (38-40.11), (35-40.11), (39-40.11), (41-40.11) = -3.11, 5.89, 9.89, 8.89, -4.11, -2.11, -5.11, -1.11, 0.89
3. Square each deviation:
Squares: (-3.11)^2, (5.89)^2, (9.89)^2, (8.89)^2, (-4.11)^2, (-2.11)^2, (-5.11)^2, (-1.11)^2, (0.89)^2 = 9.6721, 34.6921, 97.6121, 78.8521, 16.8921, 4.4521, 26.1121, 1.2321, 0.7921
4. Calculate the average of the squared deviations (variance):
Variance = (9.6721 + 34.6921 + 97.6121 + 78.8521 + 16.8921 + 4.4521 + 26.1121 + 1.2321 + 0.7921) / 9 = 270.3087 / 9 = 30.03
5. Calculate the square root of the variance to get the standard deviation:
Standard Deviation = √(30.03) = 5.48
Comparing the standard deviations, Ara has a lower standard deviation of 5.27, while Ron has a slightly higher standard deviation of 5.48. Therefore, Ara has a more consistent biking record as there is less variation in the distances she biked over the 4 hours.