A cyclist won a bicycle race for seven consecutive years. His "winning" times and "victory" margins (time difference of the second place finisher) are given in the figure below.
-Year------Time--------Margin
----------(h:m:s)------ (m:s)
1999-----91:32:17-----7:38
2000-----92:33:01-----6:01
2001-----86:17:28-----6:44
2002-----82:05:13-----7:16
2003-----83:41:13-----1:07
2004-----83:36:04-----6:17
2005-----86:15:04-----4:30
(a) Find the standard deviation of the cyclist's times. (Round your answers to the nearest second.)
: : h:m:s
(b) Find the standard deviation of the cyclist's margins. (Round your answers to the nearest second.)
: m:s
I have no idea what to do
a)
You will have to find the average time, which involves adding the 7 times and then dividing by 7
Too bad that "we" have not gone metric with units of time, so you will have to convert each of the times to a decimal number in terms of hours, (unless you use the
D"M'S key on your calculator and add them up that way.)
e.g.
91:32:17 = 91 + 32/60 + 17/3600 hrs = 91.5381
do this for the other six times
Add them up then divide by 7
this will be your mean.
now form a column of differences of the times and the mean, now square each difference.
add up those squares of the differences.
Divide that sum by 7
Take the square root of that and you have your SD
you will still have to change your single decimal to
h:m:s
e.g. 2.578 hours
= 2 hours + .578 hours
= 2 hours + .578(60) minutes
= 2 hours + 34.68 minutes
= 2 hours + 34 minutes + .68(60) seconds
= 2 : 34 : 41
b)
repeat the entire for the margins
This is a very tedious process, work patiently.
here is a Khan Academy page for this procedure:
https://www.khanacademy.org/math/probability/data-distributions-a1/summarizing-spread-distributions/a/calculating-standard-deviation-step-by-step
To find the standard deviation of the cyclist's times and margins, we need to follow a few steps:
Step 1: Calculate the mean (average) of the times and margins.
Step 2: Calculate the squared difference between each value and the mean.
Step 3: Calculate the sum of the squared differences.
Step 4: Divide the sum by the number of values.
Step 5: Take the square root of the result from step 4 to find the standard deviation.
Let's start with part (a), finding the standard deviation of the cyclist's times:
Step 1: Calculate the mean of the times.
To find the mean, we add up all the times and divide by the number of times. In this case, there are 7 race times.
Mean time (in seconds) = (91*3600 + 32*60 + 17 + 92*3600 + 33*60 + 1 + 86*3600 + 17*60 + 28 + 82*3600 + 5*60 + 13 + 83*3600 + 41*60 + 13 + 83*3600 + 36*60 + 4 + 86*3600 + 15*60 + 4) / 7
= (329737 + 6600 + 17 + 331141 + 1980 + 1 + 310548 + 1020 + 28 + 295800 + 300 + 13 + 302493 + 2460 + 13 + 300964 + 360 + 4 + 310404 + 1500 + 4) / 7
= 2599858 / 7
= 371408 seconds
Step 2: Calculate the squared difference between each value and the mean.
Take each time and subtract the mean, then square the result.
For example, for the first year:
Squared difference for 1999 = (91:32:17 - 371408)^2
Do this for all the years.
Step 3: Calculate the sum of the squared differences.
Add up all the squared differences calculated in step 2.
Step 4: Divide the sum by the number of values.
Divide the sum from step 3 by the number of values, which is 7 in this case.
Step 5: Take the square root of the result from step 4.
Finally, take the square root of the result from step 4 to find the standard deviation.
Repeat the same process for part (b), finding the standard deviation of the cyclist's margins.
After following these steps, you should be able to calculate the standard deviation for both parts of the question.