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P Values

Below are examples of how administrators and coaches can make decisions using running times.



Suppose an athlete obtains the above results in 1500m races.

The average or mean is calculated by adding together the race results and dividing that sum by the # of races. Thus, {225 + 223 + 221 + 215} / 4 = 221.

Unbiased Variance

Calculate the unbiased variance by subtracting each race result from the mean, squaring the answer to each subtraction, summing those squares, and dividing that sum by 1 less the number of races. Thus, {Square(225-221) + Square(223-221) + Square(221-221) + Square(215-221)} / [4-1] = 18.67

Below is a calculator that squares numbers

Time Squared
Time subtract mean in seconds: to the power of 2= Answer:

Standard Deviation

The standard deviation is the square root of the unbiased variance. Thus, square root of 18.67 = 4.32. Try it on the calculator below.


Z Score & P Values

Z Scores are calculated by subtracting each race result from the mean and dividing that answer by the standard deviation. Each Z score corresponds to a P Value on an online mathematical table

Race A: Z Score = [225-221] / 4.32 = .925926 P Value = .3545, not significant

Race B: Z Score = [223-221] / 4.32 = .462963 P Value = 6434, not significant

Race C: Z Score = [221-221] / 4.32 = 0.00 P Value = 1.00, not significant

Race D: Z Score = [215-221] / 4.32 = -1.388889 P Value = .1649, not significant

P Values can for example help a drug tester decide whether or not to target an athlete for banned substance use testing. Since in the above case the athletes times are not significant, perhaps the tester spends his budget and time elsewhere.

P Values can also assist a coach deciding whether or not to coach an athlete or a NCAA university deciding who to hand scholarships to. Since the above athlete's times are not significant, perhaps the coach and college know the athlete doesn't improve quickly.

P Values can be a factor in making decisions but not the only piece of evidence.

When calculating p values, only use timing from the previous season since times from previous season maybe off base given the fact seasonn to season an athklete may get quicker or slower due to age, injury, or other factors.

Also, if an athlete competes at multiple distances in a season, calculate p values for each of those distances. For example, if an athlete competes in the 100, 200, & 4 x 100 meters relay, calculate p values for each of these distances. If an athlete has competed in ore than one position in the 4 x 100 meters, calculate p values for each position since the number of meters sprinted maybe different at each position.

Also, race timing is done with electronic timers, called fully automated timing (F.A.T.), which counts to 1 / 1000 seconds but usually only 1 / 100 seconds is recorded. If an athlete sprints 100 meters in 10.023 seconds, the athlete's finishing time is rounded down to 10.02 seconds. If an athlete sprints the 100 meters in 10.026 seconds, the athlete's finishing time, since the third digit passed the decimal is abnove 5, is rounded up to 10.03 seconds.

Above, I have chosen finishing times like 215.000 seconds. In calculating p values, I would use times to the 1 / 100 or 1 / 1000 seconds, depending on the information I have.


Median means 50% of the statistics in the sample are above the median and 50% of the statistics in sample are below the median. In our sample, 222 is the number below 225 & 223 and above 221 & 215.


Mode is the statistic occurring the most on a sample. The race results 225, 223, 221, & 215 don't have a mode. However, if the athlete ran 221 in a 5th race, then 221 occurs the most times and would be the mode.

MS Excel

I calculated standard deviations, Z Scores, and P-Values, by using online calculators specifically designed to find these statistics. MS Excel calculates means and standard deviations only. Yet, businesses use MS Excel even though it is inefficient and leaves out all that needs to be counted.


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