[Math] How to calculate standard deviation in case of known x values, assuming a normal distribution

normal distributionstandard deviationstatistics

I have learned that 17% of the vehicles within a certain population drive less than 7. 500 km per year. On average, a given vehicle in that population will drive 13.300 km per year.

Assuming a normal distribution, how (mathematically, using a graphic calculator, or using Excel) can I calculate the standard deviation for the population at hand1?

Best Answer

Following @lulu's Comment, a 'grid search' for $\sigma$ would work, but there is a way to get an approximate solution by solving an explicit equation.

You know that $P(X < 7500) = 0.17$ Thus $$P\left(\frac{X-\mu}{\sigma} < \frac{7500-13300}{\sigma}\right) = 0.17$$ and using your calculator or printed tables you can find that $\frac{-5800}{\sigma} = -0.9541653$ and solve for $\sigma.$ I used R statistical software (where the inverse of the standard normal CDF is denoted qnorm) as follows:

qnorm(.17)
## -0.9541653

Addendum: If you suspect that $\sigma$ is between 1000 and 10,000, then here's how a grid search in R for $\sigma$, denoted sg, might work to give 6079.

sg = 1000:10000                      # list of integers from 1000 to 10000
pr=pnorm(7500, 13300, sg)            # P(X <= 7500) for X ~ NORM(13300, sg)
sg[abs(pr-.17) == min(abs(pr-.17))]  # search for sg with pr closest to .17
6079
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