How can you estimate the mean and standard deviation if your given two percentiles and their values

Question

The 16th percentile of a normal distributed variable has a value of 25 and the 97.5th percentile has a value of 40.

 

Answer 1

The easiest way to answer this question is to set up a system of equations using the definition of the z-score. The z-score is defined as:

z = (x - μ)/σ

Where x is the value, μ is the mean, and σ is the standard diviation.  For a normal distribution, the precentile can be calculated from the z-score.  These value of the percentile for each z-score can be found in various tables (see:http://www.regentsprep.org/Regents/math/algtrig/ATS7/ZChart.htm).  For the 15.9th percentile, the z-score is -1.0 and for the 97.7th precentile, the z-score is +2.0.  Thus, we have the following system of equations:

-1.0 = (25 - μ)/σ   =>   μ - σ = 25

+2.0 = (40 - μ)/σ   =>   μ + 2σ = 40

Solving this system of equations, one finds the mean to be 30 and the standard diviation to be 5.