How do I solve for mean and standard deviation using two percentiles?

Question

Consider a normal distribution curve where 60-th percentile is at 17 and the 5-th percentile is at 9. Use this information to find the mean,  , and the standard deviation,  , of the distribution.

Answer 1

use standardized z equation z = (x - mean)/ SD

z for 60th percentile is 0.253

z for 5th percentile is -1.645

so for 60th percentile, x is 17: 0.253 = (17 - mean)/SD

so for 5th percentile, x is 9: -1.645 = (9 - mean)/SD

two equations 2 unknowns:

0.253*SD + mean = 17

-1.645*SD + mean = 9

difference of two equations:

1.898*SD = 8

SD = 4.215

substitute SD = 4.215 into 1st equation:

0.253 * 4.215 + mean = 17

mean = 15.93