# Estimate Mean and Standard Deviation from Percentiles

I have a derived dataset that specifies the percentiles from the 10th to 95th in increments of 5 along with the total number of data points. Is there a way to estimate the mean of the original dataset?

• sum the percentiles multiplied by $0.05$ and the first multiplied by $0.1$, i think that should work. – Manuel Jun 24 '14 at 22:17
• If the data below the 10th or above the 95th percentile can be arbitrary, your estimate may be arbitrarily bad. On the other hand, if the distribution is sufficiently nice, you might get a very good estimate. – Glen_b -Reinstate Monica Jun 25 '14 at 2:46

## 1 Answer

Yes you would weight each incremental Midpoint by its probability and sum up. The probability is 5% each time as given, the first increment has 10%.

This is equivalent to the definition of mean:

$E(X)=\sum_{i=1}^n x_i p_i$

For the variance you have:

$V(X)=\sum_{i=1}^n (x_i-E(X))^2p_i$

And standarddeviation $\sigma=\sqrt {V(X)}$.

• please notice that first incremental point should be multipplied by $\%10$ – Manuel Jun 24 '14 at 22:19
• yes, first increment has 10%, but the last one 5% (95th percentile).. – emcor Jun 24 '14 at 22:20