In a data the number i is repeated i times for i = 1,2, ……, n. then the mean of the data is

$\mathrm{i}=1, 2, 3, ....... \mathrm{n}$

i is repeated i times

$\begin{array}{rcl}\sum _{\mathrm{i}=1}^{\mathrm{n}} {\mathrm{i}}^2& =& \frac{\mathrm{n}(\mathrm{n}+1) (2\mathrm{n}+1)}{6}\\ \sum _{\mathrm{i}=1}^{\mathrm{n}} \mathrm{i}& =& \frac{\mathrm{n}(\mathrm{n}+1)}{2}\\ \mathrm{Mean}& =& \frac{\sum {\mathrm{i}}^2}{\sum \mathrm{i}}=\frac{{\displaystyle \frac{\mathrm{n}(\mathrm{n}+1) (2\mathrm{n}+1)}{6}}}{{\displaystyle \frac{\mathrm{n}(\mathrm{n}+1)}{2}}}\\ & =& \frac{2\mathrm{n}+1}{3}\end{array}$