Statistics and Probability: JEE Main 2023 Mathematics question

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-baqh{text-align:center;vertical-align:top} ${x_i}$ 0 1 2 3 4 5 ${f_i}$ $k + 2$ $2k$ ${k^2} - 1$ ${k^2} - 1$ ${k^2} + 1$ $k - 3$ where $\sum f_{i}=62$. If $[x]$ denotes the greatest integer $\leq x$, then $\left[\mu^{2}+\sigma^{2}\right]$ is equal to :