Calculate the sum of frequencies

The sum of all frequencies is $\sum f_i = 48 + X + 31 + 23 = 102 + X$.

$\sum f_i = 102 + X$

Calculate the sum of products

The sum of products of $x_i$ and $f_i$ is $\sum f_i x_i = (55 \times 48) + (60 \times X) + (70 \times 31) + (95 \times 23)$. Calculating these: $2640 + 60X + 2170 + 2185 = 6995 + 60X$.

$\sum f_i x_i = 6995 + 60X$

Apply mean formula

Using $\overline{x} = 61$, we have $61 = \frac{6995 + 60X}{102 + X}$. Multiply both sides by $(102 + X)$ to get $61(102 + X) = 6995 + 60X$.

$6222 + 61X = 6995 + 60X$

Solve for X

Isolate X: $61X - 60X = 6995 - 6222$, which gives $X = 773$.

$X = 773$