Step 1: Set up the tabular products $f_i x_i$ and sums

Calculate the sum of frequencies $\sum f_i$ and the sum of products $\sum f_i x_i$ containing the unknown $X$.

$\sum f_i = 19 + X + 13 + 24 = X + 56$

Step 2: Calculate product sum $\sum f_i x_i$

Multiply each $x_i$ by its corresponding frequency $f_i$: $20 \times 19 = 380$, $30 \times X = 30X$, $40 \times 13 = 520$, $85 \times 24 = 2040$.

$\sum f_i x_i = 380 + 30X + 520 + 2040 = 30X + 2940$

Step 3: Apply the mean formula

Substitute the given mean $\overline{x} = 37$ into the mean formula and solve for $X$.

$37 = \frac{30X + 2940}{X + 56}$

Step 4: Solve the linear equation for $X$

Cross-multiply and solve: $37(X + 56) = 30X + 2940 \implies 37X + 2072 = 30X + 2940 \implies 7X = 868 \implies X = 124$.

$X = 124$