To solve for the missing frequency (denoted as $x$) based on the information provided, we need to use the formula for the calculation of the median:

$\text{Median} = L + \left( \frac{\frac{N}{2} - Cf}{f} \right) \times H$

Definitions:

• $Q_1$: The first quartile (provided as 51.75)
• $L$: The lower boundary of the class containing $Q_1$ (given as 52)
• $Cf$: Cumulative frequency before the $Q_1$ class (given as 8)
• $f$: Frequency of the class containing $Q_1$ (denoted as $x$)
• $N$: Total frequency (calculated as $4 \times (36 + x)$)
• $H$: Class width (4)

Step 1: Find $N$

Given $\frac{N}{4} = 36 + x$, the total frequency $N$ is:

$N = 4 \times (36 + x) = 144 + 4x$

Step 2: Use the median formula

Substitute the known values into the formula:

$51.75 = 52 + \left( \frac{\frac{N}{4} - Cf}{f} \right) \times H$

Substitute $N = 144 + 4x$, $Cf = 8$, $H = 4$, and $\frac{N}{4} = 36 + x$:

$51.75 = 52 + \left( \frac{(36 + x) - 8}{x} \right) \times 4$

Simplify:

$51.75 = 52 + \left( \frac{28 + x}{x} \right) \times 4$

Step 3: Solve for $x$

Rearrange the equation:

$51.75 - 52 = \left( \frac{28 + x}{x} \right) \times 4$

$-0.25 = \left( \frac{28 + x}{x} \right) \times 4$

$-0.25 \div 4 = \frac{28 + x}{x}$

$-0.0625 = \frac{28 + x}{x}$

Multiply through by $x$:

$-0.0625x = 28 + x$

Combine like terms:

$-0.0625x - x = 28$

$-1.0625x = 28$

Solve for $x$:

$x = \frac{28}{-1.0625}$

$x \approx -26.36$

Since a frequency cannot be negative, please check whether the given values or assumptions are correct. Let me know if you'd like me to recheck the calculations or interpret your data differently.

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Related Questions:

1. Can you confirm the exact value of $Q_1$ or if it's calculated differently?
2. Is the value of $L$ meant to be the lower class boundary or midpoint of the class?
3. Should $H$ represent the class width, or is it specific to a different scale?
4. Can you confirm whether $Cf = 8$ includes all classes before the target one?
5. Are there any additional constraints for the total frequency $N$?

Tip: Double-check the definition of each term to ensure the correct application of the formula for quartiles.