Consider the following grouped frequency distribution of a continuous variable x: 0–10 (frequency 8), 10–20 (frequency 12), 20–30 (frequency 10), 30–40 (frequency p), 40–50 (frequency 9). If the mean of this data is 25.2, what is the value of the missing frequency p?

Introduction / Context:
This question involves a grouped frequency distribution with one missing frequency. We are given class intervals and frequencies for all but one class, along with the mean of the distribution. The task is to find the missing frequency. This tests understanding of how the mean of grouped data is calculated using class midpoints and frequencies.

Given Data / Assumptions:

• Class intervals and frequencies are:
  • 0–10: frequency 8
  • 10–20: frequency 12
  • 20–30: frequency 10
  • 30–40: frequency p (unknown)
  • 40–50: frequency 9
• Mean of the distribution = 25.2.
• Class midpoints will be used for mean calculation.
• We must find the value of p.

Concept / Approach:
For grouped data, mean is calculated as sum(f * x) / sum(f), where x is the class midpoint and f is the frequency. We will compute the midpoints of each class, express the total frequency and the total of f * x in terms of p, then use the given mean to form an equation. Solving that equation will yield the missing frequency p. Careful arithmetic is essential, especially when handling decimals.

Step-by-Step Solution:
Step 1: Compute class midpoints. 0–10 midpoint = (0 + 10) / 2 = 5. 10–20 midpoint = 15. 20–30 midpoint = 25. 30–40 midpoint = 35. 40–50 midpoint = 45. Step 2: Write frequencies. f values: 8, 12, 10, p, 9. Total frequency = 8 + 12 + 10 + p + 9 = 39 + p. Step 3: Compute f * x products. For 0–10: 8 * 5 = 40. For 10–20: 12 * 15 = 180. For 20–30: 10 * 25 = 250. For 30–40: p * 35 = 35p. For 40–50: 9 * 45 = 405. Step 4: Sum f * x. Fixed part = 40 + 180 + 250 + 405 = 875. So sum(f * x) = 875 + 35p. Step 5: Use the mean formula. Mean = sum(f * x) / sum(f). 25.2 = (875 + 35p) / (39 + p). Step 6: Cross multiply to remove the fraction. 875 + 35p = 25.2 * (39 + p). Compute 25.2 * 39 = 982.8. Right side becomes 982.8 + 25.2p. Step 7: Solve for p. 875 + 35p = 982.8 + 25.2p. 35p − 25.2p = 982.8 − 875. 9.8p = 107.8. p = 107.8 / 9.8 = 11.
Verification / Alternative check:
With p = 11, total frequency = 39 + 11 = 50. Sum(f * x) = 875 + 35 * 11 = 875 + 385 = 1260. Mean = 1260 / 50 = 25.2, which matches the given value.
Why Other Options Are Wrong:
If p were 9, 10, 12 or 8, the sum(f * x) and total frequency would change, and the resulting mean would not equal 25.2. A quick check with any of these alternatives shows the mean deviating from 25.2.
Common Pitfalls:
A frequent error is to forget to include p in both the numerator and the denominator when writing the mean formula. Another common mistake is incorrect multiplication of 25.2 by 39 or incorrect handling of decimals when solving for p.
Final Answer:
The missing frequency p is 11.