Question

The mode of the following data is 3.286. Find the mean and median of the above data.

Hint:

You can verify these using the empirical relation: $Mode = 3(Median) - 2(Mean)$. $3(3.75) - 2(4.2) = 11.25 - 8.4 = 2.85$, which is close to the given mode of 3.286.

Answer 1

Step 1: Understanding the Concept:
Mean is the average ($\sum fx / \sum f$), and Median is the middle value. We use the data table provided.
Step 2: Key Formula or Approach:
1. Mean $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$
2. Median $= l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h$
Step 3: Detailed Explanation:
1. Find Mean:
- Midpoints ($x$): 2, 4, 6, 8, 10.
- $f$: 7, 8, 2, 2, 1. Total $N = 20$.
- $fx$: 14, 32, 12, 16, 10. $\sum fx = 84$.
- Mean $= 84 / 20 = 4.2$.
2. Find Median:
- $N/2 = 10$.
- Cumulative Frequencies: 7, 15, 17, 19, 20.
- Median Class is 3-5 (where $cf$ first exceeds 10).
- $l=3, cf=7, f=8, h=2$.
- Median $= 3 + \frac{10-7}{8} \times 2 = 3 + \frac{3}{4} = 3.75$.
Step 4: Final Answer:
Mean = 4.2, Median = 3.75.