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: Basic Idea:
The Mean represents the average value of the dataset, calculated using the formula $\sum fx / \sum f$.
The Median represents the central value of the data and is calculated using the grouped data median formula.

Step 2: Formula Used:
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: Complete Calculation:
(A) Calculation of Mean:
- Class midpoints ($x$): 2, 4, 6, 8, 10
- Corresponding frequencies ($f$): 7, 8, 2, 2, 1
- Total frequency $N = 7 + 8 + 2 + 2 + 1 = 20$
- Multiply $f$ and $x$ to get $fx$: 14, 32, 12, 16, 10
- $\sum fx = 84$
- Mean $= \frac{84}{20} = 4.2$

(B) Calculation of Median:
- First find $\frac{N}{2} = \frac{20}{2} = 10$
- Cumulative frequencies: 7, 15, 17, 19, 20
- The cumulative frequency just greater than 10 is 15, so the Median class is 3–5
- Here, $l = 3$, $cf = 7$, $f = 8$, $h = 2$
- Median $= 3 + \left( \frac{10 - 7}{8} \right) \times 2$
- Median $= 3 + \frac{3}{8} \times 2$
- Median $= 3 + \frac{6}{8} = 3.75$

Step 4: Final Result:
Mean = 4.2
Median = 3.75