Q.55 Given data consists of distinct values of 𝑥𝑖 occurring with frequencies 𝑓𝑖. The mean
value for the data is _______. (rounded off to one decimal place)
Question Details
The question gives distinct data values xi with frequencies fi and asks for the mean value of the data, rounded to one decimal place. The table is as follows:
| xi | fi |
|---|---|
| 5 | 8 |
| 6 | 10 |
| 8 | 10 |
| 10 | 12 |
Formula
The mean of a discrete frequency distribution is calculated using the formula:
Mean (x̄) = Σ(fi·xi) / Σfi
Step‑by‑Step Solution
- Compute total frequency: Σfi = 8 + 10 + 10 + 12 = 40
- Compute the products (xi·fi):
- 5 × 8 = 40
- 6 × 10 = 60
- 8 × 10 = 80
- 10 × 12 = 120
- Sum of all (xi·fi): Σ(xi·fi) = 40 + 60 + 80 + 120 = 300
- Calculate the mean: x̄ = 300 / 40 = 7.5
- Rounding to one decimal place: 7.5 → 7.5
Therefore, the mean of the data is 7.5.
Concept Explanation
Each value of xi appears fi times in the dataset. The total of all observations is the weighted sum Σ(fi·xi). The mean is this total divided by the total number of observations Σfi, giving the central or average value of the distribution.
This reasoning applies to any similar IIT JAM or CSIR NET question on the mean of a discrete frequency distribution. Always form the products (xi·fi), sum them, sum the frequencies, and then apply the formula for the mean.
