Introduction
Understanding how to calculate mean, variance, and standard deviation is fundamental in statistics. These measures help summarize the distribution of data in terms of central tendency and spread. In this question, we are given a frequency distribution (presumably in tabular form) for the profits earned by 100 companies during 1987–88 and we have to calculate:
- a) Mean
- b) Variance
- c) Standard Deviation
Let’s walk through how to calculate each of these values using the assumed data since the actual table is not provided in the question. For explanation purposes, we will use an assumed data set with class intervals, midpoints, and frequencies.
Assumed Frequency Table
Let’s assume the profits and frequencies are as follows:
| Profit (Rs. Lakhs) | Midpoint (x) | Frequency (f) | fx | x² | fx² |
|---|---|---|---|---|---|
| 0–10 | 5 | 10 | 50 | 25 | 250 |
| 10–20 | 15 | 20 | 300 | 225 | 4500 |
| 20–30 | 25 | 30 | 750 | 625 | 18750 |
| 30–40 | 35 | 25 | 875 | 1225 | 30625 |
| 40–50 | 45 | 15 | 675 | 2025 | 30375 |
Total Frequency (Σf) = 10 + 20 + 30 + 25 + 15 = 100
Σfx = 50 + 300 + 750 + 875 + 675 = 2650
Σfx² = 250 + 4500 + 18750 + 30625 + 30375 = 84400
a) Mean
Mean (μ) = Σfx / Σf = 2650 / 100 = Rs. 26.5 Lakhs
b) Variance
Variance (σ²) is calculated as:
σ² = [Σfx² / Σf] – (Mean)²
σ² = (84400 / 100) – (26.5)² = 844 – 702.25 = 141.75
Variance = Rs. 141.75 Lakhs²
c) Standard Deviation
Standard Deviation (σ) = √Variance = √141.75 ≈ 11.9 Lakhs
Conclusion
From the above data and computations, we get:
- Mean Profit = Rs. 26.5 Lakhs
- Variance = Rs. 141.75 Lakhs²
- Standard Deviation ≈ Rs. 11.9 Lakhs
These measures help describe the profitability of companies in a statistical way. The mean gives us a central value, while variance and standard deviation describe the dispersion or variability of profits among the companies. A higher standard deviation indicates that the profits are more spread out from the average.