Understanding Grouped Data Statistics
When dealing with large datasets, information is often organized into intervals (classes). To find the mean and standard deviation, we use the midpoint of each class ($x_i$) and its corresponding frequency ($f_i$).
Step 1: Create the Frequency Table
First, we calculate the midpoint $x_i$ for each class (average of the class boundaries) and compute necessary sums.
| Class | $f_i$ | $x_i$ | $f_i x_i$ | $f_i x_i^2$ |
|---|---|---|---|---|
| 5-10 | 2 | 7.5 | 15 | 112.5 |
| 10-15 | 9 | 12.5 | 112.5 | 1406.25 |
| 15-20 | 29 | 17.5 | 507.5 | 8881.25 |
| 20-25 | 54 | 22.5 | 1215 | 27337.5 |
| 25-30 | 11 | 27.5 | 302.5 | 8318.75 |
| 30-35 | 5 | 32.5 | 162.5 | 5281.25 |
| **Sum** | **110** | **2315** | **51337.5** |
Step 2: Calculating the Mean
The formula for the mean ($\bar{x}$) is: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{2315}{110} \approx 21.05$$
Step 3: Calculating Standard Deviation
The standard deviation ($\sigma$) is calculated using the formula: $$\sigma = \sqrt{\frac{\sum f_i x_i^2}{N} - (\bar{x})^2}$$
Where $N = \sum f_i = 110$.
- Calculate the mean of squares: $\frac{51337.5}{110} \approx 466.70$
- Square the mean: $(21.05)^2 \approx 443.10$
- Variance: $466.70 - 443.10 = 23.60$
- Standard Deviation: $\sqrt{23.60} \approx 4.86$
Thus, the mean is approximately 21.05 and the standard deviation is approximately 4.86.