1. **Problem Statement:** We are given a set of data with grouped frequency distribution and some calculations for variance and standard deviation. We need to verify the calculations and correct any mistakes using ungrouped data. 2. **Given Data:** - Age intervals: 6-14, 15-23, 24-32, 33-41 - Frequencies (C1): 11, 1, 11, 1 - Relative frequencies (Fin): 0.3, 0.46, 0.3, 0.16 - Sum of squared deviations (S²) = 859 - Standard deviation S = \sqrt{859} = 29.3 (claimed) 3. **Check the variance and standard deviation calculations:** - The formula for variance is $$S^2 = \frac{\sum (x - \bar{x})^2}{n-1}$$ where $\bar{x}$ is the mean and $n$ is the number of data points. - The user shows $$\sum (x - 25)^2 = 859$$ and calculates $$s = \sqrt{\frac{859}{6-1}} = \sqrt{171.8} = 13$$ which is correct for sample standard deviation. 4. **Check the individual squared deviations:** - The values given are: - $(17 - 25)^2 = 64$ (not 324) - $(14 - 25)^2 = 121$ - $(21 - 25)^2 = 16$ - $(25 - 25)^2 = 0$ - $(20 - 25)^2 = 25$ - $(23 - 25)^2 = 4$ - $(25 - 25)^2 = 0$ - $(34 - 25)^2 = 81$ (not 9) - $(38 - 25)^2 = 169$ - The user made mistakes in calculating $(17-25)^2$ and $(34-25)^2$. 5. **Sum of squared deviations corrected:** - Sum = 64 + 121 + 16 + 0 + 25 + 4 + 0 + 81 + 169 = 480 6. **Calculate corrected variance and standard deviation:** - Number of data points $n=9$ - Variance $$s^2 = \frac{480}{9-1} = \frac{480}{8} = 60$$ - Standard deviation $$s = \sqrt{60} \approx 7.75$$ 7. **Conclusion:** - The original sum of squared deviations 859 is incorrect due to calculation errors. - Correct sum is 480, leading to a standard deviation of approximately 7.75, not 13 or 29.3. **Taglish Explanation:** May mali sa pagcompute ng squared differences sa data mo, lalo na sa $(17-25)^2$ at $(34-25)^2$. Dapat 64 at 81 ang values, hindi 324 at 9. Kapag inayos ito, ang total squared deviation ay 480 lang, kaya ang tamang standard deviation ay mga 7.75 lang, hindi 13 o 29.3. Importante na tama ang bawat step para accurate ang resulta.