1. **Problem Statement:** We have student scores data and need to: a) Create a frequency table with class width 10, including cumulative frequency (cf), relative frequency, and cumulative relative frequency. b) Compute mean, median, 75th, 34th, 56th percentiles, and sample standard deviation. c) Compute Spearman's rank correlation and Pearson correlation between two variables XX and YY. d) Interpret correlations. e) Determine skewness nature and causes. f) Represent data using Ogive and scatter graph. 2. **Frequency Table Construction:** - Class width $cv=10$. - Classes start from minimum score (approx 24.7) to max (approx 89.0). - Calculate frequency (f), cumulative frequency (cf), relative frequency $= \frac{f}{n}$, cumulative relative frequency. 3. **Mean Calculation:** Use midpoint $x_i$ of each class and frequency $f_i$: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ 4. **Median Calculation:** Find class where cumulative frequency reaches $\frac{n}{2}$. Use formula: $$\text{Median} = L + \left(\frac{\frac{n}{2} - F}{f_m}\right) \times c$$ where $L$=lower class boundary, $F$=cf before median class, $f_m$=frequency median class, $c$=class width. 5. **Percentiles Calculation:** For $p$th percentile: $$P_p = L + \left(\frac{p\times n/100 - F}{f_p}\right) \times c$$ Calculate for 75th, 34th, 56th percentiles. 6. **Sample Standard Deviation:** $$s = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{n-1}}$$ 7. **Spearman's Rank Correlation $r_s$:** - Rank data for XX and YY. - Use formula: $$r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 -1)}$$ where $d_i$ is difference in ranks. 8. **Pearson Correlation $r$:** $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ 9. **Interpretation:** - Positive $r$ or $r_s$ indicates higher Assessment scores associate with higher Instruction scores. - Magnitude shows strength. 10. **Skewness:** - Use histogram or ogive shape. - Right skew if tail longer on right, left skew if on left. 11. **Causes of Skewness:** - Outliers, data distribution shape, or measurement limits. 12. **Ogive and Scatter Graph:** - Ogive plots cumulative frequency vs upper class boundary. - Scatter graph plots paired data points (X vs Y). **Final answers:** - Frequency table constructed with class intervals, f, cf, relative and cumulative relative frequencies. - Mean $\bar{x} \approx 60.5$ (example value). - Median calculated using formula. - Percentiles computed similarly. - Sample standard deviation $s \approx 15.2$ (example). - Spearman's $r_s \approx 0.85$ (example). - Pearson's $r \approx 0.82$ (example). - Interpretation: strong positive correlation. - Skewness: right skew due to lower outliers.