1. **State the problem:** We have recovery times for 12 patients: $16, 9, 13, 6, 10, 8, 15, 7, 11, 5, 12, 4$ days. We will calculate the mean, deviations, variance, standard deviation, coefficient of variation, z-score, percentiles, median, and interquartile range. 2. **Calculate the mean recovery time ($\bar{x}$):** $$\bar{x} = \frac{16 + 9 + 13 + 6 + 10 + 8 + 15 + 7 + 11 + 5 + 12 + 4}{12} = \frac{116}{12} = 9.67$$ 3. **Calculate deviations $(x - \bar{x})$ and squared deviations $(x - \bar{x})^2$ for each $x$: ** | x | $x - \bar{x}$ | $(x - \bar{x})^2$ | |----|-------|-----------| | 16 | $16 - 9.67 = 6.33$ | $6.33^2 = 40.07$ | | 9 | $9 - 9.67 = -0.67$ | $(-0.67)^2 = 0.45$ | | 13 | $13 - 9.67 = 3.33$ | $3.33^2 = 11.09$ | | 6 | $6 - 9.67 = -3.67$ | $(-3.67)^2 = 13.47$ | | 10 | $10 - 9.67 = 0.33$ | $0.33^2 = 0.11$ | | 8 | $8 - 9.67 = -1.67$ | $(-1.67)^2 = 2.78$ | | 15 | $15 - 9.67 = 5.33$ | $5.33^2 = 28.39$ | | 7 | $7 - 9.67 = -2.67$ | $(-2.67)^2 = 7.11$ | | 11 | $11 - 9.67 = 1.33$ | $1.33^2 = 1.77$ | | 5 | $5 - 9.67 = -4.67$ | $(-4.67)^2 = 21.81$ | | 12 | $12 - 9.67 = 2.33$ | $2.33^2 = 5.43$ | | 4 | $4 - 9.67 = -5.67$ | $(-5.67)^2 = 32.13$ | 4. **Totals:** $$\sum (x) = 116$$ $$\sum(x - \bar{x}) = 0$$ (by definition of mean) $$\sum (x - \bar{x})^2 = 164.61$$ 5. **Calculate variance ($s^2$):** Using sample variance formula: $$s^2 = \frac{\sum (x - \bar{x})^2}{n-1} = \frac{164.61}{11} = 14.96$$ 6. **Calculate standard deviation ($s$):** $$s = \sqrt{14.96} = 3.87$$ 7. **Calculate coefficient of variation (CoV):** $$CoV = \frac{s}{\bar{x}} \times 100 = \frac{3.87}{9.67} \times 100 = 40.04\%$$ 8. **Calculate the z-score for $x=16$: ** $$z = \frac{16 - 9.67}{3.87} = \frac{6.33}{3.87} = 1.63$$ A z-score of 1.63 is not typically considered significantly high (usually above 2 or below -2). 9. **Calculate the percentile rank of value 10:** Values less than 10: 9,6,8,7,5,4 (6 values) $$\text{Percentile} = \frac{6}{12} \times 100 = 50^{th}\text{ percentile (approx.)}$$ 10. **Find quartiles:** Sort data: 4,5,6,7,8,9,10,11,12,13,15,16 - Q1 (25th percentile) is median of first 6 values: median of 4,5,6,7,8,9 Median between 6 and 7: $$Q1 = \frac{6 + 7}{2} = 6.5$$ - Q3 (75th percentile) is median of last 6 values: median of 10,11,12,13,15,16 Median between 12 and 13: $$Q3 = \frac{12 + 13}{2} = 12.5$$ 11. **Median (50th percentile):** Median is average of 6th and 7th values in sorted data (9 and 10): $$\text{Median} = \frac{9 + 10}{2} = 9.5$$ Yes, this is the median. 12. **Interquartile range (IQR):** $$IQR = Q3 - Q1 = 12.5 - 6.5 = 6$$ **Final table:** | x | $x - \bar{x}$ | $(x - \bar{x})^2$ | |----|---------|-----------| | 16 | 6.33 | 40.07 | | 9 | -0.67 | 0.45 | | 13 | 3.33 | 11.09 | | 6 | -3.67 | 13.47 | | 10 | 0.33 | 0.11 | | 8 | -1.67 | 2.78 | | 15 | 5.33 | 28.39 | | 7 | -2.67 | 7.11 | | 11 | 1.33 | 1.77 | | 5 | -4.67 | 21.81 | | 12 | 2.33 | 5.43 | | 4 | -5.67 | 32.13 | |Total|-0.00 (approx) | 164.61 | **Summary answers:** - Mean = 9.67 - Variance = 14.96 - Std deviation = 3.87 - CoV = 40.04% - Z-score of 16 = 1.63 (not significantly high) - Percentile of 10 ≈ 50th - Q1 = 6.5 - Q3 = 12.5 - Median = 9.5 - IQR = 6