1. **Problem Statement:** Given the dataset of student test scores: 45, 62, 75, 85, 90, 93, 95, 95, 100, find the range, 1st and 3rd quartiles, 25th and 50th percentiles, mean, median, mode, variance, and standard deviation. Also, comment on any equalities observed between quartiles and percentiles. 2. **Range:** The range is the difference between the maximum and minimum values. $$\text{Range} = 100 - 45 = 55$$ 3. **Quartiles and Percentiles:** - Sort the data (already sorted): 45, 62, 75, 85, 90, 93, 95, 95, 100 - Number of data points $n=9$ - 1st quartile ($Q_1$) is the 25th percentile, 3rd quartile ($Q_3$) is the 75th percentile. To find $Q_1$ (25th percentile): Position = $0.25 \times (n+1) = 0.25 \times 10 = 2.5$ Interpolate between 2nd and 3rd values: $$Q_1 = 62 + 0.5 \times (75 - 62) = 62 + 6.5 = 68.5$$ To find $Q_3$ (75th percentile): Position = $0.75 \times (n+1) = 0.75 \times 10 = 7.5$ Interpolate between 7th and 8th values: $$Q_3 = 95 + 0.5 \times (95 - 95) = 95$$ The 25th percentile is $68.5$ (same as $Q_1$), and the 50th percentile is the median. 4. **Median (50th percentile):** Since $n=9$ is odd, median is the middle value (5th value): $$\text{Median} = 90$$ 5. **Mode:** The most frequent value(s) in the dataset. Values 95 appears twice, others once. $$\text{Mode} = 95$$ 6. **Mean:** Sum all values and divide by $n$. $$\text{Mean} = \frac{45 + 62 + 75 + 85 + 90 + 93 + 95 + 95 + 100}{9} = \frac{740}{9} \approx 82.22$$ 7. **Variance and Standard Deviation:** Variance formula: $$\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$$ Calculate squared deviations: $(45-82.22)^2 = 1380.93$ $(62-82.22)^2 = 408.84$ $(75-82.22)^2 = 52.18$ $(85-82.22)^2 = 7.72$ $(90-82.22)^2 = 60.52$ $(93-82.22)^2 = 115.92$ $(95-82.22)^2 = 163.36$ $(95-82.22)^2 = 163.36$ $(100-82.22)^2 = 316.72$ Sum of squared deviations: $$1380.93 + 408.84 + 52.18 + 7.72 + 60.52 + 115.92 + 163.36 + 163.36 + 316.72 = 2669.55$$ Variance: $$\sigma^2 = \frac{2669.55}{9} \approx 296.62$$ Standard deviation: $$\sigma = \sqrt{296.62} \approx 17.22$$ 8. **Comments on Equalities:** - The 25th percentile equals the 1st quartile ($68.5$), which is expected since quartiles are specific percentiles. - The 50th percentile equals the median ($90$), confirming the median is the middle value. - The 3rd quartile ($95$) is the 75th percentile, showing quartiles correspond to specific percentiles. - These equalities occur because quartiles are defined as specific percentiles (25th, 50th, 75th), so their values align.