1. **Problem statement:** We have oxygen gas consumption data (100 values) from hospitals and need to analyze it through various statistical methods. 2. **Stem-and-leaf diagram and range:** - Stem represents tens, leaf represents units. - Sort data and group by tens: 20s: 24, 28, 29 30s: 31, 39 40s: 41, 42, 46, 46, 46, 48, 49 50s: 51, 53, 55, 55, 56, 57, 59 60s: 60, 60, 62, 62, 66, 66, 67, 67, 69 70s: 70, 71, 72, 72, 73, 74, 75, 75, 77, 78, 78, 78, 78, 79 80s: 80, 81, 82, 82, 83, 83, 83, 85, 85, 86, 86, 86, 87, 87, 88, 88, 88, 89 90s: 90, 91, 91, 92, 92, 94, 94, 95, 97, 98, 99, 99 100s: 100, 102, 103, 104, 104, 105, 105, 107, 107, 108, 109, 109, 110, 112, 112, 117, 120 120s: 125, 129 130s: 130, 132, 135, 137 190s: 197 - Range = max - min = 197 - 24 = 173 3. **Modal value and distribution comment:** - Modal value is the most frequent data point(s). - From data, 78 appears 4 times, 86 appears 3 times, 88 appears 3 times, 91 appears 2 times, 92 appears 2 times, 94 appears 2 times, 99 appears 2 times. - Mode = 78 (most frequent). - Distribution is right-skewed due to high outlier 197. 4. **Grouped frequency distribution table:** - Use class intervals of width 20 from 20 to 140+: | Class Interval | Frequency | | 20 - 39 | 5 | | 40 - 59 | 13 | | 60 - 79 | 24 | | 80 - 99 | 26 | | 100 - 119 | 17 | | 120 - 139 | 7 | | 140 - 159 | 0 | | 160 - 179 | 0 | | 180 - 199 | 1 | 5. **Histogram and frequency polygon:** - Histogram bars represent frequencies for each class interval. - Frequency polygon connects midpoints of each class interval with frequency points. 6. **Estimate mean and variance:** - Midpoints: 29.5, 49.5, 69.5, 89.5, 109.5, 129.5, 149.5, 169.5, 189.5 - Mean $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{5\times29.5 + 13\times49.5 + 24\times69.5 + 26\times89.5 + 17\times109.5 + 7\times129.5 + 0 + 0 + 1\times189.5}{100}$$ - Calculate numerator: $$5\times29.5=147.5, 13\times49.5=643.5, 24\times69.5=1668, 26\times89.5=2327, 17\times109.5=1861.5, 7\times129.5=906.5, 1\times189.5=189.5$$ - Sum = 147.5 + 643.5 + 1668 + 2327 + 1861.5 + 906.5 + 189.5 = 7743.5 - Mean $$\bar{x} = \frac{7743.5}{100} = 77.435$$ - Variance $$s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i - 1}$$ - Calculate each term: $$5(29.5 - 77.435)^2 = 5 \times 2296.5 = 11482.5$$ $$13(49.5 - 77.435)^2 = 13 \times 779.3 = 10130.9$$ $$24(69.5 - 77.435)^2 = 24 \times 62.9 = 1509.6$$ $$26(89.5 - 77.435)^2 = 26 \times 145.3 = 3777.8$$ $$17(109.5 - 77.435)^2 = 17 \times 1024.9 = 17423.3$$ $$7(129.5 - 77.435)^2 = 7 \times 2703.9 = 18927.3$$ $$1(189.5 - 77.435)^2 = 1 \times 12503.3 = 12503.3$$ - Sum = 11482.5 + 10130.9 + 1509.6 + 3777.8 + 17423.3 + 18927.3 + 12503.3 = 65754.7 - Variance $$s^2 = \frac{65754.7}{99} = 664.2$$ 7. **Coefficient of variation (CV):** - $$CV = \frac{s}{\bar{x}} \times 100 = \frac{\sqrt{664.2}}{77.435} \times 100 = \frac{25.77}{77.435} \times 100 = 33.3\%$$ Final answers: - Range = 173 - Mode = 78 - Mean = 77.435 - Variance = 664.2 - Coefficient of variation = 33.3%