1. **Stating the problem:** We have grouped data representing grades in intervals with frequencies: - 0 people: 50% or below - 1 person: 50%-60% - 4 people: 60%-70% - 11 people: 70%-80% - 27 people: 80%-90% - 7 people: 90% or above We need to find the mean, median, mode, range, interquartile range (IQR), and standard deviation of this data. 2. **Assign midpoints to each interval:** - 50% or below: midpoint = 50 (assuming 50 as max for this group) - 50%-60%: midpoint = \frac{50+60}{2} = 55 - 60%-70%: midpoint = 65 - 70%-80%: midpoint = 75 - 80%-90%: midpoint = 85 - 90% or above: midpoint = 95 (assuming 100 max, midpoint = \frac{90+100}{2} = 95) 3. **Calculate total number of people:** $$N = 0 + 1 + 4 + 11 + 27 + 7 = 50$$ 4. **Calculate the mean:** $$\text{Mean} = \frac{\sum (f \times x)}{N}$$ where $f$ is frequency and $x$ is midpoint. Calculate sum of $f \times x$: $$0 \times 50 + 1 \times 55 + 4 \times 65 + 11 \times 75 + 27 \times 85 + 7 \times 95 = 0 + 55 + 260 + 825 + 2295 + 665 = 4100$$ So, $$\text{Mean} = \frac{4100}{50} = 82$$ 5. **Calculate the median:** Median position = $\frac{N+1}{2} = \frac{50+1}{2} = 25.5$th person. Cumulative frequencies: - 50% or below: 0 - 50%-60%: 1 - 60%-70%: 5 (1+4) - 70%-80%: 16 (5+11) - 80%-90%: 43 (16+27) - 90% or above: 50 (43+7) The 25.5th person lies in the 80%-90% group. Median = midpoint of 80%-90% = 85 6. **Calculate the mode:** Mode is the class with highest frequency: 80%-90% with 27 people. Mode = midpoint = 85 7. **Calculate the range:** Range = max value - min value = 100 - 50 = 50 8. **Calculate the interquartile range (IQR):** - Q1 position = $\frac{N+1}{4} = \frac{51}{4} = 12.75$th person - Q3 position = $3 \times \frac{N+1}{4} = 38.25$th person Locate Q1: - 60%-70% cumulative: 5 - 70%-80% cumulative: 16 Q1 lies in 70%-80% group. Locate Q3: - 80%-90% cumulative: 43 Q3 lies in 80%-90% group. Approximate Q1 = 75 (midpoint of 70%-80%) Approximate Q3 = 85 (midpoint of 80%-90%) IQR = Q3 - Q1 = 85 - 75 = 10 9. **Calculate the standard deviation:** Use formula: $$\sigma = \sqrt{\frac{\sum f(x - \mu)^2}{N}}$$ where $\mu = 82$ (mean). Calculate each term: - For 50: $(50 - 82)^2 = 1024$, frequency 0, contribution = 0 - For 55: $(55 - 82)^2 = 729$, frequency 1, contribution = 729 - For 65: $(65 - 82)^2 = 289$, frequency 4, contribution = 1156 - For 75: $(75 - 82)^2 = 49$, frequency 11, contribution = 539 - For 85: $(85 - 82)^2 = 9$, frequency 27, contribution = 243 - For 95: $(95 - 82)^2 = 169$, frequency 7, contribution = 1183 Sum contributions: $$0 + 729 + 1156 + 539 + 243 + 1183 = 3850$$ Standard deviation: $$\sigma = \sqrt{\frac{3850}{50}} = \sqrt{77} \approx 8.775$$ **Final answers:** - Mean = 82 - Median = 85 - Mode = 85 - Range = 50 - Interquartile Range = 10 - Standard Deviation $\approx$ 8.775