1. **Problem Statement:** Given the quiz scores: 78, 85, 90, 75, 85, 92, 95, 85, 80, 70, compute the mean, median, mode, range, variance, and standard deviation. 2. **Mean (Average):** The mean is calculated by summing all scores and dividing by the number of scores. $$\text{Mean} = \frac{78 + 85 + 90 + 75 + 85 + 92 + 95 + 85 + 80 + 70}{10}$$ $$= \frac{835}{10} = 83.5$$ 3. **Median:** Arrange the scores in ascending order: $$70, 75, 78, 80, 85, 85, 85, 90, 92, 95$$ Since there are 10 scores (even number), median is the average of the 5th and 6th scores: $$\text{Median} = \frac{85 + 85}{2} = 85$$ 4. **Mode:** The mode is the most frequent score. Here, 85 appears 3 times, more than any other score. $$\text{Mode} = 85$$ 5. **Range:** The range is the difference between the highest and lowest scores. $$\text{Range} = 95 - 70 = 25$$ 6. **Variance:** Variance measures the average squared deviation from the mean. First, calculate each deviation from the mean and square it: $$\begin{aligned} (78 - 83.5)^2 &= 30.25 \\ (85 - 83.5)^2 &= 2.25 \\ (90 - 83.5)^2 &= 42.25 \\ (75 - 83.5)^2 &= 72.25 \\ (85 - 83.5)^2 &= 2.25 \\ (92 - 83.5)^2 &= 72.25 \\ (95 - 83.5)^2 &= 132.25 \\ (85 - 83.5)^2 &= 2.25 \\ (80 - 83.5)^2 &= 12.25 \\ (70 - 83.5)^2 &= 182.25 \end{aligned}$$ Sum these squared deviations: $$30.25 + 2.25 + 42.25 + 72.25 + 2.25 + 72.25 + 132.25 + 2.25 + 12.25 + 182.25 = 548.5$$ Divide by number of scores (population variance): $$\text{Variance} = \frac{548.5}{10} = 54.85$$ 7. **Standard Deviation:** The standard deviation is the square root of the variance. $$\text{Standard Deviation} = \sqrt{54.85} \approx 7.41$$ **Final answers:** - Mean = 83.5 - Median = 85 - Mode = 85 - Range = 25 - Variance = 54.85 - Standard Deviation = 7.41 --- **Guide Questions:** 1. The mode and median (both 85) best represent the group's performance because they are less affected by extreme values than the mean. 2. If one student scored 100, the new mean would be: $$\frac{835 + 100}{11} = \frac{935}{11} \approx 85.0$$ This increases the mean, showing sensitivity to high scores. 3. The variance and standard deviation indicate moderate spread; data is somewhat consistent but with some variability.