1. **State the problem:** We are given a set of numbers and asked to find the mean, median, mode, range, and interquartile range (IQR). 2. **List the data:** 80, 82, 72, 90, 100, 95, 88, 75, 76, 85, 96, 85, 85, 78, 92 3. **Mean:** The mean is the sum of all numbers divided by the count of numbers. $$\text{Mean} = \frac{80 + 82 + 72 + 90 + 100 + 95 + 88 + 75 + 76 + 85 + 96 + 85 + 85 + 78 + 92}{15}$$ Calculate the sum: $$80 + 82 + 72 + 90 + 100 + 95 + 88 + 75 + 76 + 85 + 96 + 85 + 85 + 78 + 92 = 1295$$ Divide by 15: $$\text{Mean} = \frac{1295}{15}$$ Show cancellation: $$\text{Mean} = \frac{\cancel{1295}}{\cancel{15}} = 86.333...$$ So, the mean is approximately $86.33$. 4. **Median:** The median is the middle value when the data is ordered. Order the data: $$72, 75, 76, 78, 80, 82, 85, 85, 85, 88, 90, 92, 95, 96, 100$$ Since there are 15 numbers, the median is the 8th number: $$\text{Median} = 85$$ 5. **Mode:** The mode is the number that appears most frequently. From the ordered list, $85$ appears 3 times, more than any other number. $$\text{Mode} = 85$$ 6. **Range:** The range is the difference between the maximum and minimum values. $$\text{Range} = 100 - 72 = 28$$ 7. **Interquartile Range (IQR):** The IQR is the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$). - $Q_1$ is the median of the lower half (first 7 numbers): $$72, 75, 76, 78, 80, 82, 85$$ Median of lower half is the 4th number: $$Q_1 = 78$$ - $Q_3$ is the median of the upper half (last 7 numbers): $$85, 88, 90, 92, 95, 96, 100$$ Median of upper half is the 4th number: $$Q_3 = 92$$ Calculate IQR: $$\text{IQR} = Q_3 - Q_1 = 92 - 78 = 14$$ **Final answers:** - Mean: $86.33$ - Median: $85$ - Mode: $85$ - Range: $28$ - Interquartile Range: $14$