1. **State the problem:** Find the mean, median, and standard deviation of the data set: 12, 15, 18, 20, 22, 25, 28. 2. **Mean (average):** The mean is calculated by summing all values and dividing by the number of values. $$\text{Mean} = \frac{12 + 15 + 18 + 20 + 22 + 25 + 28}{7}$$ Calculate the sum: $$12 + 15 + 18 + 20 + 22 + 25 + 28 = 140$$ So, $$\text{Mean} = \frac{140}{7}$$ Show cancellation: $$\frac{\cancel{140}}{\cancel{7}} = 20$$ Therefore, $$\text{Mean} = 20$$ 3. **Median:** The median is the middle value when data is ordered. The data is already ordered: 12, 15, 18, 20, 22, 25, 28. Since there are 7 values (odd number), the median is the 4th value. $$\text{Median} = 20$$ 4. **Standard deviation:** Use the formula for sample standard deviation: $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$$ Where $n=7$ and $\bar{x}=20$. Calculate each squared deviation: $$(12-20)^2 = (-8)^2 = 64$$ $$(15-20)^2 = (-5)^2 = 25$$ $$(18-20)^2 = (-2)^2 = 4$$ $$(20-20)^2 = 0^2 = 0$$ $$(22-20)^2 = 2^2 = 4$$ $$(25-20)^2 = 5^2 = 25$$ $$(28-20)^2 = 8^2 = 64$$ Sum these: $$64 + 25 + 4 + 0 + 4 + 25 + 64 = 186$$ Divide by $n-1=6$: $$\frac{186}{6}$$ Show cancellation: $$\frac{\cancel{186}}{\cancel{6}} = 31$$ Take the square root: $$s = \sqrt{31} \approx 5.57$$ **Final answers:** - Mean = 20 - Median = 20 - Standard deviation $\approx 5.57$