1. The problem is to find the standard deviation of the data set {12, 15, 16, 12}. 2. First, find the mean (average) of the data. $$\text{Mean} = \frac{12+15+16+12}{4} = \frac{55}{4} = 13.75$$ 3. Calculate the squared differences from the mean for each data point. - For 12: $$(12 - 13.75)^2 = (-1.75)^2 = 3.0625$$ - For 15: $$(15 - 13.75)^2 = (1.25)^2 = 1.5625$$ - For 16: $$(16 - 13.75)^2 = (2.25)^2 = 5.0625$$ - For 12: $$(12 - 13.75)^2 = (-1.75)^2 = 3.0625$$ 4. Find the variance by averaging these squared differences. $$\text{Variance} = \frac{3.0625 + 1.5625 + 5.0625 + 3.0625}{4} = \frac{12.75}{4} = 3.1875$$ 5. The standard deviation is the square root of the variance. $$\text{Standard Deviation} = \sqrt{3.1875} \approx 1.785$$ Therefore, the standard deviation of the data set is approximately $1.79$.