1. **Stating the problem:** We are given a list of squared deviations from the mean, $(x - \bar{x})^2$, and we need to find their sum, which is the numerator in the standard deviation formula. 2. **List of values:** 6.25, 2.25, 72.25, 110.25, 156.25, 0.25, 992.25, 72.25, 182.25, 240.25, 132.25, 132.25, 12.25, 552.25, 342.25, 870.25, 650.25, 42.25, 2.25, 20.25, 182.25, 72.25 3. **Formula used:** The sum of squared deviations is $$\sum (x - \bar{x})^2 = 6.25 + 2.25 + 72.25 + 110.25 + 156.25 + 0.25 + 992.25 + 72.25 + 182.25 + 240.25 + 132.25 + 132.25 + 12.25 + 552.25 + 342.25 + 870.25 + 650.25 + 42.25 + 2.25 + 20.25 + 182.25 + 72.25$$ 4. **Calculation:** Adding all these values step-by-step, $$6.25 + 2.25 = 8.5$$ $$8.5 + 72.25 = 80.75$$ $$80.75 + 110.25 = 191$$ $$191 + 156.25 = 347.25$$ $$347.25 + 0.25 = 347.5$$ $$347.5 + 992.25 = 1339.75$$ $$1339.75 + 72.25 = 1412$$ $$1412 + 182.25 = 1594.25$$ $$1594.25 + 240.25 = 1834.5$$ $$1834.5 + 132.25 = 1966.75$$ $$1966.75 + 132.25 = 2099$$ $$2099 + 12.25 = 2111.25$$ $$2111.25 + 552.25 = 2663.5$$ $$2663.5 + 342.25 = 3005.75$$ $$3005.75 + 870.25 = 3876$$ $$3876 + 650.25 = 4526.25$$ $$4526.25 + 42.25 = 4568.5$$ $$4568.5 + 2.25 = 4570.75$$ $$4570.75 + 20.25 = 4591$$ $$4591 + 182.25 = 4773.25$$ $$4773.25 + 72.25 = 4845.5$$ 5. **Final answer:** The sum of the squared deviations is $$\boxed{4845.5}$$ This sum is used as the numerator in the standard deviation formula $\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$ where $n$ is the number of data points.