Sum Squared Differences 25239C

1. **State the problem:** Calculate the sum of squared differences from $0.90$ for the given values: $9.0, 8.0, 7.0, 2.0, 3.0, 6.0, 19.0, 15.0, 53.0, 4.0, 0.0, 0.0, -4.0, -33.0, -36.0, -9.0, -4.0, -13.0, -7.0, -2.0$. 2. **Formula used:** For each value $x_i$, compute $(x_i - 0.90)^2$ and sum all results: $$\text{Sum} = \sum_{i=1}^{20} (x_i - 0.90)^2$$ 3. **Calculate each squared difference:** - $(9.0 - 0.90)^2 = (8.1)^2 = 65.61$ - $(8.0 - 0.90)^2 = (7.1)^2 = 50.41$ - $(7.0 - 0.90)^2 = (6.1)^2 = 37.21$ - $(2.0 - 0.90)^2 = (1.1)^2 = 1.21$ - $(3.0 - 0.90)^2 = (2.1)^2 = 4.41$ - $(6.0 - 0.90)^2 = (5.1)^2 = 26.01$ - $(19.0 - 0.90)^2 = (18.1)^2 = 327.61$ - $(15.0 - 0.90)^2 = (14.1)^2 = 198.81$ - $(53.0 - 0.90)^2 = (52.1)^2 = 2714.41$ - $(4.0 - 0.90)^2 = (3.1)^2 = 9.61$ - $(0.0 - 0.90)^2 = (-0.9)^2 = 0.81$ - $(0.0 - 0.90)^2 = (-0.9)^2 = 0.81$ - $((-4.0) - 0.90)^2 = (-4.9)^2 = 24.01$ - $((-33.0) - 0.90)^2 = (-33.9)^2 = 1149.21$ - $((-36.0) - 0.90)^2 = (-36.9)^2 = 1361.61$ - $((-9.0) - 0.90)^2 = (-9.9)^2 = 98.01$ - $((-4.0) - 0.90)^2 = (-4.9)^2 = 24.01$ - $((-13.0) - 0.90)^2 = (-13.9)^2 = 193.21$ - $((-7.0) - 0.90)^2 = (-7.9)^2 = 62.41$ - $((-2.0) - 0.90)^2 = (-2.9)^2 = 8.41$ 4. **Sum all values:** $$65.61 + 50.41 + 37.21 + 1.21 + 4.41 + 26.01 + 327.61 + 198.81 + 2714.41 + 9.61 + 0.81 + 0.81 + 24.01 + 1149.21 + 1361.61 + 98.01 + 24.01 + 193.21 + 62.41 + 8.41 = 7156.95$$ 5. **Check the user's given sum:** The user states the sum is $635.78$ or $636$ cm, which is much smaller than our calculated $7156.95$. This suggests the user might have made a mistake or used a different reference value. 6. **Conclusion:** The correct sum of squared differences from $0.90$ for the given values is $7156.95$. **Final answer:** $$\sum_{i=1}^{20} (x_i - 0.90)^2 = 7156.95$$