1. **Stating the problem:** We have values of $X = 13, 18, 23, 28, 33, 38, 43, 48$ and their corresponding frequencies $f = 1, 2, 5, 11, 12, 11, 5, 1$. We need to find: - $f x$ - $(x - \bar{x})$ - $f (x - \bar{x})^2$, $f (x - \bar{x})^3$, $f (x - \bar{x})^4$ - Mode 2. **Calculate the mean $\bar{x}$:** $$\bar{x} = \frac{\sum f x}{\sum f}$$ Calculate $\sum f x$: $$\sum f x = 1\times13 + 2\times18 + 5\times23 + 11\times28 + 12\times33 + 11\times38 + 5\times43 + 1\times48$$ $$= 13 + 36 + 115 + 308 + 396 + 418 + 215 + 48 = 1549$$ Calculate $\sum f$: $$\sum f = 1 + 2 + 5 + 11 + 12 + 11 + 5 + 1 = 48$$ So, $$\bar{x} = \frac{1549}{48} \approx 32.27$$ 3. **Calculate $f x$:** Already calculated as $\sum f x = 1549$. 4. **Calculate $(x - \bar{x})$ for each $x$:** $$13 - 32.27 = -19.27$$ $$18 - 32.27 = -14.27$$ $$23 - 32.27 = -9.27$$ $$28 - 32.27 = -4.27$$ $$33 - 32.27 = 0.73$$ $$38 - 32.27 = 5.73$$ $$43 - 32.27 = 10.73$$ $$48 - 32.27 = 15.73$$ 5. **Calculate $f (x - \bar{x})^2$, $f (x - \bar{x})^3$, $f (x - \bar{x})^4$:** Calculate powers and multiply by $f$: | $x$ | $f$ | $x - \bar{x}$ | $(x - \bar{x})^2$ | $f (x - \bar{x})^2$ | $(x - \bar{x})^3$ | $f (x - \bar{x})^3$ | $(x - \bar{x})^4$ | $f (x - \bar{x})^4$ | |---|---|---|---|---|---|---|---|---| | 13 | 1 | -19.27 | 371.33 | 371.33 | -7155.62 | -7155.62 | 137,889.5 | 137,889.5 | | 18 | 2 | -14.27 | 203.68 | 407.36 | -2907.5 | -5815.0 | 41,481.3 | 82,962.6 | | 23 | 5 | -9.27 | 85.94 | 429.7 | -797.1 | -3985.5 | 7,392.3 | 36,961.5 | | 28 | 11 | -4.27 | 18.23 | 200.53 | -77.8 | -855.8 | 332.0 | 3,652.0 | | 33 | 12 | 0.73 | 0.53 | 6.36 | 0.39 | 4.68 | 0.28 | 3.36 | | 38 | 11 | 5.73 | 32.83 | 361.13 | 188.2 | 2070.2 | 1,078.0 | 11,858.0 | | 43 | 5 | 10.73 | 115.15 | 575.75 | 1,235.5 | 6,177.5 | 13,255.0 | 66,275.0 | | 48 | 1 | 15.73 | 247.36 | 247.36 | 3,890.3 | 3,890.3 | 61,183.0 | 61,183.0 | Sum each column: $$\sum f (x - \bar{x})^2 = 2,599.22$$ $$\sum f (x - \bar{x})^3 = -1,588.9$$ $$\sum f (x - \bar{x})^4 = 310,785.0$$ 6. **Find the mode:** Mode is the $x$ value with the highest frequency. Here, $f$ max is 12 at $x = 33$. **Final answers:** - $\sum f x = 1549$ - $x - \bar{x}$ values as above - $\sum f (x - \bar{x})^2 = 2599.22$ - $\sum f (x - \bar{x})^3 = -1588.9$ - $\sum f (x - \bar{x})^4 = 310785.0$ - Mode = 33