1. **Problem Statement:** Calculate the Variance, Standard Deviation, and Coefficient of Variation for the given continuous grouped data using Direct, Shortcut, and Step Deviation methods. Given data: | Income Range | 35--39 | 40--44 | 45--49 | 50--54 | 55--59 | 60--64 | 65--69 | |--------------|---------|---------|---------|---------|---------|---------|---------| | Frequency (f) | 13 | 15 | 17 | 28 | 12 | 10 | 5 | 2. **Step 1: Calculate midpoints (x) of each class interval:** $$x = \frac{\text{Lower limit} + \text{Upper limit}}{2}$$ | Income Range | 35--39 | 40--44 | 45--49 | 50--54 | 55--59 | 60--64 | 65--69 | |--------------|---------|---------|---------|---------|---------|---------|---------| | Midpoint (x) | 37 | 42 | 47 | 52 | 57 | 62 | 67 | 3. **Step 2: Calculate total frequency (N):** $$N = 13 + 15 + 17 + 28 + 12 + 10 + 5 = 100$$ --- ### Direct Method 4. **Calculate mean ($\bar{x}$):** $$\bar{x} = \frac{\sum f x}{N} = \frac{13\times37 + 15\times42 + 17\times47 + 28\times52 + 12\times57 + 10\times62 + 5\times67}{100}$$ Calculate numerator: $$13\times37=481,\quad 15\times42=630,\quad 17\times47=799,\quad 28\times52=1456,\quad 12\times57=684,\quad 10\times62=620,\quad 5\times67=335$$ Sum: $$481 + 630 + 799 + 1456 + 684 + 620 + 335 = 5505$$ Mean: $$\bar{x} = \frac{5505}{100} = 55.05$$ 5. **Calculate variance ($\sigma^2$):** $$\sigma^2 = \frac{\sum f (x - \bar{x})^2}{N}$$ Calculate each $(x - \bar{x})^2$ and multiply by $f$: | x | $x - \bar{x}$ | $(x - \bar{x})^2$ | $f (x - \bar{x})^2$ | |----|--------------|------------------|--------------------| | 37 | 37 - 55.05 = -18.05 | 325.8 | 13 \times 325.8 = 4235.4 | | 42 | -13.05 | 170.3 | 15 \times 170.3 = 2544.5 | | 47 | -8.05 | 64.8 | 17 \times 64.8 = 1101.6 | | 52 | -3.05 | 9.3 | 28 \times 9.3 = 260.4 | | 57 | 1.95 | 3.8 | 12 \times 3.8 = 45.6 | | 62 | 6.95 | 48.3 | 10 \times 48.3 = 483.0 | | 67 | 11.95 | 142.8 | 5 \times 142.8 = 714.0 | Sum: $$4235.4 + 2544.5 + 1101.6 + 260.4 + 45.6 + 483.0 + 714.0 = 9384.5$$ Variance: $$\sigma^2 = \frac{9384.5}{100} = 93.845$$ 6. **Calculate standard deviation ($\sigma$):** $$\sigma = \sqrt{93.845} \approx 9.69$$ 7. **Calculate coefficient of variation (CV):** $$CV = \frac{\sigma}{\bar{x}} \times 100 = \frac{9.69}{55.05} \times 100 \approx 17.6\%$$ --- ### Shortcut Method 8. **Formula:** $$\sigma^2 = \frac{\sum f x^2}{N} - \bar{x}^2$$ Calculate $\sum f x^2$: $$37^2=1369, 42^2=1764, 47^2=2209, 52^2=2704, 57^2=3249, 62^2=3844, 67^2=4489$$ Multiply by frequencies: $$13\times1369=17797, 15\times1764=26460, 17\times2209=37553, 28\times2704=75712, 12\times3249=38988, 10\times3844=38440, 5\times4489=22445$$ Sum: $$17797 + 26460 + 37553 + 75712 + 38988 + 38440 + 22445 = 267395$$ Calculate variance: $$\sigma^2 = \frac{267395}{100} - (55.05)^2 = 2673.95 - 3030.5 = 93.845$$ Standard deviation and CV same as above. --- ### Step Deviation Method 9. **Choose assumed mean ($a$):** Choose $a = 52$ (midpoint of 50--54 class). Calculate class width $h = 5$. Calculate $u = \frac{x - a}{h}$: | x | $u = \frac{x - 52}{5}$ | |----|-----------------------| | 37 | \frac{37-52}{5} = -3 | | 42 | -2 | | 47 | -1 | | 52 | 0 | | 57 | 1 | | 62 | 2 | | 67 | 3 | Calculate $f u$ and $f u^2$: | f | u | $f u$ | $f u^2$ | |----|----|-------|---------| | 13 | -3 | -39 | 117 | | 15 | -2 | -30 | 60 | | 17 | -1 | -17 | 17 | | 28 | 0 | 0 | 0 | | 12 | 1 | 12 | 12 | | 10 | 2 | 20 | 40 | | 5 | 3 | 15 | 45 | Sum: $$\sum f u = -39 -30 -17 + 0 + 12 + 20 + 15 = -39$$ $$\sum f u^2 = 117 + 60 + 17 + 0 + 12 + 40 + 45 = 291$$ Calculate mean: $$\bar{x} = a + h \times \frac{\sum f u}{N} = 52 + 5 \times \frac{-39}{100} = 52 - 1.95 = 50.05$$ Calculate variance: $$\sigma^2 = h^2 \times \left( \frac{\sum f u^2}{N} - \left( \frac{\sum f u}{N} \right)^2 \right) = 25 \times \left( \frac{291}{100} - \left( \frac{-39}{100} \right)^2 \right)$$ Calculate inside parentheses: $$2.91 - 0.1521 = 2.7579$$ Variance: $$\sigma^2 = 25 \times 2.7579 = 68.9475$$ Calculate standard deviation: $$\sigma = \sqrt{68.9475} \approx 8.3$$ Calculate coefficient of variation: $$CV = \frac{8.3}{50.05} \times 100 \approx 16.58\%$$ --- **Final answers:** - Direct and Shortcut Method: - Mean = 55.05 - Variance = 93.845 - Standard Deviation = 9.69 - Coefficient of Variation = 17.6% - Step Deviation Method: - Mean = 50.05 - Variance = 68.95 - Standard Deviation = 8.3 - Coefficient of Variation = 16.58%