Mathematics 10 — Exercise 11.2 · Q11

Full question

The table gives monthly sales forecasts in millions of rupees. Calculate their variance and standard deviation to describe their variability.

Step-by-step solution

1. Multiply the value (or class midpoint) by its frequency.

   $$1\times 15=15$$

2. Square that value.

   $${15}^2=225$$

3. Multiply the square by its frequency.

   $$1\times 225=225$$

4. Multiply the value (or class midpoint) by its frequency.

   $$1\times 18=18$$

5. Square that value.

   $${18}^2=324$$

6. Multiply the square by its frequency.

   $$1\times 324=324$$

7. Multiply the value (or class midpoint) by its frequency.

   $$1\times 14=14$$

8. Square that value.

   $${14}^2=196$$

9. Multiply the square by its frequency.

   $$1\times 196=196$$

10. Multiply the value (or class midpoint) by its frequency.

    $$1\times 20=20$$

11. Square that value.

    $${20}^2=400$$

12. Multiply the square by its frequency.

    $$1\times 400=400$$

13. Multiply the value (or class midpoint) by its frequency.

    $$1\times 13=13$$

14. Square that value.

    $${13}^2=169$$

15. Multiply the square by its frequency.

    $$1\times 169=169$$

16. Add the frequencies to find n.

    $$1+1+1+1+1=5$$

17. Add the f×x column.

    $$15+18+14+20+13=80$$

18. Add the f×x² column.

    $$225+324+196+400+169=1314$$

19. Mean = Σfx/Σf. Divide the first total by n.

    $$\frac{80}{5}$$

20. Calculate this operation before combining the remaining terms.

    $$80\times 1=80$$

21. Calculate this operation before combining the remaining terms.

    $$1\times 5=5$$

22. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{80\times 1}{1\times 5}$$

23. Calculate these two products separately.

    $$\frac{80}{5}$$

24. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

25. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{5\times 1}{1\times 5}$$

26. Calculate these two products separately.

    $$\frac{5}{5}$$

27. Show the two divisions explicitly.

    $$\frac{\frac{5}{5}}{\frac{5}{5}}$$

28. Write the reduced fraction.

    $$\frac{1}{1}$$

29. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{5}=1$$

30. Calculate this operation before combining the remaining terms.

    $$\frac{1}{1}=1$$

31. Calculate this operation before combining the remaining terms.

    $$16\times 1=16$$

32. Show the two divisions explicitly.

    $$\frac{\frac{80}{5}}{\frac{5}{5}}$$

33. Evaluate the numerator division.

    $$\frac{80}{5}=16$$

34. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

35. Write the reduced fraction.

    $$16\times \frac{1}{1}$$

36. Calculate this operation before combining the remaining terms.

    $$80\times \frac{1}{5}=16$$

37. Population variance = Σfx²/n − (mean)². Keep the exact mean before rounding.

    $$\frac{1314}{5}-{16}^2$$

38. Calculate this operation before combining the remaining terms.

    $$1314\times 1=1314$$

39. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1314\times 1}{1\times 5}$$

40. Calculate these two products separately.

    $$\frac{1314}{5}$$

41. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 5}$$

42. Calculate these two products separately.

    $$\frac{1}{5}$$

43. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{5}=\frac{1}{5}$$

44. Calculate this operation before combining the remaining terms.

    $$1314\times \frac{1}{5}=\frac{1314}{5}$$

45. Calculate this operation before combining the remaining terms.

    $${16}^2=256$$

46. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 256=-256$$

47. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=5$$

48. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1314}{5}=\frac{1314\times 1}{5\times 1}$$

49. Calculate the converted numerator; the denominator is now common.

    $$\frac{1314}{5}$$

50. Calculate this operation before combining the remaining terms.

    $$1\times 1=1$$

51. Calculate this operation before combining the remaining terms.

    $$\left(-256\right)\times 1=-256$$

52. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-256\times \frac{1}{1}=\frac{\left(-256\right)\times 5}{1\times 5}$$

53. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1280\right)\times 1}{1\times 5}$$

54. Show the two divisions explicitly.

    $$\frac{-\frac{1280}{5}}{\frac{5}{5}}$$

55. Evaluate the numerator division.

    $$-\frac{1280}{5}=-256$$

56. Write the reduced fraction.

    $$-256\times \frac{1}{1}$$

57. Calculate this operation before combining the remaining terms.

    $$\left(-1280\right)\times \frac{1}{5}=-256$$

58. Calculate the converted numerator; the denominator is now common.

    $$-\frac{1280}{5}$$

59. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 1280=-1280$$

60. Calculate this operation before combining the remaining terms.

    $$1314-1280=34$$

61. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{34\times 1}{1\times 5}$$

62. Calculate these two products separately.

    $$\frac{34}{5}$$

63. Calculate this operation before combining the remaining terms.

    $$34\times \frac{1}{5}=\frac{34}{5}$$

64. Keep the common denominator and add the signed numerators.

    $$\frac{1314-1280}{5}=\frac{34}{5}$$

65. Calculate this operation before combining the remaining terms.

    $$\frac{1314}{5}-256=\frac{34}{5}$$

66. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{34}{5}}=\frac{1}{5}\times \sqrt{170}$$

67. Standard deviation is the POSITIVE square root of the variance.

    $$\sqrt{\frac{34}{5}}$$

68. Round the standard deviation only now.

    $$2.60768$$

69. Range is maximum minus minimum.

    $$20-13=7$$

70. The standard deviation measures how far forecasts typically lie from their mean; these are forecasts, not forecast errors, so this does not measure accuracy.

Answer

mean=16; variance=34/5; standard deviation≈2.60768; range=7