Mathematics 10 — Exercise 11.2 · Q7

Full question

Rainfall in mm with the frequencies shown: find variance and standard deviation.

Step-by-step solution

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

   $$5\times 42=210$$

2. Square that value.

   $${42}^2=1764$$

3. Multiply the square by its frequency.

   $$5\times 1764=8820$$

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

   $$3\times 51=153$$

5. Square that value.

   $${51}^2=2601$$

6. Multiply the square by its frequency.

   $$3\times 2601=7803$$

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

   $$4\times 54=216$$

8. Square that value.

   $${54}^2=2916$$

9. Multiply the square by its frequency.

   $$4\times 2916=11664$$

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

    $$9\times 61=549$$

11. Square that value.

    $${61}^2=3721$$

12. Multiply the square by its frequency.

    $$9\times 3721=33489$$

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

    $$5\times 63=315$$

14. Square that value.

    $${63}^2=3969$$

15. Multiply the square by its frequency.

    $$5\times 3969=19845$$

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

    $$4\times 71=284$$

17. Square that value.

    $${71}^2=5041$$

18. Multiply the square by its frequency.

    $$4\times 5041=20164$$

19. Add the frequencies to find n.

    $$5+3+4+9+5+4=30$$

20. Add the f×x column.

    $$210+153+216+549+315+284=1727$$

21. Add the f×x² column.

    $$8820+7803+11664+33489+19845+20164=101785$$

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

    $$\frac{1727}{30}$$

23. Calculate this operation before combining the remaining terms.

    $$1727\times 1=1727$$

24. Calculate this operation before combining the remaining terms.

    $$1\times 30=30$$

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

    $$\frac{1727\times 1}{1\times 30}$$

26. Calculate these two products separately.

    $$\frac{1727}{30}$$

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

    $$\frac{1\times 1}{1\times 30}$$

28. Calculate these two products separately.

    $$\frac{1}{30}$$

29. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{30}=\frac{1}{30}$$

30. Calculate this operation before combining the remaining terms.

    $$1727\times \frac{1}{30}=\frac{1727}{30}$$

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

    $$\frac{101785}{30}-{\left(\frac{1727}{30}\right)}^2$$

32. Calculate this operation before combining the remaining terms.

    $$101785\times 1=101785$$

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

    $$\frac{101785\times 1}{1\times 30}$$

34. Calculate these two products separately.

    $$\frac{101785}{30}$$

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

    $$\mathrm{GCD}=5$$

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

    $$\frac{101785\times 1}{1\times 5}$$

37. Calculate these two products separately.

    $$\frac{101785}{5}$$

38. Show the two divisions explicitly.

    $$\frac{\frac{101785}{5}}{\frac{5}{5}}$$

39. Evaluate the denominator division.

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

40. Write the reduced fraction.

    $$20357\times \frac{1}{1}$$

41. Calculate this operation before combining the remaining terms.

    $$101785\times \frac{1}{5}=20357$$

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

    $$\frac{30\times 1}{1\times 5}$$

43. Calculate these two products separately.

    $$\frac{30}{5}$$

44. Show the two divisions explicitly.

    $$\frac{\frac{30}{5}}{\frac{5}{5}}$$

45. Write the reduced fraction.

    $$6\times \frac{1}{1}$$

46. Calculate this operation before combining the remaining terms.

    $$30\times \frac{1}{5}=6$$

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

    $$\frac{20357\times 1}{1\times 6}$$

48. Calculate this operation before combining the remaining terms.

    $$20357\times \frac{1}{6}=\frac{20357}{6}$$

49. Show the two divisions explicitly.

    $$\frac{\frac{101785}{5}}{\frac{30}{5}}$$

50. Evaluate the numerator division.

    $$\frac{101785}{5}=20357$$

51. Evaluate the denominator division.

    $$\frac{30}{5}=6$$

52. Write the reduced fraction.

    $$\frac{20357}{6}$$

53. Calculate this operation before combining the remaining terms.

    $$101785\times \frac{1}{30}=\frac{20357}{6}$$

54. Calculate this operation before combining the remaining terms.

    $${\left(\frac{1727}{30}\right)}^2=\frac{2982529}{900}$$

55. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 2982529=-2982529$$

56. Calculate this operation before combining the remaining terms.

    $$1\times 900=900$$

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

    $$\frac{\left(-2982529\right)\times 1}{1\times 900}$$

58. Calculate this operation before combining the remaining terms.

    $$\left(-2982529\right)\times \frac{1}{900}=-\frac{2982529}{900}$$

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

    $$\frac{\left(-1\right)\times 2982529}{1\times 900}$$

60. Calculate these two products separately.

    $$-\frac{2982529}{900}$$

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

    $$\frac{2982529\times 1}{1\times 900}$$

62. Calculate these two products separately.

    $$\frac{2982529}{900}$$

63. Calculate this operation before combining the remaining terms.

    $$2982529\times \frac{1}{900}=\frac{2982529}{900}$$

64. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{2982529}{900}=-\frac{2982529}{900}$$

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

    $$\mathrm{LCD}=900$$

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

    $$\frac{20357}{6}=\frac{20357\times 150}{6\times 150}$$

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

    $$\frac{3053550}{900}$$

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

    $$-\frac{2982529}{900}=\frac{\left(-2982529\right)\times 1}{900\times 1}$$

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

    $$-\frac{2982529}{900}$$

70. Calculate this operation before combining the remaining terms.

    $$3053550-2982529=71021$$

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

    $$\frac{71021\times 1}{1\times 900}$$

72. Calculate these two products separately.

    $$\frac{71021}{900}$$

73. Calculate this operation before combining the remaining terms.

    $$71021\times \frac{1}{900}=\frac{71021}{900}$$

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

    $$\frac{3053550-2982529}{900}=\frac{71021}{900}$$

75. Calculate this operation before combining the remaining terms.

    $$\frac{20357}{6}-\frac{2982529}{900}=\frac{71021}{900}$$

76. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{71021}{900}}=\frac{1}{30}\times \sqrt{71021}$$

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

    $$\sqrt{\frac{71021}{900}}$$

78. Round the standard deviation only now.

    $$8.88326$$

79. Range is maximum minus minimum.

    $$71-42=29$$

Answer

mean=1727/30; variance=71021/900; standard deviation≈8.88326; range=29