Mathematics 10 — Exercise Review · Q9

Full question

The table shows the daily wages of 40 workers. Calculate the arithmetic mean, variance and standard deviation.

Step-by-step solution

1. Calculate this operation before combining the remaining terms.

   $$1000+1200=2200$$

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

   $$\frac{2200\times 1}{1\times 2}$$

3. Calculate these two products separately.

   $$\frac{2200}{2}$$

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

   $$\mathrm{GCD}=2$$

5. Show the two divisions explicitly.

   $$\frac{\frac{2200}{2}}{\frac{2}{2}}$$

6. Evaluate the numerator division.

   $$\frac{2200}{2}=1100$$

7. Evaluate the denominator division.

   $$\frac{2}{2}=1$$

8. Write the reduced fraction.

   $$1100\times \frac{1}{1}$$

9. Calculate this operation before combining the remaining terms.

   $$2200\times \frac{1}{2}=1100$$

10. Class 1000–1200 midpoint: add its boundaries and divide by 2.

    $$\frac{1000+1200}{2}=1100$$

11. Calculate this operation before combining the remaining terms.

    $$1200+1400=2600$$

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

    $$\frac{2600\times 1}{1\times 2}$$

13. Calculate these two products separately.

    $$\frac{2600}{2}$$

14. Show the two divisions explicitly.

    $$\frac{\frac{2600}{2}}{\frac{2}{2}}$$

15. Evaluate the numerator division.

    $$\frac{2600}{2}=1300$$

16. Write the reduced fraction.

    $$1300\times \frac{1}{1}$$

17. Calculate this operation before combining the remaining terms.

    $$2600\times \frac{1}{2}=1300$$

18. Class 1200–1400 midpoint: add its boundaries and divide by 2.

    $$\frac{1200+1400}{2}=1300$$

19. Calculate this operation before combining the remaining terms.

    $$1400+1600=3000$$

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

    $$\frac{3000\times 1}{1\times 2}$$

21. Calculate these two products separately.

    $$\frac{3000}{2}$$

22. Show the two divisions explicitly.

    $$\frac{\frac{3000}{2}}{\frac{2}{2}}$$

23. Evaluate the numerator division.

    $$\frac{3000}{2}=1500$$

24. Write the reduced fraction.

    $$1500\times \frac{1}{1}$$

25. Calculate this operation before combining the remaining terms.

    $$3000\times \frac{1}{2}=1500$$

26. Class 1400–1600 midpoint: add its boundaries and divide by 2.

    $$\frac{1400+1600}{2}=1500$$

27. Calculate this operation before combining the remaining terms.

    $$1600+1800=3400$$

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

    $$\frac{3400\times 1}{1\times 2}$$

29. Calculate these two products separately.

    $$\frac{3400}{2}$$

30. Show the two divisions explicitly.

    $$\frac{\frac{3400}{2}}{\frac{2}{2}}$$

31. Evaluate the numerator division.

    $$\frac{3400}{2}=1700$$

32. Write the reduced fraction.

    $$1700\times \frac{1}{1}$$

33. Calculate this operation before combining the remaining terms.

    $$3400\times \frac{1}{2}=1700$$

34. Class 1600–1800 midpoint: add its boundaries and divide by 2.

    $$\frac{1600+1800}{2}=1700$$

35. Calculate this operation before combining the remaining terms.

    $$1800+2000=3800$$

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

    $$\frac{3800\times 1}{1\times 2}$$

37. Calculate these two products separately.

    $$\frac{3800}{2}$$

38. Show the two divisions explicitly.

    $$\frac{\frac{3800}{2}}{\frac{2}{2}}$$

39. Evaluate the numerator division.

    $$\frac{3800}{2}=1900$$

40. Write the reduced fraction.

    $$1900\times \frac{1}{1}$$

41. Calculate this operation before combining the remaining terms.

    $$3800\times \frac{1}{2}=1900$$

42. Class 1800–2000 midpoint: add its boundaries and divide by 2.

    $$\frac{1800+2000}{2}=1900$$

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

    $$6\times 1100=6600$$

44. Square that value.

    $${1100}^2=1210000$$

45. Multiply the square by its frequency.

    $$6\times 1210000=7260000$$

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

    $$10\times 1300=13000$$

47. Square that value.

    $${1300}^2=1690000$$

48. Multiply the square by its frequency.

    $$10\times 1690000=16900000$$

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

    $$12\times 1500=18000$$

50. Square that value.

    $${1500}^2=2250000$$

51. Multiply the square by its frequency.

    $$12\times 2250000=27000000$$

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

    $$8\times 1700=13600$$

53. Square that value.

    $${1700}^2=2890000$$

54. Multiply the square by its frequency.

    $$8\times 2890000=23120000$$

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

    $$4\times 1900=7600$$

56. Square that value.

    $${1900}^2=3610000$$

57. Multiply the square by its frequency.

    $$4\times 3610000=14440000$$

58. Add the frequencies to find n.

    $$6+10+12+8+4=40$$

59. Add the f×x column.

    $$6600+13000+18000+13600+7600=58800$$

60. Add the f×x² column.

    $$7260000+16900000+27000000+23120000+14440000=88720000$$

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

    $$\frac{58800}{40}$$

62. Calculate this operation before combining the remaining terms.

    $$58800\times 1=58800$$

63. Calculate this operation before combining the remaining terms.

    $$1\times 40=40$$

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

    $$\frac{58800\times 1}{1\times 40}$$

65. Calculate these two products separately.

    $$\frac{58800}{40}$$

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

    $$\mathrm{GCD}=40$$

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

    $$\frac{40\times 1}{1\times 40}$$

68. Calculate these two products separately.

    $$\frac{40}{40}$$

69. Show the two divisions explicitly.

    $$\frac{\frac{40}{40}}{\frac{40}{40}}$$

70. Write the reduced fraction.

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

71. Calculate this operation before combining the remaining terms.

    $$40\times \frac{1}{40}=1$$

72. Calculate this operation before combining the remaining terms.

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

73. Calculate this operation before combining the remaining terms.

    $$1470\times 1=1470$$

74. Show the two divisions explicitly.

    $$\frac{\frac{58800}{40}}{\frac{40}{40}}$$

75. Evaluate the numerator division.

    $$\frac{58800}{40}=1470$$

76. Evaluate the denominator division.

    $$\frac{40}{40}=1$$

77. Write the reduced fraction.

    $$1470\times \frac{1}{1}$$

78. Calculate this operation before combining the remaining terms.

    $$58800\times \frac{1}{40}=1470$$

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

    $$\frac{88720000}{40}-{1470}^2$$

80. Calculate this operation before combining the remaining terms.

    $$88720000\times 1=88720000$$

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

    $$\frac{88720000\times 1}{1\times 40}$$

82. Calculate these two products separately.

    $$\frac{88720000}{40}$$

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

    $$\mathrm{GCD}=40$$

84. Calculate this operation before combining the remaining terms.

    $$2218000\times 1=2218000$$

85. Show the two divisions explicitly.

    $$\frac{\frac{88720000}{40}}{\frac{40}{40}}$$

86. Evaluate the numerator division.

    $$\frac{88720000}{40}=2218000$$

87. Evaluate the denominator division.

    $$\frac{40}{40}=1$$

88. Write the reduced fraction.

    $$2218000\times \frac{1}{1}$$

89. Calculate this operation before combining the remaining terms.

    $$88720000\times \frac{1}{40}=2218000$$

90. Calculate this operation before combining the remaining terms.

    $${1470}^2=2160900$$

91. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 2160900=-2160900$$

92. Calculate this operation before combining the remaining terms.

    $$2218000-2160900=57100$$

93. Extract a perfect-square factor from under the root.

    $$57100=100\times 571$$

94. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{57100}=10\times \sqrt{571}$$

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

    $$\sqrt{57100}$$

96. Round the standard deviation only now.

    $$238.956$$

97. Grouped-data range uses the highest upper boundary minus the lowest lower boundary with nonzero frequency.

    $$2000-1000=1000$$

Answer

mean=1470; variance=57100; standard deviation≈238.95606; range=1000