Mathematics 10 — Exercise 11.2 · Q6

Full question

Find range,variance and standard deviation of the FIRST 23 odd natural numbers.

Step-by-step solution

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

   $$1\times 1=1$$

2. Square that value.

   $$1^2=1$$

3. Multiply the square by its frequency.

   $$1\times 1=1$$

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

   $$1\times 3=3$$

5. Square that value.

   $$3^2=9$$

6. Multiply the square by its frequency.

   $$1\times 9=9$$

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

   $$1\times 5=5$$

8. Square that value.

   $$5^2=25$$

9. Multiply the square by its frequency.

   $$1\times 25=25$$

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

    $$1\times 7=7$$

11. Square that value.

    $$7^2=49$$

12. Multiply the square by its frequency.

    $$1\times 49=49$$

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

    $$1\times 9=9$$

14. Square that value.

    $$9^2=81$$

15. Multiply the square by its frequency.

    $$1\times 81=81$$

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

    $$1\times 11=11$$

17. Square that value.

    $${11}^2=121$$

18. Multiply the square by its frequency.

    $$1\times 121=121$$

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

    $$1\times 13=13$$

20. Square that value.

    $${13}^2=169$$

21. Multiply the square by its frequency.

    $$1\times 169=169$$

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

    $$1\times 15=15$$

23. Square that value.

    $${15}^2=225$$

24. Multiply the square by its frequency.

    $$1\times 225=225$$

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

    $$1\times 17=17$$

26. Square that value.

    $${17}^2=289$$

27. Multiply the square by its frequency.

    $$1\times 289=289$$

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

    $$1\times 19=19$$

29. Square that value.

    $${19}^2=361$$

30. Multiply the square by its frequency.

    $$1\times 361=361$$

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

    $$1\times 21=21$$

32. Square that value.

    $${21}^2=441$$

33. Multiply the square by its frequency.

    $$1\times 441=441$$

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

    $$1\times 23=23$$

35. Square that value.

    $${23}^2=529$$

36. Multiply the square by its frequency.

    $$1\times 529=529$$

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

    $$1\times 25=25$$

38. Square that value.

    $${25}^2=625$$

39. Multiply the square by its frequency.

    $$1\times 625=625$$

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

    $$1\times 27=27$$

41. Square that value.

    $${27}^2=729$$

42. Multiply the square by its frequency.

    $$1\times 729=729$$

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

    $$1\times 29=29$$

44. Square that value.

    $${29}^2=841$$

45. Multiply the square by its frequency.

    $$1\times 841=841$$

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

    $$1\times 31=31$$

47. Square that value.

    $${31}^2=961$$

48. Multiply the square by its frequency.

    $$1\times 961=961$$

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

    $$1\times 33=33$$

50. Square that value.

    $${33}^2=1089$$

51. Multiply the square by its frequency.

    $$1\times 1089=1089$$

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

    $$1\times 35=35$$

53. Square that value.

    $${35}^2=1225$$

54. Multiply the square by its frequency.

    $$1\times 1225=1225$$

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

    $$1\times 37=37$$

56. Square that value.

    $${37}^2=1369$$

57. Multiply the square by its frequency.

    $$1\times 1369=1369$$

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

    $$1\times 39=39$$

59. Square that value.

    $${39}^2=1521$$

60. Multiply the square by its frequency.

    $$1\times 1521=1521$$

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

    $$1\times 41=41$$

62. Square that value.

    $${41}^2=1681$$

63. Multiply the square by its frequency.

    $$1\times 1681=1681$$

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

    $$1\times 43=43$$

65. Square that value.

    $${43}^2=1849$$

66. Multiply the square by its frequency.

    $$1\times 1849=1849$$

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

    $$1\times 45=45$$

68. Square that value.

    $${45}^2=2025$$

69. Multiply the square by its frequency.

    $$1\times 2025=2025$$

70. Add the frequencies to find n.

    $$1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=23$$

71. Add the f×x column.

    $$1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31+33+35+37+39+41+43+45=529$$

72. Add the f×x² column.

    $$1+9+25+49+81+121+169+225+289+361+441+529+625+729+841+961+1089+1225+1369+1521+1681+1849+2025=16215$$

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

    $$\frac{529}{23}$$

74. Calculate this operation before combining the remaining terms.

    $$529\times 1=529$$

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

    $$\frac{529\times 1}{1\times 23}$$

76. Calculate these two products separately.

    $$\frac{529}{23}$$

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

    $$\mathrm{GCD}=23$$

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

    $$\frac{23\times 1}{1\times 23}$$

79. Calculate these two products separately.

    $$\frac{23}{23}$$

80. Show the two divisions explicitly.

    $$\frac{\frac{23}{23}}{\frac{23}{23}}$$

81. Write the reduced fraction.

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

82. Calculate this operation before combining the remaining terms.

    $$23\times \frac{1}{23}=1$$

83. Calculate this operation before combining the remaining terms.

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

84. Calculate this operation before combining the remaining terms.

    $$23\times 1=23$$

85. Show the two divisions explicitly.

    $$\frac{\frac{529}{23}}{\frac{23}{23}}$$

86. Evaluate the numerator division.

    $$\frac{529}{23}=23$$

87. Evaluate the denominator division.

    $$\frac{23}{23}=1$$

88. Write the reduced fraction.

    $$23\times \frac{1}{1}$$

89. Calculate this operation before combining the remaining terms.

    $$529\times \frac{1}{23}=23$$

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

    $$\frac{16215}{23}-{23}^2$$

91. Calculate this operation before combining the remaining terms.

    $$16215\times 1=16215$$

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

    $$\frac{16215\times 1}{1\times 23}$$

93. Calculate these two products separately.

    $$\frac{16215}{23}$$

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

    $$\mathrm{GCD}=23$$

95. Calculate this operation before combining the remaining terms.

    $$705\times 1=705$$

96. Show the two divisions explicitly.

    $$\frac{\frac{16215}{23}}{\frac{23}{23}}$$

97. Evaluate the numerator division.

    $$\frac{16215}{23}=705$$

98. Evaluate the denominator division.

    $$\frac{23}{23}=1$$

99. Write the reduced fraction.

    $$705\times \frac{1}{1}$$

100. Calculate this operation before combining the remaining terms.

     $$16215\times \frac{1}{23}=705$$

101. Calculate this operation before combining the remaining terms.

     $${23}^2=529$$

102. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 529=-529$$

103. Calculate this operation before combining the remaining terms.

     $$705-529=176$$

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

     $$176=16\times 11$$

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

     $$\sqrt{176}=4\times \sqrt{11}$$

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

     $$\sqrt{176}$$

107. Round the standard deviation only now.

     $$13.2665$$

108. Range is maximum minus minimum.

     $$45-1=44$$

Answer

mean=23; variance=176; standard deviation≈13.26650; range=44