Mathematics 10 — Exercise Review · Q8

Full question

Monthly expenses in thousands of rupees: find means and standard deviations. Which family spends more on average and which has more stable expenses? A: [45, 47, 46, 48, 46, 47]. B: [38, 52, 40, 50, 42, 49].

Step-by-step solution

1. Calculate the statistics for A independently.

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

   $$1\times 45=45$$

3. Square that value.

   $${45}^2=2025$$

4. Multiply the square by its frequency.

   $$1\times 2025=2025$$

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

   $$1\times 47=47$$

6. Square that value.

   $${47}^2=2209$$

7. Multiply the square by its frequency.

   $$1\times 2209=2209$$

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

   $$1\times 46=46$$

9. Square that value.

   $${46}^2=2116$$

10. Multiply the square by its frequency.

    $$1\times 2116=2116$$

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

    $$1\times 48=48$$

12. Square that value.

    $${48}^2=2304$$

13. Multiply the square by its frequency.

    $$1\times 2304=2304$$

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

    $$1\times 46=46$$

15. Square that value.

    $${46}^2=2116$$

16. Multiply the square by its frequency.

    $$1\times 2116=2116$$

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

    $$1\times 47=47$$

18. Square that value.

    $${47}^2=2209$$

19. Multiply the square by its frequency.

    $$1\times 2209=2209$$

20. Add the frequencies to find n.

    $$1+1+1+1+1+1=6$$

21. Add the f×x column.

    $$45+47+46+48+46+47=279$$

22. Add the f×x² column.

    $$2025+2209+2116+2304+2116+2209=12979$$

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

    $$\frac{279}{6}$$

24. Calculate this operation before combining the remaining terms.

    $$279\times 1=279$$

25. Calculate this operation before combining the remaining terms.

    $$1\times 6=6$$

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

    $$\frac{279\times 1}{1\times 6}$$

27. Calculate these two products separately.

    $$\frac{279}{6}$$

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

    $$\mathrm{GCD}=3$$

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

    $$\frac{279\times 1}{1\times 3}$$

30. Calculate these two products separately.

    $$\frac{279}{3}$$

31. Show the two divisions explicitly.

    $$\frac{\frac{279}{3}}{\frac{3}{3}}$$

32. Evaluate the denominator division.

    $$\frac{3}{3}=1$$

33. Write the reduced fraction.

    $$93\times \frac{1}{1}$$

34. Calculate this operation before combining the remaining terms.

    $$279\times \frac{1}{3}=93$$

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

    $$\frac{6\times 1}{1\times 3}$$

36. Calculate these two products separately.

    $$\frac{6}{3}$$

37. Show the two divisions explicitly.

    $$\frac{\frac{6}{3}}{\frac{3}{3}}$$

38. Write the reduced fraction.

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

39. Calculate this operation before combining the remaining terms.

    $$6\times \frac{1}{3}=2$$

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

    $$\frac{93\times 1}{1\times 2}$$

41. Calculate this operation before combining the remaining terms.

    $$93\times \frac{1}{2}=\frac{93}{2}$$

42. Show the two divisions explicitly.

    $$\frac{\frac{279}{3}}{\frac{6}{3}}$$

43. Evaluate the numerator division.

    $$\frac{279}{3}=93$$

44. Evaluate the denominator division.

    $$\frac{6}{3}=2$$

45. Write the reduced fraction.

    $$\frac{93}{2}$$

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

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

47. Calculate these two products separately.

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

48. Calculate this operation before combining the remaining terms.

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

49. Calculate this operation before combining the remaining terms.

    $$279\times \frac{1}{6}=\frac{93}{2}$$

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

    $$\frac{12979}{6}-{\left(\frac{93}{2}\right)}^2$$

51. Calculate this operation before combining the remaining terms.

    $$12979\times 1=12979$$

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

    $$\frac{12979\times 1}{1\times 6}$$

53. Calculate these two products separately.

    $$\frac{12979}{6}$$

54. Calculate this operation before combining the remaining terms.

    $$12979\times \frac{1}{6}=\frac{12979}{6}$$

55. Calculate this operation before combining the remaining terms.

    $${\left(\frac{93}{2}\right)}^2=\frac{8649}{4}$$

56. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 8649=-8649$$

57. Calculate this operation before combining the remaining terms.

    $$1\times 4=4$$

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

    $$\frac{\left(-8649\right)\times 1}{1\times 4}$$

59. Calculate this operation before combining the remaining terms.

    $$\left(-8649\right)\times \frac{1}{4}=-\frac{8649}{4}$$

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

    $$\frac{\left(-1\right)\times 8649}{1\times 4}$$

61. Calculate these two products separately.

    $$-\frac{8649}{4}$$

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

    $$\frac{8649\times 1}{1\times 4}$$

63. Calculate these two products separately.

    $$\frac{8649}{4}$$

64. Calculate this operation before combining the remaining terms.

    $$8649\times \frac{1}{4}=\frac{8649}{4}$$

65. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{8649}{4}=-\frac{8649}{4}$$

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

    $$\mathrm{LCD}=12$$

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

    $$\frac{12979}{6}=\frac{12979\times 2}{6\times 2}$$

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

    $$\frac{25958}{12}$$

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

    $$-\frac{8649}{4}=\frac{\left(-8649\right)\times 3}{4\times 3}$$

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

    $$\frac{\left(-25947\right)\times 1}{1\times 12}$$

71. Show the two divisions explicitly.

    $$\frac{-\frac{25947}{3}}{\frac{12}{3}}$$

72. Evaluate the numerator division.

    $$-\frac{25947}{3}=-8649$$

73. Evaluate the denominator division.

    $$\frac{12}{3}=4$$

74. Calculate this operation before combining the remaining terms.

    $$\left(-25947\right)\times \frac{1}{12}=-\frac{8649}{4}$$

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

    $$-\frac{25947}{12}$$

76. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 25947=-25947$$

77. Calculate this operation before combining the remaining terms.

    $$25958-25947=11$$

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

    $$\frac{11\times 1}{1\times 12}$$

79. Calculate these two products separately.

    $$\frac{11}{12}$$

80. Calculate this operation before combining the remaining terms.

    $$11\times \frac{1}{12}=\frac{11}{12}$$

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

    $$\frac{25958-25947}{12}=\frac{11}{12}$$

82. Calculate this operation before combining the remaining terms.

    $$\frac{12979}{6}-\frac{8649}{4}=\frac{11}{12}$$

83. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{11}{12}}=\frac{1}{6}\times \sqrt{33}$$

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

    $$\sqrt{\frac{11}{12}}$$

85. Round the standard deviation only now.

    $$0.957427$$

86. Group A: mean=46.50000, standard deviation=0.95743.

87. Calculate the statistics for B independently.

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

    $$1\times 38=38$$

89. Square that value.

    $${38}^2=1444$$

90. Multiply the square by its frequency.

    $$1\times 1444=1444$$

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

    $$1\times 52=52$$

92. Square that value.

    $${52}^2=2704$$

93. Multiply the square by its frequency.

    $$1\times 2704=2704$$

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

    $$1\times 40=40$$

95. Square that value.

    $${40}^2=1600$$

96. Multiply the square by its frequency.

    $$1\times 1600=1600$$

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

    $$1\times 50=50$$

98. Square that value.

    $${50}^2=2500$$

99. Multiply the square by its frequency.

    $$1\times 2500=2500$$

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

     $$1\times 42=42$$

101. Square that value.

     $${42}^2=1764$$

102. Multiply the square by its frequency.

     $$1\times 1764=1764$$

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

     $$1\times 49=49$$

104. Square that value.

     $${49}^2=2401$$

105. Multiply the square by its frequency.

     $$1\times 2401=2401$$

106. Add the frequencies to find n.

     $$1+1+1+1+1+1=6$$

107. Add the f×x column.

     $$38+52+40+50+42+49=271$$

108. Add the f×x² column.

     $$1444+2704+1600+2500+1764+2401=12413$$

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

     $$\frac{271}{6}$$

110. Calculate this operation before combining the remaining terms.

     $$271\times 1=271$$

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

     $$\frac{271\times 1}{1\times 6}$$

112. Calculate these two products separately.

     $$\frac{271}{6}$$

113. Calculate this operation before combining the remaining terms.

     $$271\times \frac{1}{6}=\frac{271}{6}$$

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

     $$\frac{12413}{6}-{\left(\frac{271}{6}\right)}^2$$

115. Calculate this operation before combining the remaining terms.

     $$12413\times 1=12413$$

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

     $$\frac{12413\times 1}{1\times 6}$$

117. Calculate these two products separately.

     $$\frac{12413}{6}$$

118. Calculate this operation before combining the remaining terms.

     $$12413\times \frac{1}{6}=\frac{12413}{6}$$

119. Calculate this operation before combining the remaining terms.

     $${\left(\frac{271}{6}\right)}^2=\frac{73441}{36}$$

120. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 73441=-73441$$

121. Calculate this operation before combining the remaining terms.

     $$1\times 36=36$$

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

     $$\frac{\left(-73441\right)\times 1}{1\times 36}$$

123. Calculate this operation before combining the remaining terms.

     $$\left(-73441\right)\times \frac{1}{36}=-\frac{73441}{36}$$

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

     $$\frac{\left(-1\right)\times 73441}{1\times 36}$$

125. Calculate these two products separately.

     $$-\frac{73441}{36}$$

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

     $$\frac{73441\times 1}{1\times 36}$$

127. Calculate these two products separately.

     $$\frac{73441}{36}$$

128. Calculate this operation before combining the remaining terms.

     $$73441\times \frac{1}{36}=\frac{73441}{36}$$

129. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{73441}{36}=-\frac{73441}{36}$$

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

     $$\mathrm{LCD}=36$$

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

     $$\frac{12413}{6}=\frac{12413\times 6}{6\times 6}$$

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

     $$\frac{74478}{36}$$

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

     $$-\frac{73441}{36}=\frac{\left(-73441\right)\times 1}{36\times 1}$$

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

     $$-\frac{73441}{36}$$

135. Calculate this operation before combining the remaining terms.

     $$74478-73441=1037$$

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

     $$\frac{1037\times 1}{1\times 36}$$

137. Calculate these two products separately.

     $$\frac{1037}{36}$$

138. Calculate this operation before combining the remaining terms.

     $$1037\times \frac{1}{36}=\frac{1037}{36}$$

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

     $$\frac{74478-73441}{36}=\frac{1037}{36}$$

140. Calculate this operation before combining the remaining terms.

     $$\frac{12413}{6}-\frac{73441}{36}=\frac{1037}{36}$$

141. Calculate this operation before combining the remaining terms.

     $$\sqrt{\frac{1037}{36}}=\frac{1}{6}\times \sqrt{1037}$$

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

     $$\sqrt{\frac{1037}{36}}$$

143. Round the standard deviation only now.

     $$5.36708$$

144. Group B: mean=45.16667, standard deviation=5.36708.

145. Compare means for average performance or expenditure; compare standard deviations for consistency. A smaller standard deviation means less variation.

Answer

Higher average: A. More consistent: A.