Mathematics 10 — Exercise 11.2 · Q12

Full question

The table gives unemployment rates, as percentages, for five provinces. Calculate the standard deviation and discuss how evenly the rates are distributed.

Step-by-step solution

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

   $$\frac{26\times 1}{1\times 5}$$

2. Calculate these two products separately.

   $$\frac{26}{5}$$

3. Calculate this operation before combining the remaining terms.

   $$26\times \frac{1}{5}=\frac{26}{5}$$

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

   $$\frac{1\times 26}{1\times 5}$$

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

   $$1\times \frac{26}{5}=\frac{26}{5}$$

6. Square that value.

   $${\left(\frac{26}{5}\right)}^2=\frac{676}{25}$$

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

   $$\frac{676\times 1}{1\times 25}$$

8. Calculate these two products separately.

   $$\frac{676}{25}$$

9. Calculate this operation before combining the remaining terms.

   $$676\times \frac{1}{25}=\frac{676}{25}$$

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

    $$\frac{1\times 676}{1\times 25}$$

11. Multiply the square by its frequency.

    $$1\times \frac{676}{25}=\frac{676}{25}$$

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

    $$1\times 6=6$$

13. Square that value.

    $$6^2=36$$

14. Multiply the square by its frequency.

    $$1\times 36=36$$

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

    $$\frac{24\times 1}{1\times 5}$$

16. Calculate these two products separately.

    $$\frac{24}{5}$$

17. Calculate this operation before combining the remaining terms.

    $$24\times \frac{1}{5}=\frac{24}{5}$$

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

    $$\frac{1\times 24}{1\times 5}$$

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

    $$1\times \frac{24}{5}=\frac{24}{5}$$

20. Square that value.

    $${\left(\frac{24}{5}\right)}^2=\frac{576}{25}$$

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

    $$\frac{576\times 1}{1\times 25}$$

22. Calculate these two products separately.

    $$\frac{576}{25}$$

23. Calculate this operation before combining the remaining terms.

    $$576\times \frac{1}{25}=\frac{576}{25}$$

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

    $$\frac{1\times 576}{1\times 25}$$

25. Multiply the square by its frequency.

    $$1\times \frac{576}{25}=\frac{576}{25}$$

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

    $$\frac{11\times 1}{1\times 2}$$

27. Calculate these two products separately.

    $$\frac{11}{2}$$

28. Calculate this operation before combining the remaining terms.

    $$11\times \frac{1}{2}=\frac{11}{2}$$

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

    $$\frac{1\times 11}{1\times 2}$$

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

    $$1\times \frac{11}{2}=\frac{11}{2}$$

31. Square that value.

    $${\left(\frac{11}{2}\right)}^2=\frac{121}{4}$$

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

    $$\frac{121\times 1}{1\times 4}$$

33. Calculate these two products separately.

    $$\frac{121}{4}$$

34. Calculate this operation before combining the remaining terms.

    $$121\times \frac{1}{4}=\frac{121}{4}$$

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

    $$\frac{1\times 121}{1\times 4}$$

36. Multiply the square by its frequency.

    $$1\times \frac{121}{4}=\frac{121}{4}$$

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

    $$\frac{31\times 1}{1\times 5}$$

38. Calculate these two products separately.

    $$\frac{31}{5}$$

39. Calculate this operation before combining the remaining terms.

    $$31\times \frac{1}{5}=\frac{31}{5}$$

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

    $$\frac{1\times 31}{1\times 5}$$

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

    $$1\times \frac{31}{5}=\frac{31}{5}$$

42. Square that value.

    $${\left(\frac{31}{5}\right)}^2=\frac{961}{25}$$

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

    $$\frac{961\times 1}{1\times 25}$$

44. Calculate these two products separately.

    $$\frac{961}{25}$$

45. Calculate this operation before combining the remaining terms.

    $$961\times \frac{1}{25}=\frac{961}{25}$$

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

    $$\frac{1\times 961}{1\times 25}$$

47. Multiply the square by its frequency.

    $$1\times \frac{961}{25}=\frac{961}{25}$$

48. Add the frequencies to find n.

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

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

    $$\mathrm{LCD}=10$$

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

    $$\frac{26}{5}=\frac{26\times 2}{5\times 2}$$

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

    $$\frac{52}{10}$$

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

    $$6\times \frac{1}{1}=\frac{6\times 10}{1\times 10}$$

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

    $$\frac{60}{10}$$

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

    $$\frac{24}{5}=\frac{24\times 2}{5\times 2}$$

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

    $$\frac{48}{10}$$

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

    $$\frac{11}{2}=\frac{11\times 5}{2\times 5}$$

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

    $$\frac{55}{10}$$

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

    $$\frac{31}{5}=\frac{31\times 2}{5\times 2}$$

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

    $$\frac{62}{10}$$

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

    $$\frac{52+60+48+55+62}{10}=\frac{277}{10}$$

61. Add the f×x column.

    $$\frac{26}{5}+6+\frac{24}{5}+\frac{11}{2}+\frac{31}{5}=\frac{277}{10}$$

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

    $$\mathrm{LCD}=100$$

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

    $$\frac{676}{25}=\frac{676\times 4}{25\times 4}$$

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

    $$\frac{2704}{100}$$

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

    $$36\times \frac{1}{1}=\frac{36\times 100}{1\times 100}$$

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

    $$\frac{3600}{100}$$

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

    $$\frac{576}{25}=\frac{576\times 4}{25\times 4}$$

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

    $$\frac{2304}{100}$$

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

    $$\frac{121}{4}=\frac{121\times 25}{4\times 25}$$

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

    $$\frac{3025}{100}$$

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

    $$\frac{961}{25}=\frac{961\times 4}{25\times 4}$$

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

    $$\frac{3844}{100}$$

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

    $$\frac{2704+3600+2304+3025+3844}{100}=\frac{15477}{100}$$

74. Add the f×x² column.

    $$\frac{676}{25}+36+\frac{576}{25}+\frac{121}{4}+\frac{961}{25}=\frac{15477}{100}$$

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

    $$\frac{277}{50}$$

76. Calculate this operation before combining the remaining terms.

    $$277\times 1=277$$

77. Calculate this operation before combining the remaining terms.

    $$10\times 5=50$$

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

    $$\frac{277\times 1}{1\times 50}$$

79. Calculate this operation before combining the remaining terms.

    $$277\times \frac{1}{50}=\frac{277}{50}$$

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

    $$\frac{277\times 1}{10\times 5}$$

81. Calculate these two products separately.

    $$\frac{277}{50}$$

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

    $$\frac{277\times 1}{1\times 10}$$

83. Calculate these two products separately.

    $$\frac{277}{10}$$

84. Calculate this operation before combining the remaining terms.

    $$277\times \frac{1}{10}=\frac{277}{10}$$

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

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

86. Calculate these two products separately.

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

87. Calculate this operation before combining the remaining terms.

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

88. Calculate this operation before combining the remaining terms.

    $$\frac{277}{10}\times \frac{1}{5}=\frac{277}{50}$$

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

    $$\frac{15477\times 1}{1\times 500}$$

90. Calculate this operation before combining the remaining terms.

    $$15477\times \frac{1}{500}=\frac{15477}{500}$$

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

    $$\frac{15477}{500}-{\left(\frac{277}{50}\right)}^2$$

92. Calculate this operation before combining the remaining terms.

    $$15477\times 1=15477$$

93. Calculate this operation before combining the remaining terms.

    $$100\times 5=500$$

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

    $$\frac{15477\times 1}{100\times 5}$$

95. Calculate these two products separately.

    $$\frac{15477}{500}$$

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

    $$\frac{15477\times 1}{1\times 100}$$

97. Calculate these two products separately.

    $$\frac{15477}{100}$$

98. Calculate this operation before combining the remaining terms.

    $$15477\times \frac{1}{100}=\frac{15477}{100}$$

99. Calculate this operation before combining the remaining terms.

    $$\frac{15477}{100}\times \frac{1}{5}=\frac{15477}{500}$$

100. Calculate this operation before combining the remaining terms.

     $${\left(\frac{277}{50}\right)}^2=\frac{76729}{2500}$$

101. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 76729=-76729$$

102. Calculate this operation before combining the remaining terms.

     $$1\times 2500=2500$$

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

     $$\frac{\left(-76729\right)\times 1}{1\times 2500}$$

104. Calculate this operation before combining the remaining terms.

     $$\left(-76729\right)\times \frac{1}{2500}=-\frac{76729}{2500}$$

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

     $$\frac{\left(-1\right)\times 76729}{1\times 2500}$$

106. Calculate these two products separately.

     $$-\frac{76729}{2500}$$

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

     $$\frac{76729\times 1}{1\times 2500}$$

108. Calculate these two products separately.

     $$\frac{76729}{2500}$$

109. Calculate this operation before combining the remaining terms.

     $$76729\times \frac{1}{2500}=\frac{76729}{2500}$$

110. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{76729}{2500}=-\frac{76729}{2500}$$

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

     $$\mathrm{LCD}=2500$$

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

     $$\frac{15477}{500}=\frac{15477\times 5}{500\times 5}$$

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

     $$\frac{77385}{2500}$$

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

     $$-\frac{76729}{2500}=\frac{\left(-76729\right)\times 1}{2500\times 1}$$

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

     $$-\frac{76729}{2500}$$

116. Calculate this operation before combining the remaining terms.

     $$77385-76729=656$$

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

     $$\frac{656\times 1}{1\times 2500}$$

118. Calculate these two products separately.

     $$\frac{656}{2500}$$

119. Calculate this operation before combining the remaining terms.

     $$656\times \frac{1}{2500}=\frac{164}{625}$$

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

     $$\frac{77385-76729}{2500}=\frac{656}{2500}$$

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

     $$\mathrm{GCD}=4$$

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

     $$\frac{656\times 1}{1\times 4}$$

123. Calculate these two products separately.

     $$\frac{656}{4}$$

124. Show the two divisions explicitly.

     $$\frac{\frac{656}{4}}{\frac{4}{4}}$$

125. Evaluate the denominator division.

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

126. Write the reduced fraction.

     $$164\times \frac{1}{1}$$

127. Calculate this operation before combining the remaining terms.

     $$656\times \frac{1}{4}=164$$

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

     $$\frac{2500\times 1}{1\times 4}$$

129. Calculate these two products separately.

     $$\frac{2500}{4}$$

130. Show the two divisions explicitly.

     $$\frac{\frac{2500}{4}}{\frac{4}{4}}$$

131. Write the reduced fraction.

     $$625\times \frac{1}{1}$$

132. Calculate this operation before combining the remaining terms.

     $$2500\times \frac{1}{4}=625$$

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

     $$\frac{164\times 1}{1\times 625}$$

134. Calculate this operation before combining the remaining terms.

     $$164\times \frac{1}{625}=\frac{164}{625}$$

135. Show the two divisions explicitly.

     $$\frac{\frac{656}{4}}{\frac{2500}{4}}$$

136. Evaluate the numerator division.

     $$\frac{656}{4}=164$$

137. Evaluate the denominator division.

     $$\frac{2500}{4}=625$$

138. Write the reduced fraction.

     $$\frac{164}{625}$$

139. Calculate this operation before combining the remaining terms.

     $$\frac{15477}{500}-\frac{76729}{2500}=\frac{164}{625}$$

140. Calculate this operation before combining the remaining terms.

     $$\sqrt{\frac{164}{625}}=\frac{2}{25}\times \sqrt{41}$$

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

     $$\sqrt{\frac{164}{625}}$$

142. Round the standard deviation only now.

     $$0.51225$$

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

     $$\frac{\left(-1\right)\times 24}{1\times 5}$$

144. Calculate these two products separately.

     $$-\frac{24}{5}$$

145. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{24}{5}=-\frac{24}{5}$$

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

     $$\mathrm{LCD}=5$$

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

     $$\frac{31}{5}=\frac{31\times 1}{5\times 1}$$

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

     $$-\frac{24}{5}=\frac{\left(-24\right)\times 1}{5\times 1}$$

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

     $$\frac{31-24}{5}=\frac{7}{5}$$

150. Range is maximum minus minimum.

     $$\frac{31}{5}-\frac{24}{5}=\frac{7}{5}$$

151. Rates cluster around their mean, but they are not equal. The standard deviation is measured in percentage points, and no threshold for declaring balance was specified.

Answer

mean=277/50; variance=164/625; standard deviation≈0.51225; range=7/5