Mathematics 10 — Exercise 11.2 · Q9

Full question

Monthly sales in lakhs: compare salesperson performance and consistency. A: ['5.5', '5.7', '5.4', '5.6', '5.8', '5.6']. B: ['5.5', '6.5', '4.5', '6.0', '5.0', '6.0'].

Step-by-step solution

1. Calculate the statistics for A independently.

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

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

3. Calculate these two products separately.

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

4. Calculate this operation before combining the remaining terms.

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

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

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

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

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

7. Square that value.

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

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

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

9. Calculate these two products separately.

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

10. Calculate this operation before combining the remaining terms.

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

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

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

12. Multiply the square by its frequency.

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

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

    $$\frac{57\times 1}{1\times 10}$$

14. Calculate these two products separately.

    $$\frac{57}{10}$$

15. Calculate this operation before combining the remaining terms.

    $$57\times \frac{1}{10}=\frac{57}{10}$$

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

    $$\frac{1\times 57}{1\times 10}$$

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

    $$1\times \frac{57}{10}=\frac{57}{10}$$

18. Square that value.

    $${\left(\frac{57}{10}\right)}^2=\frac{3249}{100}$$

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

    $$\frac{3249\times 1}{1\times 100}$$

20. Calculate these two products separately.

    $$\frac{3249}{100}$$

21. Calculate this operation before combining the remaining terms.

    $$3249\times \frac{1}{100}=\frac{3249}{100}$$

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

    $$\frac{1\times 3249}{1\times 100}$$

23. Multiply the square by its frequency.

    $$1\times \frac{3249}{100}=\frac{3249}{100}$$

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

    $$\frac{27\times 1}{1\times 5}$$

25. Calculate these two products separately.

    $$\frac{27}{5}$$

26. Calculate this operation before combining the remaining terms.

    $$27\times \frac{1}{5}=\frac{27}{5}$$

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

    $$\frac{1\times 27}{1\times 5}$$

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

    $$1\times \frac{27}{5}=\frac{27}{5}$$

29. Square that value.

    $${\left(\frac{27}{5}\right)}^2=\frac{729}{25}$$

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

    $$\frac{729\times 1}{1\times 25}$$

31. Calculate these two products separately.

    $$\frac{729}{25}$$

32. Calculate this operation before combining the remaining terms.

    $$729\times \frac{1}{25}=\frac{729}{25}$$

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

    $$\frac{1\times 729}{1\times 25}$$

34. Multiply the square by its frequency.

    $$1\times \frac{729}{25}=\frac{729}{25}$$

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

    $$\frac{28\times 1}{1\times 5}$$

36. Calculate this operation before combining the remaining terms.

    $$28\times \frac{1}{5}=\frac{28}{5}$$

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

    $$\frac{1\times 28}{1\times 5}$$

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

    $$1\times \frac{28}{5}=\frac{28}{5}$$

39. Square that value.

    $${\left(\frac{28}{5}\right)}^2=\frac{784}{25}$$

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

    $$\frac{784\times 1}{1\times 25}$$

41. Calculate these two products separately.

    $$\frac{784}{25}$$

42. Calculate this operation before combining the remaining terms.

    $$784\times \frac{1}{25}=\frac{784}{25}$$

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

    $$\frac{1\times 784}{1\times 25}$$

44. Multiply the square by its frequency.

    $$1\times \frac{784}{25}=\frac{784}{25}$$

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

    $$\frac{29\times 1}{1\times 5}$$

46. Calculate these two products separately.

    $$\frac{29}{5}$$

47. Calculate this operation before combining the remaining terms.

    $$29\times \frac{1}{5}=\frac{29}{5}$$

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

    $$\frac{1\times 29}{1\times 5}$$

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

    $$1\times \frac{29}{5}=\frac{29}{5}$$

50. Square that value.

    $${\left(\frac{29}{5}\right)}^2=\frac{841}{25}$$

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

    $$\frac{841\times 1}{1\times 25}$$

52. Calculate these two products separately.

    $$\frac{841}{25}$$

53. Calculate this operation before combining the remaining terms.

    $$841\times \frac{1}{25}=\frac{841}{25}$$

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

    $$\frac{1\times 841}{1\times 25}$$

55. Multiply the square by its frequency.

    $$1\times \frac{841}{25}=\frac{841}{25}$$

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

    $$1\times \frac{28}{5}=\frac{28}{5}$$

57. Square that value.

    $${\left(\frac{28}{5}\right)}^2=\frac{784}{25}$$

58. Multiply the square by its frequency.

    $$1\times \frac{784}{25}=\frac{784}{25}$$

59. Add the frequencies to find n.

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

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

    $$\mathrm{LCD}=10$$

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

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

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

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

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

    $$\frac{57}{10}=\frac{57\times 1}{10\times 1}$$

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

    $$\frac{27}{5}=\frac{27\times 2}{5\times 2}$$

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

    $$\frac{54}{10}$$

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

    $$\frac{28}{5}=\frac{28\times 2}{5\times 2}$$

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

    $$\frac{56}{10}$$

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

    $$\frac{29}{5}=\frac{29\times 2}{5\times 2}$$

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

    $$\frac{58}{10}$$

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

    $$\frac{55+57+54+56+58+56}{10}=\frac{336}{10}$$

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

    $$\mathrm{GCD}=2$$

72. Show the two divisions explicitly.

    $$\frac{\frac{336}{2}}{\frac{10}{2}}$$

73. Evaluate the numerator division.

    $$\frac{336}{2}=168$$

74. Evaluate the denominator division.

    $$\frac{10}{2}=5$$

75. Write the reduced fraction.

    $$\frac{168}{5}$$

76. Add the f×x column.

    $$\frac{11}{2}+\frac{57}{10}+\frac{27}{5}+\frac{28}{5}+\frac{29}{5}+\frac{28}{5}=\frac{168}{5}$$

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

    $$\mathrm{LCD}=100$$

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

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

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

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

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

    $$\frac{3249}{100}=\frac{3249\times 1}{100\times 1}$$

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

    $$\frac{729}{25}=\frac{729\times 4}{25\times 4}$$

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

    $$\frac{2916}{100}$$

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

    $$\frac{784}{25}=\frac{784\times 4}{25\times 4}$$

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

    $$\frac{3136}{100}$$

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

    $$\frac{841}{25}=\frac{841\times 4}{25\times 4}$$

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

    $$\frac{3364}{100}$$

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

    $$\frac{3025+3249+2916+3136+3364+3136}{100}=\frac{18826}{100}$$

88. Show the two divisions explicitly.

    $$\frac{\frac{18826}{2}}{\frac{100}{2}}$$

89. Evaluate the numerator division.

    $$\frac{18826}{2}=9413$$

90. Evaluate the denominator division.

    $$\frac{100}{2}=50$$

91. Write the reduced fraction.

    $$\frac{9413}{50}$$

92. Add the f×x² column.

    $$\frac{121}{4}+\frac{3249}{100}+\frac{729}{25}+\frac{784}{25}+\frac{841}{25}+\frac{784}{25}=\frac{9413}{50}$$

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

    $$\frac{168}{30}$$

94. Calculate this operation before combining the remaining terms.

    $$168\times 1=168$$

95. Calculate this operation before combining the remaining terms.

    $$5\times 6=30$$

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

    $$\frac{168\times 1}{1\times 30}$$

97. Calculate this operation before combining the remaining terms.

    $$168\times \frac{1}{30}=\frac{28}{5}$$

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

    $$\frac{168\times 1}{5\times 6}$$

99. Calculate these two products separately.

    $$\frac{168}{30}$$

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

     $$\mathrm{GCD}=6$$

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

     $$\frac{168\times 1}{1\times 6}$$

102. Calculate these two products separately.

     $$\frac{168}{6}$$

103. Show the two divisions explicitly.

     $$\frac{\frac{168}{6}}{\frac{6}{6}}$$

104. Evaluate the denominator division.

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

105. Write the reduced fraction.

     $$28\times \frac{1}{1}$$

106. Calculate this operation before combining the remaining terms.

     $$168\times \frac{1}{6}=28$$

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

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

108. Calculate these two products separately.

     $$\frac{30}{6}$$

109. Show the two divisions explicitly.

     $$\frac{\frac{30}{6}}{\frac{6}{6}}$$

110. Write the reduced fraction.

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

111. Calculate this operation before combining the remaining terms.

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

112. Show the two divisions explicitly.

     $$\frac{\frac{168}{6}}{\frac{30}{6}}$$

113. Evaluate the numerator division.

     $$\frac{168}{6}=28$$

114. Evaluate the denominator division.

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

115. Write the reduced fraction.

     $$\frac{28}{5}$$

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

     $$\frac{168\times 1}{1\times 5}$$

117. Calculate this operation before combining the remaining terms.

     $$168\times \frac{1}{5}=\frac{168}{5}$$

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

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

119. Calculate these two products separately.

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

120. Calculate this operation before combining the remaining terms.

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

121. Calculate this operation before combining the remaining terms.

     $$\frac{168}{5}\times \frac{1}{6}=\frac{28}{5}$$

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

     $$\frac{9413\times 1}{1\times 300}$$

123. Calculate this operation before combining the remaining terms.

     $$9413\times \frac{1}{300}=\frac{9413}{300}$$

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

     $$\frac{9413}{300}-{\left(\frac{28}{5}\right)}^2$$

125. Calculate this operation before combining the remaining terms.

     $$9413\times 1=9413$$

126. Calculate this operation before combining the remaining terms.

     $$50\times 6=300$$

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

     $$\frac{9413\times 1}{50\times 6}$$

128. Calculate these two products separately.

     $$\frac{9413}{300}$$

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

     $$\frac{9413\times 1}{1\times 50}$$

130. Calculate this operation before combining the remaining terms.

     $$9413\times \frac{1}{50}=\frac{9413}{50}$$

131. Calculate this operation before combining the remaining terms.

     $$\frac{9413}{50}\times \frac{1}{6}=\frac{9413}{300}$$

132. Calculate this operation before combining the remaining terms.

     $${\left(\frac{28}{5}\right)}^2=\frac{784}{25}$$

133. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 784=-784$$

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

     $$\frac{\left(-784\right)\times 1}{1\times 25}$$

135. Calculate this operation before combining the remaining terms.

     $$\left(-784\right)\times \frac{1}{25}=-\frac{784}{25}$$

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

     $$\frac{\left(-1\right)\times 784}{1\times 25}$$

137. Calculate these two products separately.

     $$-\frac{784}{25}$$

138. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{784}{25}=-\frac{784}{25}$$

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

     $$\mathrm{LCD}=300$$

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

     $$\frac{9413}{300}=\frac{9413\times 1}{300\times 1}$$

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

     $$\frac{9413}{300}$$

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

     $$-\frac{784}{25}=\frac{\left(-784\right)\times 12}{25\times 12}$$

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

     $$\frac{\left(-9408\right)\times 1}{1\times 300}$$

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

     $$\mathrm{GCD}=12$$

145. Show the two divisions explicitly.

     $$\frac{-\frac{9408}{12}}{\frac{300}{12}}$$

146. Evaluate the numerator division.

     $$-\frac{9408}{12}=-784$$

147. Evaluate the denominator division.

     $$\frac{300}{12}=25$$

148. Calculate this operation before combining the remaining terms.

     $$\left(-9408\right)\times \frac{1}{300}=-\frac{784}{25}$$

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

     $$-\frac{9408}{300}$$

150. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 9408=-9408$$

151. Calculate this operation before combining the remaining terms.

     $$9413-9408=5$$

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

     $$\frac{5\times 1}{1\times 300}$$

153. Calculate these two products separately.

     $$\frac{5}{300}$$

154. Calculate this operation before combining the remaining terms.

     $$5\times \frac{1}{300}=\frac{1}{60}$$

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

     $$\frac{9413-9408}{300}=\frac{5}{300}$$

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

     $$\mathrm{GCD}=5$$

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

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

158. Calculate these two products separately.

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

159. Show the two divisions explicitly.

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

160. Write the reduced fraction.

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

161. Calculate this operation before combining the remaining terms.

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

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

     $$\frac{300\times 1}{1\times 5}$$

163. Calculate these two products separately.

     $$\frac{300}{5}$$

164. Show the two divisions explicitly.

     $$\frac{\frac{300}{5}}{\frac{5}{5}}$$

165. Write the reduced fraction.

     $$60\times \frac{1}{1}$$

166. Calculate this operation before combining the remaining terms.

     $$300\times \frac{1}{5}=60$$

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

     $$\frac{1\times 1}{1\times 60}$$

168. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{60}=\frac{1}{60}$$

169. Show the two divisions explicitly.

     $$\frac{\frac{5}{5}}{\frac{300}{5}}$$

170. Evaluate the numerator division.

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

171. Evaluate the denominator division.

     $$\frac{300}{5}=60$$

172. Write the reduced fraction.

     $$\frac{1}{60}$$

173. Calculate this operation before combining the remaining terms.

     $$\frac{9413}{300}-\frac{784}{25}=\frac{1}{60}$$

174. Calculate this operation before combining the remaining terms.

     $$\sqrt{\frac{1}{60}}=\frac{1}{30}\times \sqrt{15}$$

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

     $$\sqrt{\frac{1}{60}}$$

176. Round the standard deviation only now.

     $$0.129099$$

177. Group A: mean=5.60000, standard deviation=0.12910.

178. Calculate the statistics for B independently.

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

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

180. Square that value.

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

181. Multiply the square by its frequency.

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

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

     $$\frac{13\times 1}{1\times 2}$$

183. Calculate these two products separately.

     $$\frac{13}{2}$$

184. Calculate this operation before combining the remaining terms.

     $$13\times \frac{1}{2}=\frac{13}{2}$$

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

     $$\frac{1\times 13}{1\times 2}$$

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

     $$1\times \frac{13}{2}=\frac{13}{2}$$

187. Square that value.

     $${\left(\frac{13}{2}\right)}^2=\frac{169}{4}$$

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

     $$\frac{169\times 1}{1\times 4}$$

189. Calculate these two products separately.

     $$\frac{169}{4}$$

190. Calculate this operation before combining the remaining terms.

     $$169\times \frac{1}{4}=\frac{169}{4}$$

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

     $$\frac{1\times 169}{1\times 4}$$

192. Multiply the square by its frequency.

     $$1\times \frac{169}{4}=\frac{169}{4}$$

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

     $$\frac{9\times 1}{1\times 2}$$

194. Calculate these two products separately.

     $$\frac{9}{2}$$

195. Calculate this operation before combining the remaining terms.

     $$9\times \frac{1}{2}=\frac{9}{2}$$

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

     $$\frac{1\times 9}{1\times 2}$$

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

     $$1\times \frac{9}{2}=\frac{9}{2}$$

198. Square that value.

     $${\left(\frac{9}{2}\right)}^2=\frac{81}{4}$$

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

     $$\frac{81\times 1}{1\times 4}$$

200. Calculate these two products separately.

     $$\frac{81}{4}$$

201. Calculate this operation before combining the remaining terms.

     $$81\times \frac{1}{4}=\frac{81}{4}$$

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

     $$\frac{1\times 81}{1\times 4}$$

203. Multiply the square by its frequency.

     $$1\times \frac{81}{4}=\frac{81}{4}$$

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

     $$1\times 6=6$$

205. Square that value.

     $$6^2=36$$

206. Multiply the square by its frequency.

     $$1\times 36=36$$

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

     $$1\times 5=5$$

208. Square that value.

     $$5^2=25$$

209. Multiply the square by its frequency.

     $$1\times 25=25$$

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

     $$1\times 6=6$$

211. Square that value.

     $$6^2=36$$

212. Multiply the square by its frequency.

     $$1\times 36=36$$

213. Add the frequencies to find n.

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

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

     $$\mathrm{LCD}=2$$

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

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

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

     $$\frac{13}{2}=\frac{13\times 1}{2\times 1}$$

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

     $$\frac{9}{2}=\frac{9\times 1}{2\times 1}$$

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

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

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

     $$\frac{12}{2}$$

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

     $$5\times \frac{1}{1}=\frac{5\times 2}{1\times 2}$$

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

     $$\frac{10}{2}$$

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

     $$\frac{11+13+9+12+10+12}{2}=\frac{67}{2}$$

223. Add the f×x column.

     $$\frac{11}{2}+\frac{13}{2}+\frac{9}{2}+6+5+6=\frac{67}{2}$$

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

     $$\mathrm{LCD}=4$$

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

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

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

     $$\frac{169}{4}=\frac{169\times 1}{4\times 1}$$

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

     $$\frac{81}{4}=\frac{81\times 1}{4\times 1}$$

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

     $$36\times \frac{1}{1}=\frac{36\times 4}{1\times 4}$$

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

     $$\frac{144}{4}$$

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

     $$25\times \frac{1}{1}=\frac{25\times 4}{1\times 4}$$

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

     $$\frac{100}{4}$$

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

     $$\frac{121+169+81+144+100+144}{4}=\frac{759}{4}$$

233. Add the f×x² column.

     $$\frac{121}{4}+\frac{169}{4}+\frac{81}{4}+36+25+36=\frac{759}{4}$$

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

     $$\frac{67}{12}$$

235. Calculate this operation before combining the remaining terms.

     $$67\times 1=67$$

236. Calculate this operation before combining the remaining terms.

     $$2\times 6=12$$

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

     $$\frac{67\times 1}{1\times 12}$$

238. Calculate this operation before combining the remaining terms.

     $$67\times \frac{1}{12}=\frac{67}{12}$$

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

     $$\frac{67\times 1}{2\times 6}$$

240. Calculate these two products separately.

     $$\frac{67}{12}$$

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

     $$\frac{67\times 1}{1\times 2}$$

242. Calculate these two products separately.

     $$\frac{67}{2}$$

243. Calculate this operation before combining the remaining terms.

     $$67\times \frac{1}{2}=\frac{67}{2}$$

244. Calculate this operation before combining the remaining terms.

     $$\frac{67}{2}\times \frac{1}{6}=\frac{67}{12}$$

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

     $$\frac{759\times 1}{1\times 24}$$

246. Calculate this operation before combining the remaining terms.

     $$759\times \frac{1}{24}=\frac{253}{8}$$

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

     $$\frac{759}{24}-{\left(\frac{67}{12}\right)}^2$$

248. Calculate this operation before combining the remaining terms.

     $$759\times 1=759$$

249. Calculate this operation before combining the remaining terms.

     $$4\times 6=24$$

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

     $$\frac{759\times 1}{4\times 6}$$

251. Calculate these two products separately.

     $$\frac{759}{24}$$

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

     $$\mathrm{GCD}=3$$

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

     $$\frac{759\times 1}{1\times 3}$$

254. Calculate these two products separately.

     $$\frac{759}{3}$$

255. Show the two divisions explicitly.

     $$\frac{\frac{759}{3}}{\frac{3}{3}}$$

256. Evaluate the denominator division.

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

257. Write the reduced fraction.

     $$253\times \frac{1}{1}$$

258. Calculate this operation before combining the remaining terms.

     $$759\times \frac{1}{3}=253$$

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

     $$\frac{24\times 1}{1\times 3}$$

260. Calculate these two products separately.

     $$\frac{24}{3}$$

261. Show the two divisions explicitly.

     $$\frac{\frac{24}{3}}{\frac{3}{3}}$$

262. Write the reduced fraction.

     $$8\times \frac{1}{1}$$

263. Calculate this operation before combining the remaining terms.

     $$24\times \frac{1}{3}=8$$

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

     $$\frac{253\times 1}{1\times 8}$$

265. Calculate this operation before combining the remaining terms.

     $$253\times \frac{1}{8}=\frac{253}{8}$$

266. Show the two divisions explicitly.

     $$\frac{\frac{759}{3}}{\frac{24}{3}}$$

267. Evaluate the numerator division.

     $$\frac{759}{3}=253$$

268. Evaluate the denominator division.

     $$\frac{24}{3}=8$$

269. Write the reduced fraction.

     $$\frac{253}{8}$$

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

     $$\frac{759\times 1}{1\times 4}$$

271. Calculate these two products separately.

     $$\frac{759}{4}$$

272. Calculate this operation before combining the remaining terms.

     $$759\times \frac{1}{4}=\frac{759}{4}$$

273. Calculate this operation before combining the remaining terms.

     $$\frac{759}{4}\times \frac{1}{6}=\frac{253}{8}$$

274. Calculate this operation before combining the remaining terms.

     $${\left(\frac{67}{12}\right)}^2=\frac{4489}{144}$$

275. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 4489=-4489$$

276. Calculate this operation before combining the remaining terms.

     $$1\times 144=144$$

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

     $$\frac{\left(-4489\right)\times 1}{1\times 144}$$

278. Calculate this operation before combining the remaining terms.

     $$\left(-4489\right)\times \frac{1}{144}=-\frac{4489}{144}$$

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

     $$\frac{\left(-1\right)\times 4489}{1\times 144}$$

280. Calculate these two products separately.

     $$-\frac{4489}{144}$$

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

     $$\frac{4489\times 1}{1\times 144}$$

282. Calculate these two products separately.

     $$\frac{4489}{144}$$

283. Calculate this operation before combining the remaining terms.

     $$4489\times \frac{1}{144}=\frac{4489}{144}$$

284. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{4489}{144}=-\frac{4489}{144}$$

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

     $$\mathrm{LCD}=144$$

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

     $$\frac{253}{8}=\frac{253\times 18}{8\times 18}$$

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

     $$\frac{4554}{144}$$

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

     $$-\frac{4489}{144}=\frac{\left(-4489\right)\times 1}{144\times 1}$$

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

     $$-\frac{4489}{144}$$

290. Calculate this operation before combining the remaining terms.

     $$4554-4489=65$$

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

     $$\frac{65\times 1}{1\times 144}$$

292. Calculate these two products separately.

     $$\frac{65}{144}$$

293. Calculate this operation before combining the remaining terms.

     $$65\times \frac{1}{144}=\frac{65}{144}$$

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

     $$\frac{4554-4489}{144}=\frac{65}{144}$$

295. Calculate this operation before combining the remaining terms.

     $$\frac{253}{8}-\frac{4489}{144}=\frac{65}{144}$$

296. Calculate this operation before combining the remaining terms.

     $$\sqrt{\frac{65}{144}}=\frac{1}{12}\times \sqrt{65}$$

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

     $$\sqrt{\frac{65}{144}}$$

298. Round the standard deviation only now.

     $$0.671855$$

299. Group B: mean=5.58333, standard deviation=0.67185.

300. 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.