# Data handling - calculating means and standard deviations

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Introduction

ASSIGNMENT 1I

Type I

Name:

Candidate #: [ ]

School:

October 2003

## Question 1

The table below shows the height for 60 students in centimeters:

## Table 1 | |||||

177 | 175 | 137 | 155 | 150 | 166 |

132 | 146 | 179 | 140 | 169 | 177 |

141 | 148 | 130 | 176 | 135 | 130 |

157 | 172 | 178 | 143 | 143 | 136 |

132 | 166 | 130 | 151 | 145 | 178 |

131 | 171 | 160 | 140 | 179 | 166 |

145 | 142 | 177 | 176 | 132 | 135 |

164 | 179 | 161 | 145 | 134 | 179 |

139 | 149 | 135 | 142 | 172 | 148 |

159 | 160 | 137 | 130 | 130 | 164 |

- The mean () is calculated using the equation below:

=

- The standard deviation () is calculated
^{[1]}* using the equation below:

= 17.08731661

Question 2

- The table below shows the height of 60 students after adding 5 cm to each height:

## Table 2 | |||||

182 | 180 | 142 | 160 | 155 | 171 |

137 | 151 | 184 | 145 | 174 | 182 |

146 | 153 | 135 | 181 | 140 | 135 |

162 | 177 | 183 | 148 | 148 | 141 |

137 | 171 | 135 | 156 | 150 | 183 |

136 | 176 | 165 | 145 | 184 | 171 |

150 | 147 | 182 | 181 | 137 | 140 |

169 | 184 | 166 | 150 | 139 | 184 |

144 | 154 | 140 | 147 | 177 | 153 |

164 | 165 | 142 | 135 | 135 | 169 |

- The mean () is calculated using the equation below:

=

- The standard deviation () is calculated
^{[2]}* using the equation below:

= 17.08731661

- The table below shows the height of 60 students after subtracting 12 cm from each height:

Table 3 | |||||

165 | 163 | 125 | 143 | 138 | 154 |

120 | 134 | 167 | 128 | 157 | 165 |

129 | 136 | 118 | 164 | 123 | 118 |

145 | 160 | 166 | 131 | 131 | 124 |

120 | 154 | 118 | 139 | 133 | 166 |

119 | 159 | 148 | 128 | 167 | 154 |

133 | 130 | 165 | 164 | 120 | 123 |

152 | 167 | 149 | 133 | 122 | 167 |

127 | 137 | 123 | 130 | 160 | 136 |

147 | 148 | 125 | 118 | 118 | 152 |

- The mean () is calculated using the equation below:

=

- The standard deviation () is calculated using the equation below:

= 17.08731661

Adding 'a' to a set of data will result in an increase in the mean () by 'a' while keeping the standard deviation () for the data set unchangeable. Subtracting 'a' from a set of data will result a decrease in the mean by 'a'. |

## Question 3

Middle

33.8

35.4

28.2

29.6

26.0

35.2

27.0

26.0

31.4

34.4

35.6

28.6

28.6

27.2

26.4

33.2

26.0

30.2

29.0

35.6

26.2

34.2

32.0

28.0

35.8

33.2

29.0

28.4

35.4

35.2

26.4

27.0

32.8

35.8

32.2

29.0

26.8

35.8

27.8

29.8

27.0

28.4

34.4

29.6

31.8

32.0

27.4

26.0

26.0

32.8

- The mean () is calculated using the equation below:

=

- The standard deviation () is calculated
^{[4]}* using the equation below:

= 3.417463322

Multiplying a set of data by 'a', where a>1 will result in the multiplication of the standard deviation () by 'a'. The standard deviation will be divided by 'a' if If 0<a<1.
The changes caused by the multiplication of a data set by negative value (a<0) are investigated below: |

The table below shows the same data set used in this question using a = - 3, where each value is multiplied by a;

Table 6 | |||||

-531 | -525 | -411 | -465 | -450 | -498 |

-396 | -438 | -537 | -420 | -507 | -531 |

-423 | -444 | -390 | -528 | -405 | -390 |

-471 | -516 | -534 | -429 | -429 | -408 |

-396 | -498 | -390 | -453 | -435 | -534 |

-393 | -513 | -480 | -420 | -537 | -498 |

-435 | -426 | -531 | -528 | -396 | -405 |

-492 | -537 | -483 | -435 | -402 | -537 |

-417 | -447 | -405 | -426 | -516 | -444 |

-477 | -480 | -411 | -390 | -390 | -492 |

- The mean () is calculated using the equation below:

=

- The standard deviation () is calculated using the equation below:

= 51.26194983

Conclusion

-22.9

-22.9

11.1

- The mean () is calculated using the equation below:

- =

Since the subtraction of 'a' from the data set values results in a subtraction of 'a' from the mean then the transformation is done by subtracting the mean which is 152.9 from each of the values of the data set. 152.9 – 152.9 = 0 |

(b) The table below shows the new data set which has a standard deviation=1

Table 17 | |||||

10.4 | 10.2 | 8.0 | 9.1 | 8.8 | 9.7 |

7.7 | 8.5 | 10.5 | 8.2 | 9.9 | 10.4 |

8.3 | 8.7 | 7.6 | 10.3 | 7.9 | 7.6 |

9.2 | 10.1 | 10.4 | 8.4 | 8.4 | 8.0 |

7.7 | 9.7 | 7.6 | 8.8 | 8.5 | 10.4 |

7.7 | 10.0 | 9.4 | 8.2 | 10.5 | 9.7 |

8.5 | 8.3 | 10.4 | 10.3 | 7.7 | 7.9 |

9.6 | 10.5 | 9.4 | 8.5 | 7.8 | 10.5 |

8.1 | 8.7 | 7.9 | 8.3 | 10.1 | 8.7 |

9.3 | 9.4 | 8.0 | 7.6 | 7.6 | 9.6 |

- The standard deviation () is calculated using the equation below:

= 1

Since dividing the data set values by 'a' results the dividing of the standard deviation by 'a' then the transformation is done by dividing each of the data set values by the original standard deviation which is 17.1 17.1 / 17.1 = 1 |

(c) The table below shows the new data set which has a standard deviation=1

Table 18 | |||||

1.4 | 1.3 | -0.9 | 0.1 | -0.2 | 0.8 |

-1.2 | -0.4 | 1.5 | -0.8 | 0.9 | 1.4 |

-0.7 | -0.3 | -1.3 | 1.4 | -1.0 | -1.3 |

0.2 | 1.1 | 1.5 | -0.6 | -0.6 | -1.0 |

-1.2 | 0.8 | -1.3 | -0.1 | -0.5 | 1.5 |

-1.3 | 1.1 | 0.4 | -0.8 | 1.5 | 0.8 |

-0.5 | -0.6 | 1.4 | 1.4 | -1.2 | -1.0 |

0.6 | 1.5 | 0.5 | -0.5 | -1.1 | 1.5 |

-0.8 | -0.2 | -0.1 | -0.6 | 1.1 | -0.3 |

0.4 | 0.4 | -0.9 | -1.3 | -1.3 | 0.6 |

- The mean () is calculated using the equation below:

- =

- The standard deviation () is calculated using the equation below:

= 1

From parts (a) and (b) it is concluded that subtracting the mean then dividing over the standard deviation can complete the transformation. |

[1]* Using Microsoft Excel function: STDEVP

[2]* Using Microsoft Excel function: STDEVP

[3]* Using Microsoft Excel function: STDEVP

[4]* Using Microsoft Excel function: STDEVP

[5]* Using Microsoft Excel functions: MEDIAN, QUARTILE

[6]* Using Microsoft Excel functions: MEDIAN, QUARTILE

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