# The Measurement of Adolescents, II

## Some More Statistical Exercises

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## Working with the Mean and Standard Deviation

It is to be hoped that calculators will be provided to do these calculations. I believe some learning is just doing the calculations oneself. If it is done by computer much of the point is lost. Once the work gets boring and we have seen the point then the computer can be used.

Here are some sets and some questions to be asked with each one.

Draw a histogram for each set and find the mean and standard deviation.

Example | Mean | Standard dev. | |

1 | 19,20,21 | 20 | 0.81649 |

2 | -1, 0, 1 | 0 | 0.81648 |

3 | 19,20,20,21 | 20 | 0.70711 |

4 | 38,40,42 | 40 | 1.63299 |

5 | 57,60,63 | 60 | 2.44949 |

6 | 19,19,20,20,21,21 | 20 | 0.81648 |

Example | Mean | Standard dev. | |

7 | 3,5,8 | 5.33 . . . | 2.05480 |

8 | 4,7,9 | 6.66 . . . | 2.05480 |

9 | 7,12,17 | 12.0 | 4.08248 |

10 | 12,20,21,27,32,35, | ||

45,45,56,72 | 35.55 . . . | 18.04384 |

After the students can find standard deviations tell them we may expect certain percentages of the data to fall within fixed numbers of standard deviations from the mean. Here would be the place to show the class a normal curve telling them the area of an interval under the curve represents the probability of a score falling in that interval. The probability is 68% that a score lies within plus or minus one standard deviation of the mean, 95% that a score is within plus or minus two standard deviations of the mean, and 99% that a score is within three standard deviations of the mean. I would stress that these percentages are dependent upon the distribution that is being considered.