# The Measurement of Adolescents, II

## CONTENTS OF CURRICULUM UNIT 86.05.03

## AN INTRODUCTION TO ELEMENTARY STATISTICS

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## Cumulative Frequency Polygon

*cumulative*

*frequency*

*polygon*. The cumulative frequency polygon is a line graph which is used to picture cumulative frequencies of a set of numbers. The abscissa of each point is the upper limit of an interval in a frequency distribution. The ordinate of each point is the corresponding cumulative frequency. The graph starts at a frequency of zero for a group below the lowest interval in the distribution.

###
*
Exam ple
*
:

Construct a cumulative frequency polygon to represent the following scores obtained by 40 students on a mathematics test.
86 | 82 | 56 | 73 | 87 | 89 | 72 | 86 | 88 | 76 |

72 | 69 | 84 | 85 | 62 | 97 | 70 | 78 | 84 | 93 |

70 | 60 | 91 | 76 | 83 | 94 | 65 | 72 | 92 | 81 |

98 | 78 | 88 | 76 | 96 | 89 | 90 | 83 | 74 | 80 |

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Solution
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:

Make a frequency distribution for the scores, then draw the graph.
Cumulative | % of Cumulative | ||

Scores | Frequency | Frequency | Frequency |

96-100 | 3 | 40 | 100.0 |

9195 | 4 | 37 | 92.5 |

8690 | 8 | 33 | 82.5 |

8185 | 7 | 25 | 62.5 |

7680 | 6 | 18 | 45.0 |

7175 | 5 | 12 | 30.0 |

6670 | 3 | 7 | 17.5 |

6165 | 2 | 4 | 10.0 |

5660 | 2 | 2 | 5.0 |

For some purposes the cumulative frequency polygon is very valuable. On the right side of the polygon is a scale of percent that parallels the scale of cumulative frequency. On the percent scale you read 25 corresponding to an abscissa of 72. This means that 25% of the scores were 72 or lower. The figure 72 is called the 25th
*
percentile
*
. The nth percentile is that score below which n percent of the scores in the distribution will fall.

To find the score that corresponds to a percentile on the graph, draw a horizontal line through the desired percent to intersect the cumulative frequency polygon. From the point of intersection draw a vertical line to the x-axis. The score at the point of intersection of the vertical line and the x-axis corresponds to the required percentile.

The fiftieth percentile is the
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median
*
or middle score in a set of measurements. The 25th percentile is called the
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lower
*
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quartile
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, and the 75th percentile is the
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upper
*
*
quartile
*
.

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Exercises
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:

a.) Construct a histogram to represent the given data.

- 1.) During one week a dealer sold the following number of cars: Monday 12, Tuesday 15, Wednesday 5, Thursday 6, Friday 10, Saturday 12.

b.) Make a frequency polygon to represent the given data.

c.) Draw a cumulative frequency polygon to represent the given data.

- 2.) The heights in inches of 50 high school students are:

60 | 68 | 74 | 79 | 62 | 75 | 60 | 65 | 61 | 64 |

71 | 72 | 63 | 66 | 71 | 60 | 60 | 73 | 63 | 65 |

73 | 68 | 76 | 75 | 62 | 76 | 72 | 70 | 69 | 62 |

78 | 71 | 68 | 62 | 74 | 69 | 67 | 70 | 61 | 63 |

72 | 67 | 71 | 68 | 62 | 60 | 70 | 69 | 65 | 64 |

b.) Construct a histogram to represent the data.

c.) Construct a cumulative frequency polygon.

d.) Find the median height. Find the upper and lower quartiles.

e.) Determine the 8Oth percentile.

- 3.) Forty students have the following IQ scores:

120 | 100 | 115 | 126 | 82 | 108 | 114 | 95 |

150 | 92 | 140 | 88 | 98 | 116 | 134 | 138 |

98 | 87 | 110 | 92 | 106 | 96 | 126 | 102 |

80 | 82 | 100 | 128 | 110 | 100 | 118 | 84 |

88 | 98 | 94 | 85 | 124 | 90 | 80 | 112 |

b.) Construct a cumulative frequency polygon.

c.) Determine the median IQ score and the 7Oth percentile.