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Thirty pairs of values of two variables X and Y are given below. Form a bivariate frequency table. Also, find marginal frequency distributions of X and Y.

X |
110 | 88 | 91 | 115 | 97 | 85 | 85 | 91 | 120 | 95 |

Y |
500 | 800 | 870 | 599 | 625 | 650 | 905 | 700 | 850 | 824 |

X |
82 | 105 | 99 | 90 | 108 | 124 | 90 | 90 | 111 | 89 |

Y |
970 | 609 | 990 | 735 | 600 | 735 | 729 | 840 | 999 | 780 |

X |
112 | 100 | 87 | 92 | 91 | 82 | 96 | 120 | 121 | 122 |

Y |
638 | 850 | 630 | 720 | 695 | 923 | 555 | 810 | 805 | 526 |

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#### Solution

Bivariate frequency table can be prepared by taking class intervals 80 – 90, 90 – 100, …, etc for X and 500 – 600, 600 –700, …., etc for Y.

Bivariate frequency distribution is as follows:

Y/X |
80 – 90 |
90 – 100 |
100 – 110 |
110 – 120 |
120 – 130 |
Total (f_{y}) |

500 – 600 |
– | I (1) | – | II (2) | I (1) | 4 |

600 – 700 |
II (2) | II (2) | II (2) | I (1) | – | 7 |

700 – 800 |
I (1) | IIII (4) | – | – | I (1) | 6 |

800 – 900 |
I (1) | III (3) | I (1) | – | III (3) | 8 |

900 – 1000 |
III (3) | I (1) | – | I (1) | – | 5 |

Total (f_{x}) |
7 | 11 | 3 | 4 | 5 | 30 |

Marginal frequency distribution of X:

X |
80 – 90 | 90 – 100 | 100 – 110 | 110 – 120 | 120 – 130 | Total |

frequency |
7 | 11 | 3 | 4 | 5 | 30 |

Marginal frequency distribution of Y:

Y |
500 – 600 | 600 – 700 | 700 – 800 | 800 – 900 | 900 – 100 | Total |

Frequency |
4 | 7 | 6 | 8 | 5 | 30 |

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