Frequency is the measure of how how often an event appears. The frequency distribution table is a tool which can help in comparing the frequencies of different events occurrence. There are two type of frequency distributions which are used in statistics: grouped frequency distribution and ungrouped frequency distribution. The first one is utilized for a high quantity of data, when it is impossible to list all of them in the table (“Grouped Frequency Distributions”, 2016).

In order to develop a group frequency distribution, the data is grouped to several classes. The next five rules should be followed in order to create classes (“Grouped Frequency Distributions”, 2016):

- The boundaries of the class should not overlap each other;
- Each data should be included in any of the classes;
- The boundaries between any neighbor classes should not have any gaps;
- The length of each class mush be the same;
- The length of the class should be odd.

The next steps are to be accomplished in order to create a grouped frequency distribution (“Statistics: Grouped Frequency Distributions”, 2016):

- The larges and the smallest values should be determined;
- The range, equaled to the difference between the maximum and the minimum values, should be calculated;
- The amount of desired classes should be selected;
- The length of the class should be found as a ratio of the range (from the second step) and the number of classes (from the third step);
- The staring point, less or equaled to the minimum value (from the first step) should be selected for the first class. In order to determine the starting points for the further classes the length of the class should be added to the starting point of the previous class;
- The upper limit of the class should be calculated by subtracting 1 from the lower limit of the further class;
- The data should be tallied;
- The frequencies are to be calculated;
- The cumulative frequencies are to be estimated;
- The relative frequencies and relative cumulative frequencies should be found.

**Problem #1.** The heart rate measurements were performed in one of the health care center in order to find the resting heart rate for Men. The data is presented in the table below.

Man’s number | Resting heart rate | Man’s number | Resting heart rate | Man’s number | Resting heart rate | Man’s number | Resting heart rate | Man’s number | Resting heart rate |
---|---|---|---|---|---|---|---|---|---|

#1 | 62 | #10 | 53 | #19 | 74 | #28 | 59 | #37 | 55 |

#2 | 61 | #11 | 64 | #20 | 54 | #29 | 68 | #38 | 58 |

#3 | 51 | #12 | 75 | #21 | 65 | #30 | 77 | #39 | 78 |

#4 | 74 | #13 | 81 | #22 | 76 | #31 | 78 | #40 | 67 |

#5 | 82 | #14 | 52 | #23 | 51 | #32 | 57 | #41 | 68 |

#6 | 49 | #15 | 63 | #24 | 72 | #33 | 56 | #42 | 69 |

#7 | 57 | #16 | 74 | #25 | 66 | #34 | 76 | #43 | 58 |

#8 | 66 | #17 | 53 | #26 | 57 | #35 | 81 | #44 | 64 |

#9 | 78 | #18 | 64 | #27 | 68 | #36 | 64 | #45 | 65 |

**Solution:**

Step #1. The minimum value of Resting heart rate is: 49 beats per minute. The maximum value of Resting heart rate is: 82 beats per minute.

Step #2. The Range is: R=82-49+1=34

Step #3. The width of classes is 4

Step #4. The lowest apparent limit is 48.

Step #5. The number of groups is: 34/5=7.

The results of calculations are shown in the Grouped frequency distribution table below.

Interval | Real or exact limits | Mid-point | f | p | % | Cf | Cp | C% |
---|---|---|---|---|---|---|---|---|

78-82 | 77.5-82.5 | 80 | 6 | 6/45=0.133 | 13,3 | 45=N | 1 | 100 |

73-77 | 72.5-77.5 | 75 | 7 | 0.156 | 15,6 | 39 | 0.867 | 86,7 |

68-72 | 67.5-72.5 | 70 | 5 | 0.111 | 11,1 | 32 | 0.711 | 71,1 |

63-67 | 62.5-67.5 | 65 | 10 | 0.222 | 22,2 | 27 | 0.6 | 60 |

58-62 | 57.5-62.5 | 60 | 5 | 0.111 | 11,1 | 17 | 0.378 | 37,8 |

53-57 | 52.5-57.5 | 55 | 8 | 0,178 | 17,8 | 12 | 0.267 | 26,7 |

48-52 | 47.5-52.5 | 50 | 4 | 0,089 | 8,9 | 4 | 0.089 | 8,9 |

Σf=45 | Σp=1 | Σ%=100 |

**Problem #2.** The grades which students gained after passing the exam are shown in the table below. Create the Grouped frequency distribution table.

68 | 78 | 94 | 54 | 67 | 77 | 96 | 100 | 97 | 86 |

87 | 76 | 77 | 98 | 99 | 78 | 65 | 64 | 79 | 90 |

80 | 79 | 72 | 71 | 64 | 62 | 53 | 100 | 65 | 84 |

**Solution:** The same steps as in the previous problem were accomplished.

- min=53; max=100
- Range: 100-53=47
- The width of the classes: 5
- The lowest apparent limit is 50
- The number of groups is: 47/5=10

The results of calculations are shown in the Grouped frequency distribution table below.

Interval | Real or exact limits | Mid-point | f | p | % | Cf | Cp | C% |
---|---|---|---|---|---|---|---|---|

95-100 | 94.5-100.5 | 97.5 | 7 | 0.233 | 23,3 | 30=N | 1 | 100 |

89-94 | 88.5-94.5 | 91.5 | 2 | 0.067 | 6,7 | 23 | 0,767 | 76.7 |

83-88 | 82.5-88.5 | 85.5 | 3 | 0.100 | 10 | 21 | 0,700 | 70 |

77-82 | 76.5-82.5 | 79.5 | 7 | 0.233 | 23,3 | 18 | 0,600 | 60 |

71-76 | 70.5-76.5 | 73.5 | 3 | 0.100 | 10 | 11 | 0,367 | 36.7 |

65-70 | 65.5-70.5 | 67.5 | 4 | 0.133 | 13,3 | 8 | 0,267 | 26.7 |

59-64 | 58.5-64.5 | 61.5 | 2 | 0.067 | 6,7 | 4 | 0,133 | 13.3 |

53-58 | 52.5-58.5 | 55.5 | 2 | 0.067 | 6,7 | 2 | 0,067 | 6.7 |

Σf=30 | Σp=1 | Σ%=100 |

**References**

Grouped Frequency Distributions. (2016). Mathsolutions.50webs.com. Retrieved 18 May 2016, from http://mathsolutions.50webs.com/freqdist.html

Statistics: Grouped Frequency Distributions. (2016). People.richland.edu. Retrieved 18 May 2016, from https://people.richland.edu/james/lecture/m170/ch02-grp.html

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