## Frequency Distribution

Frequency Distribution

A frequency distribution is a tabular arrangement of data whereby the data is grouped into different intervals, and then the number of observations that belong to each interval is determined. Data that is presented in this manner are known as grouped data.

The smallest value that can belong to a given interval is called the lower class limit, while the largest value that can belong to the interval is called the upper class limit. The difference between the upper class limit and the lower class limit is defined to be the class width. When designing the intervals to be used in a frequency distribution, it is preferable that the class widths of all intervals be the same.

The relative frequency distribution and percentage frequency distribution are variants of the frequency distribution. The relative frequency distribution is similar to the frequency distribution, except that instead of the number of observations belonging to a particular interval, the ratio of the number of observations in the interval to the total number of observations, also known as the relative frequency, is determined. The percentage frequency distribution is arrived at by multiplying the relative frequencies of each interval by 100%.

The cumulative frequency distribution is obtained by computing the cumulative frequency, defined as the total frequency of all values less than the upper class limit of a particular interval, for all intervals. From a frequency distribution, this can be done by simply adding together the frequencies of the interval and all other preceding intervals (i.e., intervals whose values are less than the values of a particular interval). We can also calculate the relative cumulative frequency distribution and the percentage cumulative frequency distribution from the cumulative frequency distribution.

EX. Given the following set of measurements for a particular sample:

2.5 | 5.9 | 3.2 | 1.4 | 7.0 | 4.3 | 8.9 | 0.7 | 4.2 | 9.9 |

3.4 | 4.6 | 5.0 | 6.4 | 1.1 | 9.2 | 7.7 | 0.9 | 4.0 | 2.3 |

5.6 | 2.2 | 3.1 | 4.7 | 5.5 | 6.6 | 1.9 | 3.9 | 6.1 | 5.2 |

8.2 | 3.3 | 2.2 | 5.8 | 4.1 | 3.8 | 1.2 | 6.8 | 9.5 | 0.8 |

we note that the values range from 0 to 10.0. Therefore, we can create the following 10 classes:

class 1: | 0 - 1.0 | class 6: | 5.0 - 6.0 |

class 2: | 1.0 - 2.0 | class 7: | 6.0 - 7.0 |

class 3: | 2.0 - 3.0 | class 8: | 7.0 - 8.0 |

class 4: | 3.0 - 4.0 | class 9: | 8.0 - 9.0 |

class 5: | 4.0 - 5.0 | class 10: | 9.0 - 10.0 |

We assume that a measurement that falls on the border between two intervals belongs to the previous interval (e.g. the value 4.0 belongs to class 4 instead of class 5). By counting the number of observations that fall into each class, we get the following frequency distribution:

Measurements | # of obj. | Relative Frequency | Percentage Frequency |

0.0 - 1.0 | 3 | 0.075 | 7.5 % |

1.0 - 2.0 | 4 | 0.100 | 10.0 % |

2.0 - 3.0 | 4 | 0.100 | 10.0 % |

3.0 - 4.0 | 7 | 0.175 | 17.5 % |

4.0 - 5.0 | 6 | 0.150 | 15.0 % |

5.0 - 6.0 | 5 | 0.125 | 12.5 % |

6.0 - 7.0 | 5 | 0.125 | 12.5 % |

7.0 - 8.0 | 1 | 0.025 | 2.5 % |

8.0 - 9.0 | 2 | 0.050 | 5.0 % |

9.0 - 10.0 | 3 | 0.075 | 7.5 % |

Totals | 40 | 1.000 | 100.0 % |

In the interval 4.0 - 5.0 of the above frequency distribution, 4.0 is the lower class limit, while 5.0 is the upper class limit (Actually, 4.0 belongs to the previous interval, but since any measurement just slightly greater than 4.0 falls into the 4.0 - 5.0 class, 4.0 is essentially the lower limit of the interval). The class width is 5.0 - 4.0 = 1.0; in fact, the class widths of all the intervals are 1.0, as is preferred.

The following table shows the cumulative frequency distribution for the above set of measurements:

Measurements | Cumulative Frequency | Relative Cumulative Frequency | Percentage Cumulative Frequency |

0.0 - 1.0 | 3 | 0.075 | 7.5 % |

1.0 - 2.0 | 7 | 0.175 | 17.5 % |

2.0 - 3.0 | 11 | 0.275 | 27.5 % |

3.0 - 4.0 | 18 | 0.450 | 45.0 % |

4.0 - 5.0 | 24 | 0.600 | 60.0 % |

5.0 - 6.0 | 29 | 0.725 | 72.5 % |

6.0 - 7.0 | 34 | 0.850 | 85.0 % |

7.0 - 8.0 | 35 | 0.875 | 87.5 % |

8.0 - 9.0 | 37 | 0.925 | 92.5 % |

9.0 - 10.0 | 40 | 1.000 | 100.0 % |

Totals | 40 | 1.000 | 100.0 % |

Frequency distribution tables can also be utilized for qualitative data. Since qualitative data is not ordinal (e.g. can be ordered), the concept of cumulative frequency does not apply when dealing with qualitative data.

EX. The following table illustrates the use of frequency distributions with qualitative data.

Favorite Color | # of persons | Relative Frequency | PercentageFrequency |

RED | 16 | 0.2000 | 20.00 % |

BLUE | 18 | 0.2250 | 22.50 % |

YELLOW | 10 | 0.1250 | 12.50 % |

GREEN | 9 | 0.1125 | 11.25 % |

ORANGE | 13 | 0.1625 | 16.25 % |

BLACK | 6 | 0.0750 | 7.50 % |

others | 8 | 0.1000 | 10.00 % |

Totals | 80 | 1.0000 | 100.00 % |