Organizing Data into a Frequency Distribution: A Guide for High School Mathematics

As an excellent high school mathematics teacher, I often find myself guiding students through the process of organizing and analyzing data. One particularly important technique is the construction of a frequency distribution, which can provide valuable insights into the underlying patterns and characteristics of a data set.

In this article, we will explore the steps involved in creating a frequency distribution for a set of 53 observations, where the minimum value is 42 and the maximum value is 129.

Determining the Number of Classes

The first step in creating a frequency distribution is to decide on the number of classes, or intervals, into which the data will be divided. The choice of the number of classes can have a significant impact on the clarity and usefulness of the distribution.

To determine the optimal number of classes, we can use the following formula:

Number of classes = 1 + 3.322 log(n)

Where n is the total number of observations in the data set.

Plugging in the values for our data set, we get:

Number of classes = 1 + 3.322 log(53)
Number of classes = 1 + 3.322 × 1.724
Number of classes ≈ 7.72

Since we cannot have a fractional number of classes, we will round this up to 8 classes.

Constructing the Frequency Distribution

Now that we have determined the number of classes, we can proceed to construct the frequency distribution table. To do this, we need to follow these steps:

1. Determine the class intervals: We need to divide the range of the data (from the minimum value of 42 to the maximum value of 129) into 8 equal-sized intervals. To calculate the class width, we use the formula:

   Class width = (Maximum value - Minimum value) / Number of classes
   Class width = (129 - 42) / 8
   Class width = 87 / 8
   Class width ≈ 10.875

Since we cannot have a fractional class width, we will round this up to 11.

The class intervals will be:

• 42 - 52
• 53 - 63
• 64 - 74
• 75 - 85
• 86 - 96
• 97 - 107
• 108 - 118
• 119 - 129

1. Tally the frequency of each class: For each observation in the data set, determine which class interval it falls into and keep a tally of the number of observations in each class.

2. Construct the frequency distribution table: The frequency distribution table will have the following columns:

• Class interval
• Frequency (number of observations in each class)
• Relative frequency (frequency divided by the total number of observations)
• Cumulative frequency (running total of the frequencies)
• Cumulative relative frequency (running total of the relative frequencies)

The completed frequency distribution table for our data set would look something like this:

| Class Interval | Frequency | Relative Frequency | Cumulative Frequency | Cumulative Relative Frequency | |----------------|------------|-------------------|---------------------|------------------------------| | 42 - 52 | 5 | 0.094 | 5 | 0.094 | | 53 - 63 | 8 | 0.151 | 13 | 0.245 | | 64 - 74 | 12 | 0.226 | 25 | 0.472 | | 75 - 85 | 10 | 0.189 | 35 | 0.660 | | 86 - 96 | 8 | 0.151 | 43 | 0.811 | | 97 - 107 | 6 | 0.113 | 49 | 0.924 | | 108 - 118 | 3 | 0.057 | 52 | 0.981 | | 119 - 129 | 1 | 0.019 | 53 | 1.000 |

This frequency distribution table provides a clear and concise summary of the data, allowing us to identify the number of observations in each class, the relative frequency of each class, and the cumulative frequency and relative frequency distributions.

With this information, we can gain valuable insights into the characteristics of the data set, such as the central tendency, dispersion, and potential outliers. This knowledge can then be used to inform further data analysis and decision-making processes.