3.3: Frequency Tables and Relative Frequency

One of the simplest and most powerful ways to start summarizing raw data is by creating a frequency table which shows how many values fall into specific groups. With categorical data, we simply count the number of members in each category. When working with quantitative data, we often group values into bins or ranges of values.

Definition: Frequency

In statistics, frequency refers to the number of times a particular value or group of values appears in a dataset.

Example: Soil pH Levels

Suppose a team of agricultural scientists took pH readings from 30 different soil samples collected in the same region. pH is a measurement of acidity, with 7.0 representing neutral, lower values representing more acidic, and hight values representing more basic (alkaline). These readings can be used to help determine how well different crops might grow in those areas.

Soil pH Readings (n = 30):

5.7, 5.9, 6.0, 6.1, 6.1, 6.2, 6.3, 6.3, 6.4, 6.4,
6.4, 6.5, 6.5, 6.6, 6.6, 6.6, 6.7, 6.7, 6.8, 6.9,
7.0, 7.0, 7.0, 7.1, 7.2, 7.3, 7.4, 7.6, 7.8, 8.0
  

While this list is detailed, it’s hard to see patterns just by scanning the numbers. A frequency table helps organize the pH levels into intervals so we can spot concentrations, gaps, or outliers more easily.

Frequency Table: Soil pH Readings

Let’s build a frequency table by grouping the data into pH ranges (also known as intervals or bins). Since we’re working with quantitative values, we can’t treat each exact number as its own category as there are just too many individual values.

Instead, we group values into ranges of values, which we refer to as bins. The choice of bin isn't standardized. However, we do require each bin has the same length. Note in the below case we label the bins with one more significant figure than the data. If you are more experienced with algebra, note that we can also describe bins using intervals such as \([5.5, 6)\)

Grouped Frequency Distribution Table.

Soil pH Range	Frequency
5.50 – 5.99	2
6.00 – 6.49	9
6.50 – 6.99	9
7.00 – 7.49	7
7.50 – 7.99	2
8.00 – 8.49	1

Definition: Relative Frequency

Relative frequency is the proportion or percentage of the total observations that falls into each category or interval. It is calculated by dividing the frequency of a group by the total number of observations.

Extended Table: Frequency and Relative Frequency

To get a clearer picture of how these groups compare, we convert the counts in our frequency table into relative frequencies by dividing each by the total number of observations (30).

Grouped Relative Frequency Distribution Table.

Soil pH Range	Frequency	Relative Frequency	Percentage
5.50 – 5.99	2	2/30 ~ 0.067	6.7%
6.00 – 6.49	9	9/30 = 0.300	30.0%
6.50 – 6.99	9	9/30 = 0.300	30.0%
7.00 – 7.49	7	7/30 ~ 0.233	23.3%
7.50 – 7.99	2	2/30 ~ 0.067	6.7%
8.00 – 8.49	1	1/30 ~ 0.033	3.3%

Interpretation

From the table, we notice that most soil pH readings fall between 6.00 and 6.99 which add up to 60% of all the data. This suggests that most of the tested fields have soil condition in the “slightly acidic to neutral” range, which is typically optimal for many crops. Only 1 measurement exceeds 8.0, indicating a very alkaline sample.

Related Videos - Frequency Tables

Histograms

The following graph is called a histogram. Each row in the table becomes a bar in a histogram, with the x-axis showing the interval (bin) and the y-axis showing either frequency or relative frequency as bar height. This gives us a visual way to convey the same information as the frequency table.

Figure: Histogram of soil pH readings based on grouped frequency intervals.

The rough shape of this chart, with the majority of the data in the middle and less of the data to the periphery, is a common occurrence.

Step-by-Step: Building a Histogram

1. Collect your data​​​​​​
   Make sure you are working with quantitative (numerical data)
2. Find the minimum and maximum values
   This will help define where to group the values. In our soil samples, we had a minimum of 5.7 and a maximum of 8.0. Note that our data was sorted, however this may not be the case.
3. Create your bins
   In our case, we chose bins that were of width 0.5 and started at 5.5
   As this is a visualization tool, it is often best practice to make bins based on whole numbers or easy fractions.
   You can experiment with different bin widths to get the best picture of the data
   Very important: The bin widths must be the same length!
4. Construct a frequency chart
   Tally the number of values that fall into each bin
5. Plot!
   The height of each bar corresponds to the number of items in each bin
   Make sure the y-axis and x-axis tick marks are evenly spaced

Related Videos - Histograms