Grouped Frequency Distribution

A Grouped Frequency Distribution organises large datasets into classes (intervals) with their corresponding frequencies, making it easier to analyze and interpret data trends. This method is essential for summarising raw data in business, economics, and statistics.

Key Concepts Related to Group Frequency

1. Class Interval

A class interval groups data into ranges. For example, ages grouped as 20–29, 30–39, etc.

2. Class Frequency

Class frequency indicates how many data points or number of observations fall within a class interval. To determine the Class Frequencies note that classes must be mutually exclusive.

3. Total Frequency

The sum of all class frequencies.

4. Class Limit

For discrete variables, values can be isolated from whole numbers. For example, if you are interested in how many pieces of cloth were sold by a shop last month, the data may be 200, 2000 or 2 units but cannot take any fractional number. Therefore, Class Limits can apply to discrete variables.

• Lower-Class Limit: Smallest value in a class (e.g., 20 in 20–29)

• Upper-Class Limit: Largest value in a class (e.g., 29 in 20–29)

Note: Class Limits are occasionally established by rounding the values of a continuous variable to the nearest Class Limit for theoretical and practical ease. For instance, you wish to document the percentage scores of 10th-grade students to the closest whole number (e.g., 60, 85, 75, 90) or the nearest tenth decimal place (e.g., 59.8, 85.2, 75.0, 91.45). In this case, ensure that all observations are consistently rounded.

5. Class Boundaries

For continuous variables, all values are recorded to a closest certain unit (for example, ages are, in general, recorded, as 20 years, 24 years etc.) and to a certain decimal place (for example, age of 20 years 10 months can be recorded as 20.1 years). In such cases, we need to adjust limits to avoid gaps between the Upper-Class Limit and the Lower Class Limit of the proceeding class. The difference (d) between the Upper-Class Limit and the Lower Class Limit of the next class, is equally distributed among the limits.

• Lower Boundary: Lower limit − 0.5d

• Upper Boundary: Lower limit + 0.5d

  
  

6. Class Mark (Midpoint)

It represents the exact mid-value or mid-point of a Class Interval and is often used as the representative of the Class Interval. It can be calculated as:

Class Mark = (Lower limit + Upper limit) / 2​

7. Class Width

Class Width is the size of a class and is calculated by the difference between consecutive lower limits:

Class Width = Upper limit − Lower limit

Note: •When all the Class Limits have the same value, we can think of them as Class Width. When some Class Limits are not equal, the values of Class Width are different. For instance, if a class contains either too few (outliers) or no observations, they are merged with adjacent classes.

8. Frequency Density

Frequency per unit width is called Frequency Density. It is often used when class widths vary.

Frequency Density = Class Frequency / Class Width

Note: It shows how the frequencies are concentrated in a particular class.

9. Relative Frequency

It measures the proportion of the class frequency to the total frequency and is calculated as:

Relative Frequency = Class Frequency / Total Frequency

🔢 Numerical Example

Let's consider the following data on the ages of 50 people:

Age Group (Years)	Frequency (f)
20–29	8
30–39	15
40–49	12
50–59	10
60–69	5
Total	50

Class Interval	Class Limits	Class Boundaries	Class Mark	Frequency (f)	Class Width	Frequency Density	Relative Frequency
20–29	20 - 29	19.5 – 29.5	24.5	8	10	0.8	0.16
30–39	30 - 39	29.5 – 39.5	34.5	15	10	1.5	0.3
40–49	40 - 49	39.5 – 49.5	44.5	12	10	1.2	0.24
50–59	50 - 59	49.5 – 59.5	54.5	10	10	1	0.2
60–69	60 - 69	59.5 – 69.5	64.5	5	10	0.5	0.1

We can see that the age group 30–39 has the highest frequency (15 individuals) while Relative frequency shows that 30% of individuals are aged 30–39. Using Frequency Density, we can also draw a Frequency Polygon (Figure 3).

Figure 3: Frequency Distribution of Age Distribution