Understanding Frequency Distributions

In statistics, a frequency distribution helps organize data by showing the frequency of each value or range of values. There are different ways to present this information, such as simple frequency distribution and cumulative frequency distribution.

Cumulative vs. Simple Frequency

A simple frequency for a class interval tells us exactly how many data points fall within that specific range. A cumulative frequency, on the other hand, tells us the total number of data points that are less than or equal to the upper limit of a given class interval. It is the sum of the frequencies of that class and all classes before it.

To find the simple frequency from a cumulative frequency distribution, we subtract the cumulative frequency of the previous class from the cumulative frequency of the current class.

Converting Cumulative Frequency to Simple Frequency

The formula to calculate the simple frequency ($F_i$) for the $i$-th class from its cumulative frequency ($C_i$) and the cumulative frequency of the previous class ($C_{i-1}$) is:

\(F_i = C_i - C_{i-1}\)

For the first class, the simple frequency is equal to its cumulative frequency ($F_1 = C_1$).

Step-by-Step Calculation of Simple Frequencies

We are given the following cumulative frequency distribution:

Let's calculate the simple frequency for each class:

• Class 60-62: This is the first class. The simple frequency is equal to the cumulative frequency.
• \(F_{60-62} = 3\)

• Class 63-65: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (60-62).
• \(F_{63-65} = C_{63-65} - C_{60-62} = 20 - 3 = 17\)

• Class 66-68: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (63-65).
• \(F_{66-68} = C_{66-68} - C_{63-65} = 36 - 20 = 16\)

• Class 69-71: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (66-68).
• \(F_{69-71} = C_{69-71} - C_{66-68} = 48 - 36 = 12\)

• Class 72-74: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (69-71).
• \(F_{72-74} = C_{72-74} - C_{69-71} = 50 - 48 = 2\)

Analyzing Class Frequencies

Now we have the simple frequencies for each class interval:

Identifying the Maximum Frequency Class

We compare the simple frequencies calculated for each class: 3, 17, 16, 12, and 2. The maximum value among these frequencies is 17. This maximum frequency corresponds to the class interval 63-65.

Conclusion

The class with the maximum frequency is 63-65, as it has a simple frequency of 17, which is the highest among all the given class intervals.

Revision Table: Frequency Concepts

Additional Information: Types of Frequency Distributions

Frequency distributions can be presented in various ways:

• Ungrouped Frequency Distribution: Lists each distinct value in the data and its frequency. Suitable for discrete data with a small range of values.
• Grouped Frequency Distribution: Data is grouped into class intervals, and the frequency for each interval is listed. Suitable for continuous data or discrete data with a large range.
• Relative Frequency Distribution: Shows the proportion or percentage of observations in each class. Calculated as (Simple Frequency / Total Number of Observations).
• Cumulative Relative Frequency Distribution: Shows the proportion or percentage of observations less than or equal to the upper limit of each class.