Consider the following data for the next two (02) items that follow : Class 0-30 30-60 60-90 90-120 Frequency 4 5 7 4
What is the mode of the distribution?
72
Finding the Mode of a Grouped Data Distribution
The mode of a distribution is the value that appears most frequently. For grouped data, the mode is located within the class interval that has the highest frequency. This class is called the modal class.
Let's look at the given data:
| Class | 0-30 | 30-60 | 60-90 | 90-120 |
| Frequency | 4 | 5 | 7 | 4 |
To find the mode for this grouped frequency distribution, we first identify the modal class. The modal class is the class with the highest frequency. In this table, the highest frequency is 7, which corresponds to the class interval 60-90.
So, the modal class is 60-90.
Now, we use the formula for calculating the mode of grouped data:
Mode = \( L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \)
Where:
- \( L \) = Lower limit of the modal class
- \( f_1 \) = Frequency of the modal class
- \( f_0 \) = Frequency of the class preceding the modal class
- \( f_2 \) = Frequency of the class succeeding the modal class
- \( h \) = Class size (the width of the class interval)
From our data, the modal class is 60-90. Let's identify the values for the formula:
- \( L \) = 60 (Lower limit of 60-90)
- \( f_1 \) = 7 (Frequency of 60-90)
- \( f_0 \) = 5 (Frequency of 30-60, which precedes 60-90)
- \( f_2 \) = 4 (Frequency of 90-120, which succeeds 60-90)
- \( h \) = 90 - 60 = 30 (Class size)
Now, substitute these values into the mode formula:
Mode = \( 60 + \left( \frac{7 - 5}{2(7) - 5 - 4} \right) \times 30 \)
Mode = \( 60 + \left( \frac{2}{14 - 5 - 4} \right) \times 30 \)
Mode = \( 60 + \left( \frac{2}{14 - 9} \right) \times 30 \)
Mode = \( 60 + \left( \frac{2}{5} \right) \times 30 \)
Mode = \( 60 + \frac{2 \times 30}{5} \)
Mode = \( 60 + \frac{60}{5} \)
Mode = \( 60 + 12 \)
Mode = \( 72 \)
Thus, the mode of the distribution is 72.
Revision Table: Statistics Concepts
| Concept | Definition | Calculation for Grouped Data |
| Mean | The average of the data. | \( \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \) (where \(x_i\) is the class mark) |
| Median | The middle value when data is ordered. | \( \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h \) (where \(N = \sum f_i\), \(CF\) is cumulative frequency of preceding class, \(f\) is frequency of median class) |
| Mode | The value that occurs most frequently. | \( \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \) |
Additional Information: Measures of Central Tendency
Measures of central tendency are statistical values that describe the center point of a dataset. The most common measures are mean, median, and mode. Each measure provides a different perspective on where the center of the data lies.
- Mean: Affected by extreme values (outliers). Best used for symmetrical distributions.
- Median: Not affected by extreme values. Best used for skewed distributions or data with outliers.
- Mode: Represents the most frequent value. Can be used for all types of data, including categorical data where mean and median are not applicable. A distribution can have one mode (unimodal), two modes (bimodal), or more (multimodal).
Understanding how to calculate and interpret these measures is fundamental in statistics for summarizing and analyzing data distributions like the given frequency distribution.
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