Calculate the mode of the following frequency distribution Seconds 51-55 56-60 61-65 66-70 Frequency 2 7 8 4
61.5
Calculating the Mode of Grouped Frequency Distribution
The mode is a measure of central tendency that represents the most frequently occurring value in a dataset. For a grouped frequency distribution, the mode falls within the class interval that has the highest frequency, known as the modal class.
Identifying the Modal Class
Let's look at the given frequency distribution:
| Seconds | Frequency |
|---|---|
| 51-55 | 2 |
| 56-60 | 7 |
| 61-65 | 8 |
| 66-70 | 4 |
From the table, we can see that the highest frequency is 8, which corresponds to the class interval 61-65. Therefore, the modal class is 61-65.
Formula for Mode of Grouped Data
The formula used to calculate the mode for a grouped frequency distribution is:
\[ \text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \]Where:
- \(L\) is the lower boundary of the modal class.
- \(f_1\) is the frequency of the modal class.
- \(f_0\) is the frequency of the class preceding the modal class.
- \(f_2\) is the frequency of the class succeeding the modal class.
- \(h\) is the class size (width of the class interval).
Determining the Values
Based on the modal class 61-65 and the frequency distribution:
- The modal class is 61-65.
- The frequency of the modal class, \(f_1\), is 8.
- The class preceding the modal class is 56-60. Its frequency, \(f_0\), is 7.
- The class succeeding the modal class is 66-70. Its frequency, \(f_2\), is 4.
To find the lower boundary (\(L\)) and class size (\(h\)), we need to consider the class intervals. The intervals are 51-55, 56-60, 61-65, 66-70. There is a gap of 1 between the upper limit of one class and the lower limit of the next (e.g., 55 to 56, 60 to 61). To make the classes continuous, we adjust the boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.
- The adjusted modal class boundaries are 61 - 0.5 to 65 + 0.5, which is 60.5 to 65.5.
- The lower boundary of the modal class, \(L\), is 60.5.
- The class size, \(h\), is the difference between the upper and lower boundaries: 65.5 - 60.5 = 5. Alternatively, for inclusive intervals, \(h\) can be calculated as (Upper Limit - Lower Limit + 1) for any class, e.g., 55 - 51 + 1 = 5.
Step-by-Step Mode Calculation
Now, we substitute the values into the mode formula:
\[ L = 60.5 \] \[ f_1 = 8 \] \[ f_0 = 7 \] \[ f_2 = 4 \] \[ h = 5 \] \[ \text{Mode} = 60.5 + \left(\frac{8 - 7}{2(8) - 7 - 4}\right) \times 5 \] \[ \text{Mode} = 60.5 + \left(\frac{1}{16 - 7 - 4}\right) \times 5 \] \[ \text{Mode} = 60.5 + \left(\frac{1}{16 - 11}\right) \times 5 \] \[ \text{Mode} = 60.5 + \left(\frac{1}{5}\right) \times 5 \] \[ \text{Mode} = 60.5 + 1 \] \[ \text{Mode} = 61.5 \]The calculated mode of the frequency distribution is 61.5.
Revision Table: Key Terms for Mode Calculation
| Term | Definition/Meaning in Calculation |
|---|---|
| Mode | The value that appears most frequently in a data set. For grouped data, it's the peak of the distribution. |
| Modal Class | The class interval with the highest frequency. |
| Lower Boundary (\(L\)) | The actual lower limit of the modal class, adjusted for continuity (usually by subtracting 0.5 from the listed lower limit if there are gaps between classes). |
| Frequency (\(f\)) | The number of observations within a specific class interval. |
| \(f_1\) | Frequency of the modal class. |
| \(f_0\) | Frequency of the class immediately preceding the modal class. |
| \(f_2\) | Frequency of the class immediately succeeding the modal class. |
| Class Size (\(h\)) | The width of the class interval (difference between upper and lower boundaries). |
Additional Information: Understanding Central Tendency
The mode is one of the three main measures of central tendency, alongside the mean and the median. Each measure provides a different perspective on the center of a dataset:
- Mean: The average of all values. Calculated by summing all values and dividing by the number of values. Highly affected by extreme values (outliers).
- Median: The middle value in a dataset that has been ordered from least to greatest. It divides the data into two equal halves. Less affected by extreme values than the mean.
- Mode: The most frequent value. Useful for categorical data or when identifying the most typical value. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.
For grouped frequency distributions, the calculation methods for mean, median, and mode are slightly different from ungrouped data, involving class intervals and frequencies.
Similar Questions
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Class- interval
Frequency
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6
10-20
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50-60
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Important Questions from Measures of Central Tendency
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
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3
What is the mean age of this group of people?
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If the difference of mode and median is 36, then the difference of median and mean is:
In a Mathematics test 15 students scored 80 marks, 20 students scored 75 marks, 28 students scored 65 marks and 25 students scored 60 marks, mode of the score is:
If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
