Consider the following for the next two (02) items that follow :
A frequency distribution is as follows:
Marks 18-26 27-35 36-44 45-53 54-62 63-71 72-80 Number
of
students5 7 10 15 8 3 2
Finding the Mode of a Frequency Distribution
This solution explains how to find the mode for a given frequency distribution. The mode is the value that appears most frequently in a data set. For grouped data, we identify the modal class and use a specific formula.
Frequency Distribution Data
The provided frequency distribution of marks and the number of students is:
| Marks Interval | Number of Students (Frequency) |
| 18-26 | 5 |
| 27-35 | 7 |
| 36-44 | 10 |
| 45-53 | 15 |
| 54-62 | 8 |
| 63-71 | 3 |
| 72-80 | 2 |
Identifying the Modal Class
First, we need to find the class with the highest frequency. This is called the modal class.
- The frequencies are 5, 7, 10, 15, 8, 3, 2.
- The highest frequency is 15.
- The marks interval corresponding to the frequency 15 is 45-53.
Therefore, the modal class is 45-53.
Formula for Mode of Grouped Data
The formula used to calculate the mode (\(M_o\)) for a grouped frequency distribution is:
\(M_o = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h\)Where:
- \(L\) = Lower boundary of the modal class.
- \(f_1\) = 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.
- \(h\) = Class width (size of the class interval).
Calculating the Mode
Let's determine the values for the formula based on our data:
- Modal Class: 45-53
- \(L\) (Lower boundary): The lower limit is 45. To find the lower boundary, we subtract 0.5 from the lower limit (assuming continuous classes). So, \(L = 45 - 0.5 = 44.5\).
- \(f_1\) (Frequency of modal class): The frequency corresponding to 45-53 is 15. So, \(f_1 = 15\).
- \(f_0\) (Frequency of preceding class): The class before 45-53 is 36-44, and its frequency is 10. So, \(f_0 = 10\).
- \(f_2\) (Frequency of succeeding class): The class after 45-53 is 54-62, and its frequency is 8. So, \(f_2 = 8\).
- \(h\) (Class width): The width of each class interval is 9 (e.g., \(53 - 45 + 1 = 9\), or \(26.5 - 17.5 = 9\)). So, \(h = 9\).
Now, substitute these values into the mode formula:
\(M_o = 44.5 + \frac{(15 - 10)}{(2 \times 15 - 10 - 8)} \times 9\)Step 1: Calculate the numerator \((f_1 - f_0)\):
\(f_1 - f_0 = 15 - 10 = 5\)Step 2: Calculate the denominator \((2f_1 - f_0 - f_2)\):
\(2f_1 - f_0 - f_2 = (2 \times 15) - 10 - 8 = 30 - 10 - 8 = 12\)Step 3: Substitute these values back into the formula:
\(M_o = 44.5 + \frac{5}{12} \times 9\)Step 4: Calculate the fraction part:
\(\frac{5}{12} \times 9 = \frac{45}{12}\)Simplify the fraction:
\(\frac{45}{12} = \frac{15}{4} = 3.75\)Step 5: Add the result to the lower boundary \(L\):
\(M_o = 44.5 + 3.75\) \(M_o = 48.25\)Conclusion
The calculated mode for the given frequency distribution is 48.25.
Similar Questions
What is the median of the distribution ?
- Consider the following data :
110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements:
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0·1.
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct? - Which measure of central tendency is least affected by the presence of extreme observations in the data?
- To find the average ratio like price/unit, work done/hour, kilometre/hour under certain conditions, the suitable measure of central tendency applicable is
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Marks Number of Candidates More than 10 100 More than 20 75 More than 30 60 More than 40 40 - The arithmetic mean of 200 observations is 60. If 5 is multiplied to each observation, then what will be the new arithmetic mean?
- A distribution consists of 3 components with frequencies 45, 40 and 55 having their means 2, 2.5 and 2 respectively. What is the mean of the combined distribution?
- Which one of the following is a positional average?
- What is the value of \(u\)?
- What is the value of \(v\)?
Important Questions from Elementary Statistics
In a colony 5 families have 1 child, 7 families have 2 children, 8 families have 3 children and 3 families have 4 children.What is the mode of the number of children.
For which set of numbers do the mean, median and mode all have the same value?
The median of 5, 8, 25, 22, 34, 18 is
Find the mean of first 5 two-digit multiples of 4.
The difference between the upper limit and the lower limit of a class is called :
