Kindly check out the below video to learn about the concept of mean, median and mode. Watch the video till the end to complete the task.
Mean
The mean of the ungrouped frequency distribution can be determined using the formula:
\(\overline X = \frac{f_1 x_1 + f_2 x_2 + ... + f_n x_n}{f_1 + f_2 + ... + f_n}\) \(= \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i}\)
The mean of a grouped frequency distribution can be determined using any one of the following methods.
- Direct method
- Assumed mean method
- Step deviation method
Direct method:
The formula for finding the arithmetic mean using the direct method is given by:
\(\overline X = \frac{\sum f_ix_i}{\sum f_i}\)
Where \(i\) varies from \(1\) to \(n\), \(x_i\) is the midpoint of the class interval and \(f_i\) is the frequency.
Steps:
1. Calculate the midpoint of the class interval and name it as \(x_i\).
2. Multiply the midpoints\(x_i\) with the frequency\(f_i\) of each class interval and name it as \(f_ix_i\).
3. Find the values \(\sum f_ix_i\) and \(\sum f_i\).
4. Divide \(\sum f_ix_i\) by \(\sum f_i\) to determine the mean of the data.
Assumed mean method:
Consider if the data is very large and finding the products of the observations and then adding them becomes tedious and may result in errors. Let us use the assumed mean method to find the mean of grouped frequency to avoid such complications.
Steps:
1. Calculate the midpoint of the class interval and name it as \(x_i\).
2. From the data of \(x_i\), choose any value(preferably in the middle) as the assumed mean(\(a\)).
3. Determine the deviation \(d=x-a\) for each of the classes.
4. Multiply the deviation and frequency of each class interval and name it \(f_id_i\).
5. Find the values \(\sum f_id_i\) and \(\sum f_i\).
6. Calculate the mean by applying the formula \(\overline X = a + \frac{\sum f_id_i}{\sum f_i}\)
Step deviation method:
Let us consider the steps for finding the mean of grouped data using the step deviation method.
Steps:
1. Calculate the midpoint of the class interval and name it as \(x_i\).
2. From the data of \(x_i\), choose any value(preferably in the middle) as the assumed mean(\(a\)).
3. Determine the deviation (\(d = x_i - a\)) for each class.
4. Determine the deviation (\(u = \frac{x_i - a}{h}\) where \(h\) is the class size) for each class.
5. Multiply the frequency and \(u_i\) of each class interval and name it as \(f_iu_i\).
6. Calculate the mean by applying the formula \(\overline X = a + \left[\frac{\sum fd}{\sum f} \times h \right]\).
Median
Ungrouped frequency distribution
Let us recall what we learnt in class \(IX\) in the concept of median of ungrouped frequency distribution.
The median of an ungrouped frequency distribution can be determined using the following steps.
1. Arrange the given data in ascending or descending order.
2. Find the cumulative frequency distribution and denote \(N\) as the total frequency.
3. If \(N\) is odd, then median \(= \left(\frac{N + 1}{2} \right)^{th}\) term.
4. If \(N\) is even, then median \(= \left(\frac{(\frac{N}{2}^{th}) \text{observation} + (\frac{N}{2}+1)^{th} \text{observation}}{2} \right)\)
Grouped frequency distribution
The following steps can determine the median of the grouped frequency distribution:
1. Find the cumulative frequency distribution and denote \(n\) as the total frequency.
2. Find the \(\frac{n}{2}^{th}\) term.
3. The class which contains the cumulative frequency \(\frac{n}{2}\) is the median class.
4. The median of the class can be determined using the formula:
Median \(= l + \frac{(\frac{n}{2} - cf)}{f} \times h\)
Where \(l\) is the lower limit of the median class,
\(cf\) is the cumulative frequency of the class preceding the median class,
\(f\) is the frequency of the median class,
\(h\) is the width of the median class, and
\(n\) is the total frequency.
Mode
Grouped frequency distribution
The mode of the grouped frequency distribution can be determined using the formula:
Mode \(= l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)
The class interval with maximum frequency is called the modal class.
Where \(l\) is the lower limit 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, and
\(h\) is the width of the class interval.
Empirical relationship between mean, median and mode
In the previous topics, we have learnt how to find the mean, median and mode. From those, we can see an approximate relationship between these \(3\) measures of central tendency. And, the relationship between them is given by:
Mode \(= 3\) Median \(- 2\) Mean