1. Median for Individual Series
In an individual series, data is ungrouped and needs to be arranged in ascending order.
Steps:
– Arrange the data in ascending order.
– Let \(n\) be the total number of observations.
Formula (if \(n\) is odd):
\[\text{Median} = \text{Value of } \left( \frac{n + 1}{2} \right)^\text{th} \text{ item}\]
Formula (if \(n\) is even):
\[\text{Median} = \frac{\left(\frac{n}{2}\right)^\text{th item} + \left(\frac{n}{2} + 1\right)^\text{th item}}{2}\]
2. Median for Discrete Series
In a discrete series, values are associated with their respective frequencies.
Steps:
– Arrange the values in ascending order.
– Calculate cumulative frequency (CF).
– Find the position of the median using:
\[\frac{N + 1}{2} \quad \text{where } N = \sum f\]
Formula:
\[\text{Median} = \text{Value corresponding to } \left( \frac{N + 1}{2} \right)^\text{th} \text{ item in CF}\]
3. Median for Continuous Series
For a continuous or grouped frequency distribution:
Steps:
– Create cumulative frequency (CF) column.
– Determine \(\frac{N}{2}\), where \(N = \sum f\)
– Identify the median class (the class interval where \(\frac{N}{2}\) lies).
Formula:
\[\text{Median} = l + \left( \frac{\frac{N}{2} – CF}{f} \right) \times i\]
Where:
\(l\) = Lower boundary of the median class
\(N\) = Total frequency
\(CF\) = Cumulative frequency before the median class
\(f\) = Frequency of the median class
\(i\) = Class width (interval size)