Median (Positional Average)

The Median is a measure of central tendency that represents the Middle Value in a data set when arranged in ascending or descending order. Unlike the mean, the median is not affected by extreme values, making it particularly useful for skewed data distributions. In UGC NET Commerce, understanding the median is important for interpreting datasets where central tendency may not align with the arithmetic mean.

What is Median?

The median is the value that divides an ordered dataset into two equal halves. Half of the values lie below the median and half lie above it.

Median in Different Data Types

a. Individual Series

When raw data is given without frequency.

Steps:

1. Arrange the data in ascending order.

2. Use the following formulas:

• If n is odd:

  $$\text{Median} = \text{Value of } \left(\frac{n+1}{2}\right)^{th} \text{ item}$$
  

• If n is even:

  $$\text{Median} = \frac{\left(\frac{n}{2}\right)^{th} \text{ item} + \left(\frac{n}{2} + 1\right)^{th} \text{ item}}{2}$$

Example (Odd n):

$$n = 5, \quad \text{Median} = \text{3rd item} = 25$$

Example (Even n):

$$n = 6, \quad \text{Median} = \frac{3rd + 4th}{2} = \frac{20 + 25}{2} = 22.5$$

b. Discrete Series

Data values are associated with frequencies.

Steps:

1. Arrange the data in ascending order (if not already).

2. Find cumulative frequencies (CF).

3. Locate the $\frac{n+1}{2}^{th}$ item.

4. Identify the corresponding value of $x$ as the median.

Example:

$$n = \sum f = 10,\quad \frac{n+1}{2} = \frac{11}{2} = 5.5^{th} \text{ item}$$

Median lies in class where CF ≥ 5.5 ⇒ x = 30

c. Continuous Series (Grouped Data)

For grouped or class-interval data.

Formula:

$$\text{Median} = L + \left( \frac{\frac{n}{2} - CF}{f} \right) \times h$$

Where:

• $L$ = lower boundary of median class

• $n$ = total frequency

• $CF$ = cumulative frequency before median class

• $f$ = frequency of median class

• $h$ = class width

Steps:

1. Calculate total frequency $n$.

2. Identify $\frac{n}{2}$.

3. Find the median class (where CF just crosses $\frac{n}{2}$).

4. Apply the formula.

Example:

$$n = 35,\quad \frac{n}{2} = 17.5 \Rightarrow \text{Median class} = 20–30$$

Values:

• $L = 20$, $CF = 13$, $f=\mathrm{12, }$$h = 10$
  

$$\text{Median}=20+\left(\frac{17.5-13}{12}\right)\times 10=20+\left(\frac{4.5}{12}\right)\times 10=20+3.75=23.75$$

Merits of Median

• Not affected by extreme values or outliers.

• Can be used for ordinal and skewed data.

• Simple to understand and calculate.

• Median is always present in the dataset range.

Demerits of Median

• Does not use all observations (especially in grouped data).

• Cannot be easily used for further algebraic treatments.

• Less stable than mean in large samples.

• Grouped data requires interpolation.

When to Use Median

• When data contains extreme values or outliers.

• For skewed distributions (e.g., income, property prices).

• In ordinal-scale data (e.g., rankings, satisfaction levels).

Conclusion

The Median is a vital statistical tool that offers a realistic measure of central tendency, particularly in non-symmetrical or qualitative data sets. While it may not utilize every data point like the mean, its robustness to outliers makes it an essential concept in business statistics and practical data interpretation.