Measure of Central Tendency

Arithmetic Mean

Definition: The Mean (or average) of a dataset is the sum of all values divided by the number of values. It is generally represented by \((\bar{X})\).

Formulas for Different Series

- Individual Series: 

  \(\text{Mean (} \bar{X} \text{)} = \frac{\sum X}{N}\)

  where \( X \) represents each value and \( N \) is the total number of values.

- Discrete Series: 

  \(\text{Mean (} \bar{X} \text{)} = \frac{\sum fX}{\sum f}\)

  where \( f \) is the frequency of each value \( X \).

- Continuous Series: 

  \(\text{Mean (} \bar{X} \text{)} = \frac{\sum fM}{\sum f}\)

  where \( M \) is the midpoint of each class interval and \( f \) is the frequency of each class.

Median

Definition: The Median is the middle value of a dataset when arranged in ascending or descending order. If the dataset has an odd number of values, the median is the middle one. If it has an even number, it is the average of the two middle values. It is generally represented by \((M_d)\).

Formulas for Different Series

- Individual Series: 

  Sort the data, and find the middle value:

  - If \( N \) is odd: Median = \(\text{middle value}\)

  - If \( N \) is even: Median = \(\frac{\text{(middle value 1 + middle value 2)}}{2}\)

- Discrete Series: 

  Arrange data in ascending order and find the cumulative frequency. The median corresponds to the value for which cumulative frequency is \(\frac{N}{2}\).

- Continuous Series: 

  \(\text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h\)

  where:

  - \( L \) = lower boundary of the median class,

  - \( N \) = total frequency,

  - \( F \) = cumulative frequency before the median class,

  - \( f \) = frequency of the median class,

  - \( h \) = class interval width.