Measures of Central Tendency

Measures of Central Tendency are statistical measures that identify a single value which represents the center or typical value of a dataset. They help summarize large amounts of data into a single, easy-to-understand figure.

The three most commonly used measures are:

• Mean
• Median
• Mode

1. Mean (Arithmetic Average)

The Mean is the sum of all values divided by the number of values.

Formula

Where:

• $\sum x$ = sum of all observations
• $N$ = number of observations

For Grouped Data

Where fx = frequency × midpoint

Advantages

• Easy to calculate
• Uses all data values

Disadvantages

• Affected by extreme values (outliers)

Example

Marks: 10, 20, 30 Mean = (10 + 20 + 30) / 3 = 20

2. Median

The Median is the middle value when data is arranged in ascending or descending order.

For Ungrouped Data

• If N is odd → median = middle value
• If N is even → median = average of the two middle values

For Grouped Data

Where:

• $L$ = lower boundary of median class
• $N$ = total frequency
• $cf$ = cumulative frequency before median class
• $f$ = frequency of median class
• $h$ = class width

Advantages

• Not affected by extreme values
• Suitable for skewed data

Disadvantages

• Does not use all values
• Difficult to use for algebraic calculations

3. Mode

The Mode is the most frequently occurring value in a dataset.

For Ungrouped Data

The value with the highest frequency.

For Grouped Data

Where:

• $L$ = lower limit of modal class
• $f_1$ = frequency of modal class
• $f_0$ = frequency before modal class
• $f_2$ = frequency after modal class
• $h$ = class width

Advantages

• Easy to compute
• Useful for qualitative data (e.g., most popular item)

Disadvantages

• May not be unique (multiple modes)
• Not useful for further mathematical analysis

Comparison of Mean, Median, Mode

When to Use Which?

• Mean → when data is symmetrical and has no outliers
• Median → when data is skewed or has extreme values
• Mode → for categorical data or when finding the most common item