Mode

Measure of central tendency representing the value that appears most frequently in a dataset.

## Core concept

Mode is the observation with the highest frequency in a dataset. It indicates the value that occurs most often and represents the typical or most popular value.

• Applicable to all types of data: qualitative (categorical) and quantitative (numerical)
• Unlike mean and median, mode is not affected by extreme values
• A dataset can have no mode, one mode (unimodal), two modes (bimodal), or multiple modes (multimodal)

## Formula / rule

For ungrouped data: - Identify the value with the maximum frequency - No formula required; direct observation from frequency distribution

For grouped data (class intervals):

$$\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$

Where: - L = lower boundary of modal class (class with highest frequency) - f₁ = frequency of modal class - f₀ = frequency of class before modal class - f₂ = frequency of class after modal class - h = class width (size of interval)

## Identification of modal class

1. Locate the class with maximum frequency
2. That interval becomes the modal class
3. Apply the formula if further precision is needed

## Common exam applications

Ungrouped data example: Dataset: 5, 7, 5, 9, 5, 12, 7, 5

Frequency: 5 appears 4 times, 7 appears 2 times, 9 and 12 appear once each

Mode = 5 (highest frequency)

Grouped data example:

| Marks | Frequency | |-------|-----------| | 0–10 | 5 | | 10–20 | 12 | | 20–30 | 18 | | 30–40 | 10 | | 40–50 | 5 |

Modal class = 20–30 (frequency 18 is highest)

$$\text{Mode} = 20 + \frac{18 - 12}{2(18) - 12 - 10} \times 10$$

$$= 20 + \frac{6}{36 - 22} \times 10 = 20 + \frac{6}{14} \times 10 = 20 + 4.29 = 24.29$$

## Advantages and limitations

Advantages: - Easy to understand and compute for ungrouped data - Unaffected by extreme values (outliers) - Can be used for qualitative data - Useful for discrete data and categorical variables

Limitations: - May not be unique (bimodal/multimodal distributions) - Less useful when all values appear with equal frequency (no mode) - For grouped data, heavily dependent on class intervals chosen - Less amenable to algebraic manipulation compared to mean

## Relationship with mean and median

For a normal distribution (symmetrical): $$\text{Mean} = \text{Median} = \text{Mode}$$

For a skewed distribution: - Right-skewed: Mean > Median > Mode - Left-skewed: Mode > Median > Mean

This relationship (Karl Pearson's formula): $$\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})$$

## Common mistakes

• Confusing mode with frequency (mode is the value, not the count)
• Forgetting to identify the modal class before applying the formula
• Using modal class width incorrectly in grouped data calculations
• Ignoring the possibility of multiple modes in a dataset