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Measures of Central Tendency

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1. Mean (Arithmetic Mean)

• Geometric Mean - used for multiplicative/ratio data: $GM = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}$
• Harmonic Mean - used for rates/speeds: $HM = \frac{n}{\sum(1/x_i)}$
• Weighted Mean - assigns different weights to values: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

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2. Median

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3. Mode

• No mode - all values appear once
• Unimodal - one mode
• Bimodal - two modes
• Multimodal - more than two modes

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Comparison Table

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Relationship Between Mean, Median, and Mode

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Which Measure to Use?

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Quick Summary

• Mean - the mathematical average; best for symmetric, outlier-free data
• Median - the middle value; best for skewed data or when outliers are present
• Mode - the most frequent value; the only measure usable with categorical data

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📊 Measures of Central Tendency - Complete Detailed Guide

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What is a Measure of Central Tendency?

• Summarizes large datasets into a single meaningful value
• Enables comparison between different datasets
• Forms the foundation for further statistical analysis
• Helps in decision-making in business, research, medicine, and economics

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Classification of Measures of Central Tendency

Measures of Central Tendency
├── Mathematical Averages
│   ├── Arithmetic Mean (Simple & Weighted)
│   ├── Geometric Mean
│   └── Harmonic Mean
└── Positional Averages
    ├── Median
    └── Mode

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PART 1: MEAN

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1.1 Arithmetic Mean (AM)

A) Simple Arithmetic Mean - Ungrouped Data

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B) Arithmetic Mean - Grouped Data (3 Methods)

Method 1: Direct Method

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Method 2: Assumed Mean (Short-cut) Method

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Method 3: Step Deviation Method

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Properties of Arithmetic Mean

1. Sum of deviations from mean = 0: $\sum(x_i - \bar{x}) = 0$
2. Least sum of squared deviations: Mean minimizes $\sum(x_i - \bar{x})^2$
3. Combined mean: If two groups have means $\bar{x}_1$ and $\bar{x}2$ with sizes $n_1$ and $n_2$: $$\bar{x}{combined} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$$
4. Effect of transformation: If every value is multiplied by $k$, the mean is also multiplied by $k$
5. Unique: There is only one arithmetic mean for any dataset

Merits and Demerits of Arithmetic Mean

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1.2 Weighted Arithmetic Mean

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1.3 Geometric Mean (GM)

• Population growth rates
• Compound interest calculations
• Stock market returns
• Biological growth rates
• Index numbers

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1.4 Harmonic Mean (HM)

• Average speeds over equal distances
• Rates (price per unit, efficiency)
• Finance (P/E ratios)

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1.5 Relationship: AM, GM, HM

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PART 2: MEDIAN

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2.1 Definition

The median is a positional average - its value depends on position, not magnitude of observations.

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2.2 Median for Ungrouped Data

• Odd n: $$\text{Median} = \left(\frac{n+1}{2}\right)^{th} \text{ observation}$$
• Even n: $$\text{Median} = \frac{\left(\frac{n}{2}\right)^{th} + \left(\frac{n}{2}+1\right)^{th} \text{ observations}}{2}$$

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2.3 Median for Grouped Data

• $L$ = Lower class boundary of median class
• $N$ = Total frequency ($\sum f$)
• $cf$ = Cumulative frequency before the median class
• $f$ = Frequency of the median class
• $h$ = Class width

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2.4 Graphical Method - Ogive

1. Plot a "less than" ogive (cumulative frequency vs upper class boundary)
2. Draw a horizontal line from $N/2$ on the y-axis to the curve
3. Drop a vertical line to the x-axis - that value is the median

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2.5 Quartiles, Deciles, and Percentiles (Partition Values)

• Q₂ = D₅ = P₅₀ = Median (all refer to the same value)

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2.6 Properties of Median

1. Not affected by extreme values - most important property
2. Can be calculated for open-ended class intervals
3. Can be located graphically
4. Applicable to ordinal data
5. The sum of absolute deviations from the median is minimum: $\sum|x_i - M|$ is minimized

Merits and Demerits of Median

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PART 3: MODE

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3.1 Definition

The word "mode" comes from the French word "la mode" meaning fashion.

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3.2 Mode for Ungrouped Data

Types of Mode:

• Unimodal - one mode: {2, 3, 5, 7, 5, 7, 7} → Mode = 7
• Bimodal - two modes: {2, 3, 3, 5, 5, 7} → Modes = 3 and 5
• Multimodal - multiple modes: more than 2 values with same highest frequency
• No mode - all values appear equally often: {1, 2, 3, 4, 5}

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3.3 Mode for Grouped Data

• $L$ = Lower boundary of the modal class
• $f_1$ = Frequency of the modal class
• $f_0$ = Frequency of the class before the modal class
• $f_2$ = Frequency of the class after the modal class
• $h$ = Class width

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3.4 Properties of Mode

1. Can be used with any type of data including nominal (categorical)
2. Not affected by extreme values
3. Can be determined graphically using a histogram
4. May be non-unique (bimodal/multimodal datasets)
5. Can be used for open-ended classes (if mode is not in the open class)

Merits and Demerits of Mode

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PART 4: RELATIONSHIP AND COMPARISON

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4.1 Empirical Relationship (Karl Pearson's Formula)

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4.2 Position in Distributions

Symmetric (Normal) Distribution:

Positively Skewed (Right-skewed):

Negatively Skewed (Left-skewed):

Negatively Skewed     Symmetric       Positively Skewed
     ___                 ___                 ___
    /   \               /   \               /   \
   /     \___          /     \          ___/     \
  Mean Med Mode      Mean=Med=Mode    Mode Med Mean

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4.3 Complete Comparison Table

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PART 5: SPECIAL TOPICS

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5.1 Trimmed Mean

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5.2 Midrange

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5.3 When to Use Each Measure

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5.4 Real-World Applications

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5.5 Numerical Summary Example

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Summary

A measure of central tendency represents the "center" of a dataset. The mean is mathematically powerful but vulnerable to outliers. The median is robust and reliable for skewed data. The mode is the only measure applicable to categorical data. The geometric and harmonic means serve specialized purposes with rates and ratios. Choosing the right measure depends entirely on the type of data, shape of distribution, and the purpose of analysis.