The Three Core Measures of Central Tendency

Mean (Average)

The arithmetic average calculated by summing all values and dividing by the count of values.

Best for: Normally distributed data without extreme outliers.

Formula: Mean = Σx / n

Median (Middle Value)

The middle value in a sorted dataset, separating the higher half from the lower half.

Best for: Skewed distributions or data with outliers.

Method: Sort data, find middle position.

Mode (Most Frequent)

The value that appears most frequently in a dataset.

Best for: Categorical data or identifying common values.

Note: A dataset can have one mode, multiple modes, or no mode.

Detailed Examples

Example 1: Student Exam Scores

Dataset: Final exam scores for a class of 10 students
78, 92, 85, 88, 76, 95, 89, 85, 90, 82
Mean Calculation:
Sum = 78 + 92 + 85 + 88 + 76 + 95 + 89 + 85 + 90 + 82 = 860
Count = 10
Mean = 860 ÷ 10 = 86.0
Median Calculation:
Sorted data: 76, 78, 82, 85, 85, 88, 89, 90, 92, 95
Middle positions: 5th and 6th values (85 and 88)
Median = (85 + 88) ÷ 2 = 86.5
Mode Identification:
Frequency count: 85 appears twice, all others appear once
Mode = 85
Interpretation: The average exam score is 86.0, the middle score is 86.5, and the most frequent score is 85. In this relatively symmetric distribution, all three measures are close together.

Example 2: Household Income in a Neighborhood

Dataset: Annual household income (in thousands) for 9 households
45, 52, 48, 62, 55, 120, 50, 58, 950
Mean Calculation:
Sum = 45 + 52 + 48 + 62 + 55 + 120 + 50 + 58 + 950 = 1440
Count = 9
Mean = 1440 ÷ 9 = 160.0
Median Calculation:
Sorted data: 45, 48, 50, 52, 55, 58, 62, 120, 950
Middle position: 5th value
Median = 55
Mode Identification:
All values appear only once
Mode = No mode
Interpretation: The mean income of $160,000 is heavily skewed by one extremely high income ($950,000). The median of $55,000 better represents what a typical household earns. This demonstrates why median is preferred for income reporting.

When Measures Diverge: A Critical Insight

Consider a small company's employee salaries (in thousands):

42, 45, 48, 52, 55, 60, 65, 70, 75, 250
Mean = $77.2K | Median = $57.5K | Mode = No mode

The mean is $19.7K higher than the median due to one high salary. This shows how a single outlier can dramatically affect the mean while leaving the median relatively stable.

Theoretical Foundations

Law of Large Numbers: As sample size increases, the sample mean converges to the population mean.

Central Limit Theorem: The distribution of sample means approaches a normal distribution as sample size grows, regardless of the population's distribution.

These principles form the theoretical basis for using central tendency measures in statistical inference and hypothesis testing.

Critical Limitation

Central tendency measures must always be reported alongside measures of dispersion (range, variance, standard deviation).

Example: Two datasets with identical means [10, 10, 10, 10, 10] and [0, 5, 10, 15, 20] both have a mean of 10, but their spreads are completely different.

Practical Applications

Mean: Educational testing, scientific measurements, quality control

Median: Income distribution, real estate prices, survival analysis

Mode: Market research (most popular product), voting (most common choice), epidemiology (most frequent symptom)

The choice of measure depends on data type, distribution shape, and the presence of outliers.