Which Average Should You Use? (Mean vs. Median vs. Mode)
In everyday speech, "average" almost always refers to the arithmetic mean. However, in statistical analysis and real-world economics, relying exclusively on the arithmetic mean can produce deceptive results when data is skewed by extreme outliers.
Type of Average
Mathematical Concept
Ideal Real-World Use Case
Sensitivity to Outliers
Arithmetic Mean
Sum of all numbers divided by count (n).
Symmetric data sets: test scores, temperatures, manufacturing tolerances.
High (heavily pulled by extreme highs or lows).
Median
The exact middle number in a sorted array.
Skewed distributions: household income, home sale prices, net worth.
Zero / Robust (immune to extreme single outliers).
Mode
The value that occurs with highest frequency.
Categorical or discrete data: shoe sizes, popular inventory items, election polls.
None (reflects most common occurrence).
Geometric Mean
The n-th root of the product of n numbers.
Compounding metrics: investment portfolio CAGR returns, population growth rates.
Rates, ratios, and speeds: average trip speed across equal distances, price-to-earnings ratios.
High (very sensitive to small numbers close to zero).
Why Housing & Salaries Use the Median
If nine workers earn $50,000 and one executive earns $1,500,000, the arithmetic mean salary is $195,000—a misleading figure that misrepresents 90% of the company. The median salary is $50,000, which accurately captures the typical employee compensation.
The Pythagorean Means Relationship
For any set of positive real numbers, the arithmetic mean (A), geometric mean (G), and harmonic mean (H) strictly satisfy the inequality: A ≥ G ≥ H. Equality holds if and only if all numbers in the dataset are identical.
Frequently Asked Questions
What is the difference between an average and a mean?
In casual conversation, the terms "average" and "mean" are used interchangeably. Strictly speaking, "average" is an umbrella term for measures of central tendency (which includes the arithmetic mean, median, mode, geometric mean, and harmonic mean). The "arithmetic mean" is the specific mathematical formula summing all numbers and dividing by the total count.
How do you calculate a weighted average?
To calculate a weighted average, multiply each number by its corresponding percentage or weight factor, add all those weighted products together, and divide by the sum of all weights. For example, if Homework (worth 20%) is 90 and Final Exam (worth 80%) is 70, the weighted average is ((90 × 20) + (70 × 80)) / 100 = (1800 + 5600) / 100 = 74.00.
Can a dataset have more than one mode, or no mode?
Yes. If no number repeats in a dataset, there is no mode. If two numbers repeat with the exact same maximum frequency, the dataset is bimodal. If three or more numbers tie for top frequency, it is multimodal.
Why does the Geometric Mean differ from the Arithmetic Mean?
The arithmetic mean adds numbers, which is suitable for independent additive metrics. The geometric mean multiplies numbers and takes their root, which correctly accounts for compounding growth. For instance, if an investment gains +100% in year one (doubles) and loses −50% in year two (halves), the arithmetic mean is (+100 − 50)/2 = +25% per year, which falsely implies profit. The geometric mean correctly reveals a 0% net compound annual growth rate.
What is the difference between sample and population standard deviation?
Population standard deviation (σ) divides the sum of squared differences by n, because you have measured every single item in the entire population. Sample standard deviation (s) divides by (n − 1) (Bessel's correction) to correct for statistical bias when estimating a broader population from a smaller sample group.