Average Calculator

Calculate the mean, median, mode, and range of any set of numbers with our free online calculator.

Average Calculator

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Mean

Arithmetic Average

The arithmetic mean is the sum of all values divided by the number of values. It's what most people refer to as the "average."

Mean = (x₁ + x₂ + ... + xₙ) ÷ n

The mean is sensitive to outliers and extreme values, making it useful for datasets where all values should have equal weight.

Median

Middle Value

The median is the middle value in a sorted list of numbers. If there is an even number of observations, the median is the average of the two middle values.

The median is less affected by outliers and extreme values, making it a better measure of central tendency for skewed distributions.

Mode

Most Frequent Value

The mode is the value that appears most frequently in a dataset. A dataset may have:

  • No mode (all values appear once)
  • One mode (unimodal)
  • Two modes (bimodal)
  • Three or more modes (multimodal)

Range

Spread of Values

The range is the difference between the maximum and minimum values in a dataset.

Range = Maximum value - Minimum value

The range gives a simple measure of dispersion but is heavily influenced by outliers.

Related Concepts

Weighted Average

An average where some values contribute more than others based on their weight or importance.

Geometric Mean

The n-th root of the product of n values, useful for growth rates and ratios.

Harmonic Mean

The reciprocal of the arithmetic mean of the reciprocals, useful for rates and ratios.

Moving Average

An average that continuously updates using a fixed number of the most recent observations.

When to Use Different Averaging Methods

Different averaging methods serve different purposes and are suitable for different types of data:

  • Mean (Arithmetic average): Best for normally distributed data without extreme outliers. Examples include test scores, heights, and temperatures.
  • Median: Ideal for skewed distributions or when outliers are present. Examples include income distributions, house prices, and response times.
  • Mode: Useful for categorical data or when the most common value is important. Examples include shoe sizes, product categories, and survey responses.
  • Weighted Average: Used when some values should contribute more than others. Examples include GPAs, investment portfolios, and ratings.
  • Geometric Mean: Best for growth rates, returns, and ratios. Examples include investment returns, population growth rates, and bacteria growth.

Interesting Facts About Averages

  • The concept of the arithmetic mean dates back to the Babylonians around 300 BCE.
  • The "law of large numbers" states that as a sample size grows, its mean gets closer to the average of the entire population.
  • In a normal distribution, the mean, median, and mode are all equal.
  • The phrase "regression to the mean" refers to the phenomenon where extreme values tend to be followed by more average values.
  • The "central limit theorem" states that the distribution of sample means approximates a normal distribution as the sample size gets larger.
  • The term "average" comes from the Arabic word "awariya" meaning "damaged goods" - originally used in maritime insurance to distribute losses.

Real-world Applications

  • Education: Grade point averages, test score analysis, and performance metrics
  • Finance: Investment returns, inflation rates, and economic indicators
  • Sports: Batting averages, scoring averages, and performance statistics
  • Health: BMI calculations, vital signs monitoring, and drug dosing
  • Business: Customer satisfaction ratings, production metrics, and quality control
  • Science: Experimental results, statistical analysis, and data normalization