Mean (Average) Calculator with Examples
Enter numbers separated by commas to calculate the mean (average) instantly. Includes step-by-step examples and worked problems.
Quick Examples
Result
What is Mean (Average)? The mean, or average, is the sum of a set of numbers divided by the total count of numbers. It gives a central value for a data set.
Practical Applications: Mean calculation is widely used in statistics, education, finance, and everyday life to find typical values and trends.
Mean Calculation Examples — Step by Step
Common mean problems with full solutions.
Mean of 5, 10, 15
Mean = 10
Step 2: Count = 3
Step 3: Mean = 30 ÷ 3 = 10
Mean of test scores: 72, 85, 90, 68, 95
Mean = 82
Step 2: Count = 5
Step 3: Mean = 410 ÷ 5 = 82
This is the same as a simple average grade calculation. Weighted averages (for different credit subjects) require multiplying each score by its weight first.
Mean formula
Mean = (x₁ + x₂ + … + xₙ) / n = Σx / n
Example: mean of {2, 4, 4, 4, 5, 5, 7, 9}
= (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
Types of Averages: Mean, Median, and Mode
When people say "average," they usually mean the arithmetic mean, but there are actually three common measures of central tendency, each with distinct strengths:
- Mean (Arithmetic Average): The sum of all values divided by the number of values. It considers every data point and is the most widely used average.
- Median: The middle value when all numbers are sorted in order. If the dataset has an even number of values, the median is the average of the two middle numbers. The median is resistant to outliers, making it ideal for skewed distributions.
- Mode: The value that appears most frequently in a dataset. A set can have no mode, one mode, or multiple modes. The mode is particularly useful for categorical data.
Mean Formula: Mean = (x₁ + x₂ + ... + xₙ) / n
When to Use Each Type of Average
Choosing the right average depends on your data and what you want to communicate:
- Use the mean when your data is symmetrically distributed without extreme outliers. It works well for test scores, temperatures, and measurements where every value matters equally.
- Use the median when your data is skewed or contains outliers. Household income is a classic example -- a few extremely high earners can inflate the mean, but the median better represents the "typical" household.
- Use the mode when you want to know the most common value, especially with categorical or discrete data. For example, the most popular shoe size sold at a store is best described by the mode.
Weighted Average and Trimmed Mean
A weighted average assigns different levels of importance (weights) to different values. This is essential when not all data points contribute equally to the outcome.
Weighted Mean: (w₁x₁ + w₂x₂ + ... + wₙxₙ) / (w₁ + w₂ + ... + wₙ)
For example, if your final grade is 30% homework, 30% midterm, and 40% final exam, and you scored 90, 80, and 85 respectively, your weighted average is (0.30 x 90 + 0.30 x 80 + 0.40 x 85) = 27 + 24 + 34 = 85.
A trimmed mean removes a specified percentage of the highest and lowest values before calculating the mean. This approach reduces the influence of outliers while still using most of the data. For instance, a 10% trimmed mean on a 20-value dataset removes the 2 lowest and 2 highest values before averaging the remaining 16. Olympic judges often use this technique, discarding the highest and lowest scores to reduce bias.
Comparison of Average Types
The following table illustrates how each type of average behaves with the same dataset: 2, 3, 5, 5, 7, 9, 100.
| Type | Value | Calculation | Sensitive to Outliers? |
|---|---|---|---|
| Mean | 18.71 | (2+3+5+5+7+9+100) / 7 | Yes |
| Median | 5 | Middle value of sorted list | No |
| Mode | 5 | Most frequent value | No |
| Trimmed Mean (14%) | 5.8 | (3+5+5+7+9) / 5 | Less sensitive |
| Weighted Mean | Varies | Depends on assigned weights | Depends on weights |
Real-World Applications of Averages
Averages play a critical role across many fields:
- Education: Teachers use the mean to calculate final grades and the median to understand typical student performance, especially when a few scores are unusually high or low.
- Finance: Investors track the weighted average cost of capital (WACC), moving averages in stock prices, and average annual returns to make informed decisions.
- Healthcare: Average blood pressure readings, mean recovery times, and median survival rates help doctors assess treatments and set benchmarks.
- Sports: Batting averages, points per game, and adjusted statistics all rely on different types of averages to evaluate player performance fairly.
Step-by-Step Mean Calculation Walkthrough
Let's calculate the mean, median, and mode for the dataset: 4, 7, 13, 7, 2, 9, 7, 3, 12, 6
Step 1: Sort the data
2, 3, 4, 6, 7, 7, 7, 9, 12, 13
Step 2: Calculate the Mean
Sum = 2 + 3 + 4 + 6 + 7 + 7 + 7 + 9 + 12 + 13 = 70
Mean = 70 ÷ 10 = 7.0
Step 3: Find the Median
10 values (even number) → average the 5th and 6th values: (7 + 7) / 2 = 7.0
Step 4: Find the Mode
7 appears 3 times (more than any other value) → Mode = 7
In this dataset, mean = median = mode = 7, which indicates a perfectly symmetric distribution. This alignment is rare in real-world data — usually the three measures differ, revealing whether the data is skewed or contains outliers.
Geometric Mean vs Arithmetic Mean: When to Use Each
The arithmetic mean is the familiar "add and divide" average. The geometric mean is the nth root of the product of n numbers. They give different results, and choosing the wrong one can lead to serious financial errors.
Why Geometric Mean Matters for Investment Returns
An investment that gains +50% one year, then loses −50% the next is NOT back to even. Starting with $1,000: after +50% = $1,500, then −50% = $750.
- • Arithmetic mean of returns: (+50 + −50) / 2 = 0% (incorrectly suggests breakeven)
- • Geometric mean: √(1.50 × 0.50) − 1 = √0.75 − 1 = −13.4%/year (the truth)
| Situation | Use | Why |
|---|---|---|
| Test scores, temperatures, heights | Arithmetic Mean | Values are additive |
| Annual investment returns, CAGR | Geometric Mean | Values are multiplicative |
| Speed over equal distances | Harmonic Mean | Rate × time = distance |
| Grades with different weights | Weighted Mean | Not all values equal |
| Household income, home prices | Median | Skewed by high outliers |
Moving Averages: How They Work and Why Traders Use Them
A simple moving average (SMA) calculates the arithmetic mean of a fixed number of recent values, then "moves" forward one period at a time as new data arrives. Moving averages smooth out short-term noise to reveal underlying trends.
Example: 5-Day SMA of Stock Price
| Day | Price | 5-Day SMA | Calculation |
|---|---|---|---|
| 1 | $100 | — | Not enough data |
| 2 | $102 | — | Not enough data |
| 3 | $99 | — | Not enough data |
| 4 | $104 | — | Not enough data |
| 5 | $103 | $101.60 | (100+102+99+104+103)÷5 |
| 6 | $107 | $103.00 | (102+99+104+103+107)÷5 |
| 7 | $105 | $103.60 | (99+104+103+107+105)÷5 |
Medium-term trend. A key support level for actively managed stocks.
Long-term trend. Stocks above it are in "bull territory." A major technical level.
50-day SMA crosses above 200-day SMA = bullish signal used by institutional traders.