Mean Calculator
Find the arithmetic mean, geometric mean, harmonic mean, or weighted average of any list of numbers. Enter values separated by commas or spaces to get the mean plus a full descriptive-statistics summary: median, mode, range, variance, and standard deviation.
Formula
Worked example
For 12, 7, 19, 4, 23: arithmetic mean = (12+7+19+4+23)/5 = 65/5 = 13. Geometric mean = (12*7*19*4*23)^(1/5) = 147168^0.2 = approx 10.88. Harmonic mean = 5/(1/12+1/7+1/19+1/4+1/23) = approx 8.62.
The three classic types of mean
The arithmetic mean is the most familiar: add up all the values and divide by the count. It is the balance point of the data and the right choice for most everyday calculations. The geometric mean multiplies all values together and takes the nth root; it is ideal when your data represents rates of change or ratios, such as compound annual growth rates or index returns, because it accounts for the multiplicative nature of growth. The harmonic mean divides the count by the sum of the reciprocals; it gives the correct average for rates measured over equal distances or equal quantities, such as average speed for a round trip where each leg covers the same distance.
Weighted mean: making some values count more
A weighted arithmetic mean lets you assign more importance to certain values. Each value is multiplied by its weight, the products are summed, and that total is divided by the sum of the weights. Grade-point averages use this: a three-credit course contributes three times as much as a one-credit course. Survey data often weights responses by demographics. Enter weights in the same order as your values; any missing weights default to 1. A weight of 2 has the same effect as entering that value twice in an unweighted calculation.
Mean vs. median: which describes your data better?
The mean uses every value, so a single extreme outlier can pull it far from what is typical. The median, by contrast, is just the middle value of the sorted data, so it ignores how extreme the outer values are. On symmetric data the two are nearly equal. When they diverge, the data is skewed. Income and house prices are classic examples: a handful of very high earners or expensive properties push the arithmetic mean well above the median, which is why economists usually report median household income as the more representative figure. Reporting both gives a fuller picture, and the standard deviation tells you how spread out the values are.
Inequality of means: harmonic <= geometric <= arithmetic
For any set of positive numbers, the harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean. The three are equal only when all values are identical. This inequality is a foundational result in mathematics (the AM-GM-HM inequality) and helps you choose the right mean: if you are averaging rates and you use the arithmetic mean, you will always overestimate; if you use the harmonic mean you will always get a value at or below the true arithmetic average. Knowing which direction an error runs can be critical in engineering and finance calculations.
Which mean should I use?
| Type of mean | Formula | Best used for |
|---|---|---|
| Arithmetic | sum / n | Scores, temperatures, any additive data |
| Geometric | (x1 * x2 * ... * xn)^(1/n) | Rates of change, CAGR, index returns |
| Harmonic | n / sum(1/xi) | Speeds over equal distance, unit-price averages |
| Weighted arithmetic | sum(wi*xi) / sum(wi) | GPA, survey weights, portfolio returns |
Pick the mean that matches how your data was generated.
Frequently asked questions
What is the difference between the arithmetic mean, geometric mean, and harmonic mean?
The arithmetic mean (sum divided by count) is best for additive data like test scores. The geometric mean (nth root of the product) is best for multiplicative data like growth rates. The harmonic mean (count divided by the sum of reciprocals) is best for rates like average speed. For positive numbers, harmonic <= geometric <= arithmetic, with equality only when all values are identical.
What is the difference between the mean and the median?
The mean is the sum of all values divided by how many there are, so every value affects it. The median is the middle value when data is sorted, so it is not pulled by extreme outliers. On skewed data, such as income or house prices, the median is often a better description of what is typical.
How do I calculate a weighted mean?
Multiply each value by its weight, sum those products, then divide by the total of the weights. For example, if 70 has a weight of 3 and 90 has a weight of 1, the weighted mean is (70*3 + 90*1) / (3+1) = 300/4 = 75. Switch to Weighted arithmetic mean in the Type of mean selector and enter your weights.
Can I calculate the mean of negative numbers or decimals?
Yes, the arithmetic and weighted means work with any real numbers including negatives and decimals. The geometric and harmonic means require all positive values; they will return undefined if any value is zero or negative.
Does the order of the numbers matter?
No, for the arithmetic, geometric, and harmonic means the order does not affect the result because multiplication and addition are commutative. For the weighted mean, the order matters only in that weights must be listed in the same order as your values.
What is the population standard deviation shown here?
The population standard deviation treats your entered values as the entire population, not a sample. It is the square root of the average squared deviation from the mean. If your data is a sample from a larger population and you need the sample standard deviation, use n-1 instead of n in the denominator (Bessel's correction). Use our standard deviation calculator for more control over this.