Average Calculator: Mean, Median, Mode, Std Dev
Enter a list of numbers to instantly compute the arithmetic mean, geometric mean, harmonic mean, weighted average, median, mode, range, and standard deviation. Switch between average types with one click and see a full step-by-step breakdown.
Formula
Worked example
For 10, 20, 30, 40, 50: arithmetic mean = 150/5 = 30; geometric mean = (10x20x30x40x50)^(1/5) = 26.03; harmonic mean = 5/(1/10+1/20+1/30+1/40+1/50) = 21.90; median = 30; std dev = 14.14.
How the arithmetic mean is calculated
The arithmetic mean is the sum of all values divided by the count of values. This calculator adds every number you enter, divides by how many numbers there are, and returns that result. Alongside the mean it independently computes the median (the middle value when sorted), geometric mean, harmonic mean, mode, range, population standard deviation, sample standard deviation, and population variance, giving you a complete statistical portrait of your dataset in one step.
Geometric mean and harmonic mean
The geometric mean multiplies all n values together and takes the nth root. Because it applies to logarithmic scale it compresses the effect of large outliers and is the standard average for compound growth rates, investment returns, and biological data that spans orders of magnitude. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. It is the correct average when you are combining rates over equal intervals, for example if you drive 60 km/h for one hour and 90 km/h for another hour, the harmonic mean of 72 km/h is the speed that covers the same total distance in two hours. For any dataset with all positive values the harmonic mean is less than or equal to the geometric mean, which is less than or equal to the arithmetic mean.
Weighted average
A weighted average lets you assign each value an importance score called a weight. The result is the sum of each value multiplied by its weight, divided by the total weight. This is how grade-point averages work (a 3-credit course contributes three times as much as a 1-credit course) and how portfolio returns are computed (each asset weighted by its share of the portfolio). Enter each value and its weight separated by a colon in the weighted mode, one pair per line.
Standard deviation, variance, and spread
The standard deviation measures how far values typically fall from the mean. A small standard deviation means the values are clustered tightly; a large one means they are spread widely. Population standard deviation divides by n (all values are the complete population); sample standard deviation divides by n-1 (your list is a sample from a larger population, and the correction avoids underestimating spread). Variance is simply the standard deviation squared. Range is the simplest spread measure: maximum minus minimum, useful but sensitive to a single extreme outlier.
Mean vs. median: which should you use?
The mean treats every value equally and is the right choice when your data is roughly symmetric with no extreme outliers, such as exam scores in a well-distributed class or repeated measurements in a lab. The median is the middle value of a sorted list, making it resistant to outliers. It is preferred for skewed data like household incomes or home prices, where a few very large values would pull the mean well above what is typical. A useful rule of thumb: if the mean and median in the results above differ noticeably, the distribution is skewed and the median is likely the more honest summary. This calculator shows both so you can judge at a glance.
Mode and its limitations
The mode is the value that appears most often in a dataset. It is the only average that can be used with categorical (non-numeric) data, such as the most common survey response. In continuous numeric data where no value repeats exactly, every value occurs once and there is technically no mode (this calculator reports "No mode" in that case). When two or more values tie for highest frequency, all are reported as modes and the dataset is called bimodal or multimodal.
Limitations of the arithmetic mean
The arithmetic mean is undefined for an empty dataset and is sensitive to extreme values because every number contributes to the sum. Two datasets can share the same mean while having very different distributions. For data with natural ratios or multiplicative structure (growth rates, speeds, price indices) the geometric or harmonic mean is more appropriate. Always verify that the arithmetic mean suits your specific context before drawing conclusions from it.
Which average should you use?
| Average type | Best for | Example |
|---|---|---|
| Arithmetic mean | Symmetric data, no extreme outliers | Test scores, temperatures |
| Median | Skewed data, outliers present | House prices, incomes |
| Geometric mean | Growth rates, ratios, log-scale data | Annual return rates, populations |
| Harmonic mean | Rates over equal intervals | Average speed, fuel efficiency |
| Weighted average | Values of different importance | GPA, portfolio returns |
| Mode | Most common value in categorical/count data | Survey responses, test scores |
Choose the right measure of central tendency for your data type. When in doubt, start with arithmetic mean and check whether the median is far away.
Frequently asked questions
What is the difference between mean and average?
In everyday usage, 'average' almost always refers to the arithmetic mean. Technically, 'average' is a general term that can refer to the mean, median, or mode depending on context, but unless otherwise specified it means the arithmetic mean: sum divided by count.
Why is my mean much higher than my median?
When the mean exceeds the median significantly, your data is right-skewed, meaning a few large values are pulling the mean upward. This is common in income, real-estate, or web-traffic data. In these situations the median is usually a more representative measure of what is typical.
When should I use geometric mean instead of arithmetic mean?
Use the geometric mean when your values represent multiplicative growth or ratios: compound annual growth rates, bacteria population doublings, index performance over multiple periods. It correctly accounts for the compounding effect that arithmetic mean ignores. All values must be positive.
What is the harmonic mean used for?
The harmonic mean is the correct average for rates measured over equal intervals. If you drive the same distance at different speeds, the harmonic mean of the speeds gives your true average speed. It is also used in finance (P/E ratio averaging), networking (packet throughput), and the F1 score in machine learning.
What is the difference between population and sample standard deviation?
Population standard deviation (sigma) divides by n and applies when your list contains every member of the group you care about. Sample standard deviation (s) divides by n-1 (Bessel's correction) and applies when your list is a random sample from a larger population. In practice, use the sample version unless you are certain you have the complete dataset.
Can the mean be a value that is not in my dataset?
Yes, and this is completely normal. If you average 1, 2, and 4, the mean is 2.33, which does not appear in the original list. The mean is a computed summary statistic, not necessarily an observed value.
How do I calculate a weighted average?
Switch to the 'Weighted average' mode and enter each value followed by a colon and its weight (for example, '85:3' means the score 85 carries a weight of 3). The calculator multiplies each value by its weight, sums those products, then divides by the total weight. This is how GPA, final-exam averages, and portfolio returns are typically computed.