Calculate arithmetic, geometric, harmonic, trimmed, and weighted means plus median, mode, and modality. Includes frequency table, skewness, and visual comparison.
The mean, median, and mode calculator computes all major measures of central tendency in one place. Enter your data to get the arithmetic mean (average), median (middle value), and mode (most frequent), along with the geometric mean, harmonic mean, root-mean-square, trimmed mean, and weighted mean.
Central tendency tells you where the "center" of your data lies. Different measures are appropriate for different situations — the mean works best for symmetric data, the median is robust to outliers and skewed distributions, and the mode identifies the most common value. This calculator computes all of them simultaneously, with a visual comparison showing exactly how they relate.
You also get a frequency table, skewness metrics (Fisher and Pearson), and an optional weighted mean for situations where not all data points are equally important. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
This calculator provides a comprehensive view of central tendency that no single measure can offer alone. By computing arithmetic, geometric, harmonic, trimmed, and weighted means alongside the median and mode, it helps you choose the right measure for your specific data and application.
The visual comparison chart and skewness metrics reveal the shape of your distribution at a glance — essential for choosing the right statistical tests and reporting the most representative summary of your data.
Arithmetic mean = Σxᵢ / n. Median = middle value(s) of sorted data. Mode = most frequent value. Geometric mean = (∏xᵢ)^(1/n). Harmonic mean = n / Σ(1/xᵢ). RMS = √(Σxᵢ²/n). Trimmed mean = mean after removing top/bottom k% of values.
Result: Mean = 82.10, Median = 83, Mode = No mode
With 10 values, the mean is 821/10 = 82.10. The sorted data has no repeated values, so no mode exists. The median is the average of the 5th and 6th values in sorted order: (81+85)/2 = 83. Mean < Median suggests slight left skew.
The "best" average depends on your data and context. For symmetric data without outliers, the arithmetic mean is generally preferred — it uses all data and has the smallest standard error. For skewed data (income, property values), the median is more representative. For growth rates, use the geometric mean. For rates (speed, fuel economy), use the harmonic mean.
For unimodal, moderately skewed distributions, an approximate relationship holds: Mode ≈ 3 × Median − 2 × Mean. This is Pearson's empirical rule. It's useful for estimating the mode when you only know the mean and median, but it breaks down for multimodal or highly skewed distributions.
Weighted means appear everywhere: GPA weights credit hours, stock indices weight by market cap (S&P 500) or price (Dow Jones), and quality metrics weight by importance. When calculating weighted means, ensure weights are meaningful and sum to a positive value. Negative weights can cause paradoxical results.
The mean (average) sums all values and divides by count — it uses every data point but is sensitive to outliers. The median is the middle value when data is sorted — it's robust to extreme values. The mode is the most frequently occurring value — it's the only measure that works for categorical data and identifies peaks in distributions.
Use the median when data is skewed (income, home prices, wait times) or contains outliers. For example, a neighborhood with nine $300K homes and one $10M mansion has a mean of $1.27M but a median of $300K — the median better represents the typical home price.
If every value in your data appears exactly once, there is no mode — the data is called "amodal." This is common in continuous data with many unique values. Mode is most meaningful for discrete data or data that has been grouped into categories.
The geometric mean is used for multiplicative processes — compound interest rates, population growth rates, and any situation where you multiply rather than add. If an investment returns +10%, −5%, +20% over three years, the geometric mean gives the true average annual return, while the arithmetic mean overestimates it.
A trimmed mean removes a percentage of the highest and lowest values before computing the average. A 10% trimmed mean drops the top 10% and bottom 10%. This makes the mean more resistant to outliers while still using most of the data, unlike the median which uses only the center value.
In positively (right) skewed data, the mean is pulled higher than the median by large outliers. In negatively (left) skewed data, the mean is pulled lower. The relationship mean > median suggests right skew, mean < median suggests left skew. Pearson's skewness coefficient formalizes this as 3(mean − median)/SD.