Mean, Median, Mode & Average Calculator

About the Mean, Median, Mode & Average Calculator

Type or paste your numbers to get the mean, median, mode and range with every step shown: the sum divided by the count, the values sorted with the middle one marked, and how often each value occurs. Alongside them you get the other averages people need — the geometric, harmonic and root-mean-square means, a trimmed mean and the midrange — and a weighted mean for grades with credits or marks with percentage weights.

For grouped data (class intervals with frequencies) the calculator uses the formulas taught in school: the mean by the direct, assumed-mean or step-deviation method, the median from l + ((N/2 − cf)/f) × h and the mode from l + ((f₁ − f₀)/(2f₁ − f₀ − f₂)) × h, with a table you can check line by line and the less-than and more-than ogives, which cross at the median.

Common uses

• Checking homework and exam answers for the mean, median and mode of grouped data, step by step.
• Working out a weighted average such as a grade point average from credits, or a final mark from weighted parts.
• Choosing a fair “typical value” for skewed data such as incomes or house prices, where the median or a trimmed mean says more than the mean.
• Averaging growth rates (geometric mean) or speeds over equal distances (harmonic mean).

Which average should I use?

• Mean (Σx ÷ n): uses every value; best for symmetric data without outliers.
• Median: the middle value when the data are sorted (the mean of the two middle values when n is even). It ignores how extreme the largest and smallest values are, so it is the better “typical value” for skewed data such as incomes.
• Mode: the most frequent value; the only average for categories (the most common shoe size). A data set can have several modes, or none when every value occurs equally often.
• Geometric mean (ⁿ√(x₁ × … × xₙ)): for rates of growth and ratios — the average yearly growth over several years is the geometric mean of the yearly growth factors.
• Harmonic mean (n ÷ Σ(1/x)): for rates over equal amounts — driving 60 km/h one way and 40 km/h back averages 48 km/h, not 50.
• Root mean square (√(Σx² ÷ n)): the typical size of values that can be positive or negative, such as errors.
• Trimmed mean: the mean after cutting the same share of the smallest and largest values, a compromise between the mean and the median. Here 10% from each end means ⌊0.1 × n⌋ values from each end, as in R’s mean(x, trim = 0.1); it equals Excel’s TRIMMEAN(data, 20%), which takes the total share.

For positive data, mean ≥ geometric mean ≥ harmonic mean, with equality only when all the values are equal.

Formulas for grouped data

With class intervals, each class is represented by its class mark x = (lower limit + upper limit) ÷ 2, and N = Σf.

• Mean, direct method: x̄ = Σfx ÷ N. Assumed-mean method: x̄ = A + Σfd ÷ N with d = x − A. Step-deviation method: x̄ = A + h × Σfu ÷ N with u = (x − A) ÷ h. All three give the same mean; A and h only make the arithmetic smaller.
• Median = l + ((N/2 − cf) ÷ f) × h, where the median class is the first class whose cumulative frequency is more than N/2, l is its lower boundary, cf the cumulative frequency of the classes before it, f its frequency and h its width.
• Mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h, where the modal class has the highest frequency f₁, f₀ and f₂ are the frequencies of the classes before and after it, and l and h are its lower boundary and width.

These are the formulas of the NCERT Mathematics textbook for Class 10 (chapter “Statistics”). Inclusive classes such as 10–19, 20–29 use the boundaries half-way between them (9.5, 19.5, 29.5 …) for l. The rough relation mode ≈ 3 × median − 2 × mean is shown as a check; it only holds for moderately skewed data.

Reading the median from an ogive

A less-than ogive plots the cumulative frequency — how many values lie below each upper class boundary — and rises from 0 to N. A more-than ogive plots how many values lie at or above each lower boundary and falls from N to 0. Both pass through N/2 at the same point, so the x-coordinate where they cross is the median; reading it from a carefully drawn graph gives the same value as the median formula, because the formula assumes the values of a class are spread evenly across it — exactly what the straight lines between the points assume.

Limitations

• Grouped data give estimates: the original values are not known, and every value of a class is taken to be at its class mark (for the mean) or spread evenly across the class (for the median). Open-ended classes such as “below 10” need a limit before they can be used.
• The grouped mode formula assumes classes of equal width. When the widths differ, the calculator compares frequency densities (frequency ÷ width) instead and says so.
• Frequencies must be whole numbers; decimal weights belong in the weighted mean, which gives the weighted mean only (not a weighted median).
• A list can have up to 50,000 values. Commas separate values, so do not use thousands separators (type 12500, not 12,500).

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