How to Calculate Mean, Median, and Mode

The arithmetic mean (what most people call “the average”) is the sum of values divided by how many values you have:

Mean = (x₁ + x₂ + … + xₙ) ÷ n

Worked example

Test scores: 88, 92, 70, 95, 85.

Sum = 88 + 92 + 70 + 95 + 85 = 430. Mean = 430 ÷ 5 = 86.

Properties worth remembering

• Every value influences the mean — outliers pull it hard
• The mean can be a number that never appears in the list (average of 2 and 10 is 6)
• For percentages that all use the same denominator, averaging percents can be fine; mixing rates with different bases needs care or a weighted approach

Mean temperature across days, mean order value, and class average score are everyday fits — as long as extreme outliers are not distorting the story you want to tell.

Turning a score into “percent of total points” is separate from averaging those percents; the free percentage calculator and percentage guide cover part-whole percent math.

The median is the middle value after you sort the data from smallest to largest. Half the observations sit at or below it, half at or above (with ties handled by the usual even-n rule).

Odd count

Sorted: 70, 85, 88, 92, 95. Middle position = 3rd value → median = 88.

Even count

Sorted: 70, 85, 88, 92. Middle pair = 85 and 88 → median = (85 + 88) ÷ 2 = 86.5.

Why median resists outliers

Replace 95 with 10,000 in the five-score list: mean explodes; median stays near the middle of the other values. House prices, incomes, and reaction times often publish medians for that reason.

Example intuition: incomes $40k, $45k, $50k, $55k, $2M → mean is dominated by $2M; median $50k better describes a “typical” household in the set.

The mode is the value that appears most often. A data set can have one mode (unimodal), two (bimodal), more than two (multimodal), or no mode if all frequencies are equal.

Worked examples

• 3, 5, 5, 7, 9 → mode = 5
• 2, 2, 4, 4, 4, 6, 6 → mode = 4
• 1, 2, 3, 4 → each once → no unique mode (or “all values equally frequent”)
• Red, blue, blue, green, blue → mode = blue (works for categories, not only numbers)

When mode shines

Survey “most common choice,” shoe size stocked most often, most frequent error code in logs. Mode can be a category label; mean and median require an ordered numeric scale.

For continuous data with all unique measurements (many decimals), mode may be useless unless you bin values into ranges first.

A weighted average multiplies each value by a weight (importance, credits, volume), sums those products, and divides by the sum of weights:

Weighted mean = (Σ wᵢxᵢ) ÷ (Σ wᵢ)

Worked example (course credits)

Grades: A = 4.0 in a 3-credit class, B = 3.0 in a 4-credit class, A = 4.0 in a 1-credit lab.

Numerator = (3×4.0) + (4×3.0) + (1×4.0) = 12 + 12 + 4 = 28.

Denominator = 3 + 4 + 1 = 8.

Weighted mean = 28 ÷ 8 = 3.5.

Equal weights reduce to the ordinary mean. Wrong weights (treating every class as 1 credit when credits differ) produce a fake GPA. For full semester mapping on a 4.0 scale, the free GPA calculator and GPA guide follow credit-hour weighting end to end.

Other weighted cases

• Portfolio return: weight each asset’s return by portfolio share
• Average price: weight by units sold, not by number of SKUs
• Survey: weight responses to match population demographics

Match the measure to the data shape and the decision.

• Mean — symmetric numeric data, total mass matters (average order size for revenue math), no extreme outliers — or outliers are meaningful and should count
• Median — skewed distributions, ordinal scores, “typical” person stories (income, home prices, latency with rare spikes)
• Mode — categorical favorites, discrete peaks, inventory planning around the most common size/choice
• Weighted mean — unequal importance, volume, time spent, or credit hours

Reporting tip

For skewed data, publish mean and median together. A company can truthfully say “average salary is high” while most employees sit near a lower median — both numbers can be correct and still mislead alone.

Related: expected value in probability multiplies outcomes by probabilities (a weighted idea). Event chance itself uses different rules — the free probability calculator and probability guide cover AND/OR and odds.

Paste a clean list, choose the measure(s) you need, and add weights only when they apply.

1. Open the free average calculator.
2. Enter numbers separated the way the tool expects (commas, spaces, or new lines).
3. Remove labels and currency symbols if the parser wants plain numbers.
4. Read arithmetic mean first for a classic average.
5. Check median when outliers may be present.
6. Check mode for the most frequent value or category code.
7. Switch to weighted mode and enter matching weights (credits, quantities, percentages that sum to 100).
8. Compare mean vs median; if they diverge a lot, inspect the extremes before you act on either number.

Optional advanced means

Some tools also show geometric mean (growth rates, ratios) or harmonic mean (certain rate averages, like average speed over equal distances). Use those only when the problem type matches — default business “average” is still arithmetic mean.

Small list errors create confident wrong centers.

• Forgetting to sort before taking the median
• Averaging averages of unequal group sizes without weights
• Letting one typo outlier rewrite the mean
• Calling mode “average” in a report when readers expect the mean
• Using mean on pure categories (mean of “red/blue/green” is meaningless)
• Equal-weight GPA when credit hours differ

This guide is educational. Statistical reporting for research, finance, or compliance may require specified estimators, confidence intervals, or regulator-defined averages — follow those standards when they apply.