Average Calculator
Paste your numbers — mean, median, mode and more, the moment you stop typing.
Type or paste any list of numbers — separated by commas, spaces or new lines — and get the mean, median, mode, range, sum, count, minimum and maximum instantly. Switch to Standard for a weighted average (scores of 84 and 92 weighted 30/70 average 89.6), Detailed for the geometric mean, variance and both population and sample standard deviation, or Advanced for the grade mode that drops your lowest scores before averaging.
Exact arithmetic on the numbers you enter — means, medians and deviations are computed, not estimated. Rounding is applied only to displayed decimals (up to 4 places).
Mean, median and mode — three averages, three answers
The same list can give three different "averages"
The mean adds every value and divides by the count. The median is the middle value once the list is sorted (with an even count, the mean of the two middle values). The mode is whatever value appears most often. For the list 5, 7, 8, 8, 12, 14, 16: the sum is 70 across 7 values, so the mean is 10; the fourth of the seven sorted values is 8, the median; and 8 appears twice, so it's also the mode. Three legitimate answers — 10, 8 and 8 — from one list.
| List | Mean | Median | Mode | Range |
|---|---|---|---|---|
| 5, 7, 8, 8, 12, 14, 16 | 10 | 8 | 8 | 11 |
| 1, 2, 2, 3, 3, 4 | 2.5 | 2.5 | 2 and 3 | 3 |
| 3, 5, 9, 11 | 7 | 7 | none | 8 |
| 40,000 · 45,000 · 50,000 · 55,000 · 300,000 | 98,000 | 50,000 | none | 260,000 |
Why they disagree
The mean is the only one that uses every value's actual size, so it gets pulled toward big values — in the salary row above, one $300,000 outlier drags the mean to $98,000 while the median stays at a representative $50,000. The median only cares about position in the sorted list, and the mode only about repetition. When the mean sits well above the median, that's the signature of a tail of large values; well below, a tail of small ones.
Even count? Average the middle two. For 3, 5, 9, 11 the middle values are 5 and 9, so the median is (5 + 9) ÷ 2 = 7 — a number that isn't in the list, which is perfectly normal.
Weighted averages: when some numbers count more
Multiply, add, divide by the total weight
A weighted average is Σ(value × weight) ÷ Σ weights. The classic case is a course grade where homework counts 30% and the final exam counts 70%. Score 84 on homework and 92 on the final:
| Component | Score | Weight | Score × weight |
|---|---|---|---|
| Homework | 84 | 30 | 2,520 |
| Final exam | 92 | 70 | 6,440 |
| Total | — | 100 | 8,960 → ÷ 100 = 89.6 |
The weighted average is 89.6 — above the plain mean of 88, because the heavier-weighted exam score was the higher one. Weights don't need to sum to 100: dividing by the actual total means 3 and 7 behave exactly like 30 and 70.
Credit hours work the same way
Three courses at 95 (2 credits), 85 (4 credits) and 70 (4 credits): the products are 190, 340 and 280, totalling 810 over 10 credits — a weighted average of 81, noticeably below the unweighted mean of 83.3 because the weakest grade carries the most credits. That's precisely the arithmetic behind a GPA — our GPA Calculator layers the letter-grade points scale on top of it.
Sanity check. A weighted average always lands between your smallest and largest value — if it doesn't, a weight is on the wrong row. And with all weights equal it collapses to the plain mean.
Standard deviation, variance and the geometric mean
One worked example, both conventions
Standard deviation measures how far values typically sit from the mean. Take the default list 5, 7, 8, 8, 12, 14, 16 (mean 10) and square each deviation:
| x | x − mean | (x − mean)² |
|---|---|---|
| 5 | −5 | 25 |
| 7 | −3 | 9 |
| 8 | −2 | 4 |
| 8 | −2 | 4 |
| 12 | 2 | 4 |
| 14 | 4 | 16 |
| 16 | 6 | 36 |
| Sum of squared deviations | 98 | |
Treating the list as a whole population, the variance is 98 ÷ 7 = 14 and the standard deviation is √14 ≈ 3.742. Treating it as a sample from something larger, divide by n − 1 instead: 98 ÷ 6 ≈ 16.333, standard deviation ≈ 4.041. The n − 1 version (Bessel's correction) is always the larger of the two, and the gap shrinks as your list grows. This calculator shows both, labelled, so you never have to guess which convention a textbook or spreadsheet used — Excel's STDEV.P and STDEV.S are exactly these two.
The geometric mean: for things that multiply
The geometric mean is the n-th root of the product: GM(4, 9) = √36 = 6, not 6.5. Use it for growth rates and returns, which compound rather than add. Gain 20% then lose 10% and your two-year factor is 1.20 × 0.90 = 1.08; the geometric mean √1.08 ≈ 1.0392 says the equivalent steady rate is about 3.92% per year — the arithmetic mean's "5%" would overstate your actual growth. It's only defined when every value is positive, and it never exceeds the arithmetic mean (they're equal only when all values are identical).
Grade averages, drop-the-lowest, and picking the right average
Drop-the-lowest, quantified
Many syllabi drop each student's lowest quiz before averaging. Scores of 62, 78, 85, 90, 95 average 410 ÷ 5 = 82. Drop the 62 and the remaining four average 348 ÷ 4 = 87 — five whole points, often a letter-grade boundary. The Advanced level does this for any list: it always removes the smallest N values (the standard policy), shows you exactly which ones went, and reports the new mean and median. Because only the smallest values are removed, dropping can never lower your average.
Which average should you report?
| Situation | Use | Why |
|---|---|---|
| Test scores, costs, everyday totals | Mean | Every value counts; no wild outliers |
| Incomes, home prices, skewed data | Median | Immune to a few extreme values |
| Most common shoe size, poll answer | Mode | "Most typical single value" is the actual question |
| Growth rates, investment returns | Geometric mean | Rates compound — they multiply, not add |
| Components with different importance | Weighted mean | Weights encode the importance directly |
Reading the gap. Mean well above median → a high tail is pulling it (salaries, home prices). Mean well below median → a low tail. Mean ≈ median → roughly symmetric data, and either is a fair summary.
❓ Frequently asked Frequently asked questions
What is the difference between mean, median, and mode?
They are three different "averages," and they can disagree. Take the list 5, 7, 8, 8, 12, 14, 16. The mean is the sum divided by the count: 70 ÷ 7 = 10. The median is the middle value once the list is sorted — the fourth of seven values here, which is 8. The mode is the most frequent value — 8, which appears twice while everything else appears once. The mean uses every value's size, so big values pull it around (the 16 drags the mean two points above the median); the median only cares about position; the mode only cares about repetition. Report the one that matches the question you're answering.
When is the median better than the mean?
When the data contain outliers or are heavily skewed. Consider five salaries: $40,000, $45,000, $50,000, $55,000 and $300,000. The mean is $490,000 ÷ 5 = $98,000 — higher than what four of the five people actually earn, because the single $300,000 salary drags it up. The median is $50,000, which describes the typical person far better. That is why house prices and incomes are almost always reported as medians. Use the mean when values genuinely accumulate — costs, totals, measurements without outliers — and the median when a few extreme values would misrepresent the middle.
How do I calculate a weighted average?
Multiply each value by its weight, add those products, then divide by the total of the weights. Say homework counts 30% and the final exam 70%, and you scored 84 on homework and 92 on the exam: (84 × 30 + 92 × 70) ÷ (30 + 70) = (2,520 + 6,440) ÷ 100 = 89.6. Note the plain mean of 84 and 92 is 88 — weighting toward the higher-scored exam lifts the result. Weights don't have to sum to 100: the calculator divides by whatever they total, so weights of 3 and 7 give exactly the same answer as 30 and 70.
What is the difference between population and sample standard deviation?
The formulas are nearly identical — the only difference is what you divide by. Both start from the squared deviations from the mean. For 5, 7, 8, 8, 12, 14, 16 (mean 10), the squared deviations sum to 98. If your list is the entire population, divide by n: 98 ÷ 7 = 14, so the population standard deviation is √14 ≈ 3.742. If your list is a sample drawn from something bigger, divide by n − 1: 98 ÷ 6 ≈ 16.333, giving a sample standard deviation of ≈ 4.041. Dividing by n − 1 (Bessel's correction) compensates for the fact that a sample's own mean understates its spread. This calculator shows both, clearly labelled — pick the one that matches your data.
What is the geometric mean and when should I use it?
The geometric mean multiplies n values together and takes the n-th root: for 4 and 9 it is √(4 × 9) = √36 = 6, not the arithmetic 6.5. Use it whenever values combine by multiplying — growth rates, investment returns, percentage changes. An investment that gains 20% one year and loses 10% the next has grown by a factor of 1.20 × 0.90 = 1.08 over two years; the geometric mean √1.08 ≈ 1.0392 says the equivalent steady growth is about 3.92% per year — while the arithmetic mean of +20% and −10% wrongly suggests 5%. It is only defined for positive numbers; the calculator tells you if your list contains zeros or negatives.
Can a list have two modes — or no mode at all?
Yes to both. The mode is the most frequent value, and nothing forces there to be exactly one. In 1, 2, 2, 3, 3, 4, both 2 and 3 appear twice, so the list is bimodal with modes 2 and 3 (its mean and median are both 2.5). And if every value appears exactly once — say 3, 7, 11 — there is no mode at all. This calculator lists all tied modes, and reports "no mode" when no value repeats.
How do I find the median of an even number of values?
Sort the list and average the two middle values. For 3, 5, 9, 11 the two middle values are 5 and 9, so the median is (5 + 9) ÷ 2 = 7 — a value that isn't in the list, which is perfectly normal. With an odd count there is a single middle value: in a sorted list of 7 numbers it is the 4th. In general, for n values the median is the ((n + 1) ÷ 2)-th sorted value when n is odd, and the mean of the (n ÷ 2)-th and the next value when n is even.
What does the range tell me?
The range is simply the largest value minus the smallest — for 5, 7, 8, 8, 12, 14, 16 it is 16 − 5 = 11. It is the quickest possible read on spread, but it is fragile because it depends on only the two most extreme values: add a single 100 to that list and the range jumps from 11 to 95 while most of the data hasn't changed at all. For a spread measure that uses every value, use the standard deviation at the Detailed level.
How does drop-the-lowest grading work?
Many instructors drop each student's lowest N quiz or homework scores before averaging. With scores of 62, 78, 85, 90 and 95, the plain mean is 410 ÷ 5 = 82. Drop the lowest one (the 62) and the mean of the remaining four is 348 ÷ 4 = 87 — a full letter-grade step in many schemes. The Advanced level does this automatically: enter how many lowest scores to drop and the calculator shows which values were removed and the new mean and median. It always drops the smallest values, which is the standard policy — so dropping scores can only raise (or leave unchanged) your average.
Where these figures come from
Averages are standard mathematics, not published rates — every number on this page follows from the definitions below. The conventions (including which denominator each standard deviation uses) come from these references:
- Arithmetic mean, median and mode — definitions — Wolfram MathWorld — Arithmetic Mean, Statistical Median and Mode.
- Population vs sample standard deviation (n vs n − 1, Bessel's correction) — NIST/SEMATECH e-Handbook of Statistical Methods — Measures of Scale.
- Weighted arithmetic mean — Wikipedia — Weighted arithmetic mean.
- Geometric mean and the AM–GM relationship — Wolfram MathWorld — Geometric Mean.
Last checked: July 2026. Mathematical definitions don't change; the practical conventions noted above (which denominator spreadsheets use, drop-the-lowest grading policies) reflect usage as of this check.
Pick the question your list is raising — outliers, weights, spread or grades.
The headline is your list's arithmetic mean — the sum of every value divided by how many there are — with the median right beside it for context.
Mean above median means a tail of large values is pulling it up; below means a low tail. In the default list the mean (10) sits two above the median (8) because of the 14 and 16 at the top end.
If no value repeats, there is no mode — and if two values tie, both are modes. The breakdown lists every tied mode rather than silently picking one.
Every statistic divides by the count or count minus one, so a stray typo token changes everything. The breakdown shows the count and flags any entries that couldn't be read as numbers.
Four rules of thumb pick the right average for almost any job.
One $300,000 salary in a list of $40–55k salaries drags the mean to $98,000; the median stays at $50,000. Skewed money data almost always wants the median.
Growth rates compound. +20% then −10% is ×1.08 overall — a steady 3.92%/year (geometric), not the arithmetic 5%.
Divide squared deviations by n when your list is everything there is (a whole class's scores), and by n − 1 when it's a sample standing in for something larger (a survey of some customers). The sample version is always slightly bigger — Bessel's correction — because a sample's own mean sits artificially close to its values. Both are shown at the Detailed level, so quote the one your context demands.
The three jobs people bring to an average calculator, and the level that does each fastest.
Simple level: paste the scores and read the mean — the median and mode come free, and the chart shows where you cluster.
Standard level: values 84, 92 with weights 30, 70 → 89.6. Works for any component count — weights needn't total 100.
Advanced level: paste all quizzes, set drop = 1. 62, 78, 85, 90, 95 goes from 82 to 87 the moment the 62 disappears.
Averages feed straight into grades and change-over-time questions — these tools pick up where this one stops.
The weighted average with the GPA points scale built in — semester and cumulative.
GPA →Works backward from your current average to the score that gets your target grade.
Final Grade →Percent increase and decrease between two values — the other half of most data questions.
Percentage Increase →