Average Calculator
Plain, weighted, geometric, harmonic and trimmed mean from one list of values
Paste a list instead
Want the middle value rather than the balance point? The median, mode, spread and quartiles are on the statistics calculator.
Which mean fits your numbers
Every tile above is a correct average of the same list. Only one of them answers your question, and the choice depends on how the numbers combine in real life.
| Your numbers are | Use | Because |
|---|---|---|
| Readings of the same kind, each counting once | Mean | The totals add, so sum ÷ count is the fair share |
| Values that count unequally: exam weights, class sizes, credits | Weighted mean | Each value pulls in proportion to its weight |
| Percentages or averages from groups of different size | Weighted by group size | A plain average of them gives a small group a big vote |
| Growth rates, yearly returns, ratios that multiply | Geometric or compound | Changes compound, so the nth root of the product is the steady rate |
| Rates over a fixed distance or amount: speeds, P/E ratios, price per unit | Harmonic | More time or money sits at the slower rate |
| Panel scores or readings with a stray extreme | Trimmed mean | Dropping the ends stops one outlier moving the answer |
| Skewed data such as incomes or house prices | Median, not a mean | A few huge values drag every kind of mean upward |
For positive numbers that are not all equal, the arithmetic mean is always the largest of the three classic means and the harmonic the smallest, with the geometric in between. If your tiles ever break that order, a value was typed wrong.
Worked examples
Tap one to load it into the rows. Each is a textbook case where picking the wrong kind of average gives the wrong answer.
- Test scores: mean 71.2.
- Exam and homework: weighted mean 83.
- Two classes: weighted mean 86, not 85.
- Round trip: harmonic mean 30 km/h, not 40.
- Returns: compound average 9.98% a year, not 16.6%.
- Trimmed: 10.625 once the 4 and the 18 go.
- Pass rates: 68.39% weighted, not 70%.
The average, and which kind of mean your numbers need
Arithmetic, weighted, geometric, harmonic and trimmed means with worked examples, plus the percentage, speed and returns traps that catch most people.
An average turns a list of numbers into one number that stands for all of them. Most of the time that number is the arithmetic mean: add everything up and divide by how many there are. But the mean is a family, not a single formula, and the right member depends on how your numbers combine in real life. Test scores add, investment returns multiply, speeds divide, and a single wild judge can ruin a panel score. This calculator works out every useful mean of the same list at once, so you can see how far apart they sit and pick the one that answers your question.
The arithmetic mean, sum divided by count
Mean = (x₁ + x₂ + … + xₙ) ÷ n, written x̄ in statistics. Take 2, 7, 19, 24 and 25: they add to 77, there are five of them, and 77 ÷ 5 = 15.4. Five test scores of 56, 75, 88, 45 and 92 add to 356 and average 71.2. Negative values are allowed. The list 3, -7, 5, 13 and -2 adds to 12, so its mean is 12 ÷ 5 = 2.4.
The mean is the balance point of the data. If every value were replaced by the mean, the total would not change, which is exactly why it is the right average whenever the totals are what matter: marks that add into a final score, days that add into a year (365 ÷ 12 = 30.42 days a month), or bills that add into a budget.
Weighted average, when values count unequally
Weighted mean = Σ(w × x) ÷ Σw. Multiply each value by its weight, add the products, and divide by the total of the weights. An exam scored 90 that is worth 30% of the grade and homework scored 80 worth 70% gives (90 × 30 + 80 × 70) ÷ 100 = 83. Weights do not have to be percentages or add to 100. Scores of 70, 80 and 90 counted 2, 3 and 1 times give (140 + 240 + 90) ÷ 6 = 78.33.
The classic trap is averaging two averages. A morning class of 20 pupils averages 80 and an afternoon class of 30 averages 90. The plain average of 80 and 90 is 85, but the real average across all 50 pupils is (20 × 80 + 30 × 90) ÷ 50 = 4,300 ÷ 50 = 86, because the bigger class deserves a bigger say. Put the class size in the weight column and the calculator does it for you. The weights must be zero or positive, and they cannot all be zero, because then there is nothing to divide by.
A grade point average is the same sum with credits as the weights. An A on a 3 credit course, two Bs on 4 credit courses and a C on 2 credits is (4 × 3 + 3 × 4 + 3 × 4 + 2 × 2) ÷ 13 = 3.08. The GPA calculator handles letter grades and scales, and the grade calculator handles course weightings.
Averaging percentages without being misled
A percentage is a hidden fraction, so averaging percentages is averaging averages. Three systems installed successfully 40 times out of 50, 36 out of 45 and 30 out of 60. The rates are 80%, 80% and 50%, whose plain average is 70%. Pool the raw counts instead, 106 successes out of 155 attempts, and the true rate is 68.39%, lower because the largest group had the worst result. The calculator gets the same answer if you enter each percentage with its sample size as the weight.
The gap can be much bigger. If four people score 80% and one scores 40%, averaging the two distinct scores gives 60%, while the real average of five people is 72%. Only average percentages directly when every one of them sits on the same base.
Geometric mean, for growth that compounds
Geometric mean = (x₁ × x₂ × … × xₙ)1/n, the nth root of the product. The geometric mean of 2 and 8 is √16 = 4, and of 1, 12 and 18 it is the cube root of 216, which is 6. Every value must be above zero, which is why the tile goes blank if you enter a zero or a negative number.
Its real job is growth. Suppose $1,000 earns +10%, -12%, +90%, -30% and +25% over five years and ends at $1,609. The simple average of those five returns is 16.6% a year, which would have turned $1,000 into more than $2,100. The truthful figure is the geometric mean of the growth factors 1.10, 0.88, 1.90, 0.70 and 1.25, which is 1.0998, so 9.98% a year. Switch the calculator to % changes and it converts each percentage into a growth factor for you and shows this compound average in its own tile. Published figures are not always right either: one popular calculator gives 8.78% as the geometric mean of +15%, -10% and +25%, but (1.15 × 0.90 × 1.25)1/3 - 1 is 8.96%. For projections, the compound interest calculator runs the growth forward.
Harmonic mean, for rates and ratios
Harmonic mean = n ÷ (1/x₁ + 1/x₂ + … + 1/xₙ). Use it when you average a rate, a something per something, over a fixed amount of the thing on the bottom. Drive somewhere at 60 km/h and back at 20 km/h and your average speed is not 40. The return leg takes three times as long, and the true figure is 2 × 60 × 20 ÷ 80 = 30 km/h, the harmonic mean. The same logic averages price to earnings ratios across a portfolio and the cost per unit when you spend the same amount each time.
When the legs are not the same length, enter the distances as weights. The weighted harmonic mean, Σw ÷ Σ(w ÷ x), then gives the true average speed for a trip of 100 km at 50 km/h followed by 40 km at 70 km/h: 140 km in 2.571 hours, or 54.44 km/h.
How the three classic means relate
For positive numbers that are not all equal, the arithmetic mean is always the largest of the three Pythagorean means, the harmonic mean the smallest, and the geometric mean sits between them. They are equal only when every value is the same. For two numbers there is a neat link: the geometric mean squared equals the arithmetic mean times the harmonic mean. With the default scores the order reads 71.2, then 68.70, then 66.10.
Trimmed mean, when one value is suspect
A trimmed or truncated mean sorts the data, drops the same share from each end and averages what is left. For 4, 8, 12, 15, 9, 6, 14, 18, 12 and 9, a 10% trim removes one value from each end, the 4 and the 18, and (6 + 8 + 9 + 9 + 12 + 12 + 14 + 15) ÷ 8 = 10.625. Judged sports such as figure skating use the same idea, discarding the highest and lowest marks so one biased judge cannot swing a result, and the old LIBOR rate dropped the top 4 and bottom 4 of 18 bank submissions. Central banks watch a trimmed mean too: the Dallas Fed's Trimmed Mean PCE inflation rate was 2.3% for the 12 months to July 2026. The 25% trimmed mean has its own name, the interquartile mean.
Conventions differ, so check before you compare. This calculator trims the chosen percentage from each end, the same reading R's mean function gives its trim setting, and rounds the count down to a whole number. Excel's TRIMMEAN treats the percentage as the total to remove and rounds down to an even number: TRIMMEAN with 0.2 on 20 points drops 4, two from each end, which is only a 10% trim per end. Its own example, 4, 5, 6, 7, 2, 3, 4, 5, 1, 2 and 3 at 0.2, gives 3.778.
Quadratic mean and running averages
The quadratic mean, or root mean square, squares each value, averages the squares and takes the square root. Because no square is negative, negative and positive values cannot cancel each other out, which makes it the mean to reach for when sizes matter more than signs. For the default scores it is 73.48, a little above the arithmetic mean.
A running or cumulative average updates one value at a time with new mean = old mean + (new value - old mean) ÷ (n + 1), so you never need to add the whole list up again. The working box shows it after every row: for the default scores it reads 56, 65.5, 73, 66 and 71.2. Enter a target average and the calculator runs this backwards to tell you the next value you need. To lift a 71.2 average over five scores to 75, the sixth score has to be (75 × 6) - 356 = 94.
Blanks, zeros and what not to average
A zero is a value and pulls the mean down; a blank is not a value and is skipped. Spreadsheets agree: Excel's AVERAGE ignores empty cells, text and TRUE or FALSE in a range, but counts cells holding 0. If a missed test really scored nothing, type 0. If it was excused, leave the row empty or remove it. In the weight column, a blank next to a filled weight is counted as a weight of 1, and the tile says so.
Finally, a mean of any kind is dragged by a few huge values. Incomes, house prices and waiting times are skewed that way, and for them the middle value, the median on the statistics calculator, usually describes a typical case better.
Frequently Asked Questions
Why is my average speed for a round trip not halfway between the two speeds?
Because you spend longer at the slower speed. Out at 60 km/h and back at 20 km/h over the same distance, the return leg takes three times as long, so the true average is the harmonic mean, 2 × 60 × 20 ÷ (60 + 20) = 30 km/h, not 40. Enter the two speeds as values and read the harmonic mean tile. If the legs are different lengths, put each distance in the weight column and read the weighted harmonic tile instead.
Can I take a plain average of a column of percentages?
Only if every percentage is out of the same total. Otherwise weight each one by its base. Pass rates of 80%, 80% and 50% from groups of 50, 45 and 60 average 70% on paper, but the pooled rate is 106 out of 155, or 68.39%. Enter each percentage as a value with its group size as the weight and the weighted mean tile gives the pooled figure.
Why does the compound average of my yearly returns come out lower than the simple average?
Returns multiply rather than add, and a loss needs a bigger gain to recover. Returns of +10%, -12%, +90%, -30% and +25% average 16.6% on paper, yet $1,000 only grew to $1,609, which is 9.98% a year compounded. That lower figure is the geometric mean of the growth factors. Switch the calculator to % changes and it appears in the compound average tile.
Should a missed score be entered as a zero or left blank?
It depends on what the missing score means. A zero is a real value and pulls the average down, while a blank row is skipped completely. Excel's AVERAGE works the same way, ignoring empty cells but counting cells that hold 0. Type 0 for a test that scored nothing, and leave the row blank or delete it for one that was excused.
Do the weights in a weighted average have to be percentages that add up to 100?
No. Only the ratio between the weights matters, because the formula divides by their total. Weights of 2, 3 and 1 on scores of 70, 80 and 90 give (140 + 240 + 90) ÷ 6 = 78.33, the same as weights of 33.3%, 50% and 16.7%. Use class sizes, credits, hours or counts directly. Weights must be zero or positive, and at least one must be above zero.
What does a 10% trimmed mean remove, and does Excel mean the same thing?
Here, and in most statistics texts, it removes 10% of the values from each end after sorting, rounding the count down, then averages the rest. For ten scores that is one from the bottom and one from the top. Excel's TRIMMEAN reads the percentage as the total to remove, split between the two ends, so TRIMMEAN at 10% trims only 5% per end. Use 0.2 in Excel to match a 10% trim here.
How do I add a new number to an average without adding up the whole list again?
Use new mean = old mean + (new value - old mean) ÷ new count. If five scores average 71.2 and a sixth score of 94 arrives, the new mean is 71.2 + (94 - 71.2) ÷ 6 = 75. The working box shows this running average after every row. Enter a target average and the calculator solves it the other way round, giving the next value you need.