How to Calculate an Average
By the Calculator.hair team · Updated · 8 min read
Short answer
To calculate an average, add the values and divide by how many there are: 85, 90, 78, 92 and 85 total 430, and 430 ÷ 5 = 86. For a weighted average, multiply each value by its weight, add the results and divide by the total weight. Use the median instead when a few extreme values would pull the mean away from a typical value.
Key takeaways
- The everyday average is the mean: the sum of the values divided by the count.
- A weighted average is Σ(weight × value) ÷ Σ(weight), and it applies whenever values don't count equally.
- The median resists outliers, so it describes skewed data such as incomes better than the mean does.
- The mode is the most frequent value, and it is the only one of the three that works for categories.
- Average rate of change is (f(b) − f(a)) ÷ (b − a), the slope between two points.
How do you calculate an average?
Add all the values, then divide by the number of values. The formula is mean = sum of values ÷ count. The word average usually refers to this arithmetic mean, although the median and mode are also kinds of average.
Worked example: five test scores of 85, 90, 78, 92 and 85 add up to 430. Dividing by the count of 5 gives an average of 86. The average calculator does this from a pasted list of numbers and also shows the sum, count, median, mode, range and geometric mean.
Count every value, including zeros. A zero is a real score, so 80, 90 and 0 average to 170 ÷ 3, or about 56.67, rather than 85. Leave a value out only if it is truly missing.
What is the difference between mean, median and mode?
The mean is the sum divided by the count, the median is the middle value once the data is sorted, and the mode is the value that appears most often. All three try to describe a typical value, but they can disagree. The mean, median, mode and range calculator reports them together, along with quartiles and outliers.
To find the median, sort the values first. With an odd count, it is the middle number: 78, 85, 85, 90, 92 has a median of 85. With an even count, it is the mean of the two middle numbers: 72, 85, 88, 91 has a median of (85 + 88) ÷ 2 = 86.5.
A data set can have no mode, one mode or several. In 4, 4, 5, 5, 6, both 4 and 5 appear twice, so it has two modes, which is called bimodal. In 72, 85, 88, 91, no value repeats, so our calculators report no mode. The range, often listed alongside these, is the largest value minus the smallest. It measures spread rather than center.
| Measure | How to find it | Scores: 85, 90, 78, 92, 85 | Best for |
|---|---|---|---|
| Mean | Sum ÷ count | 430 ÷ 5 = 86 | Roughly symmetric data with no extreme values |
| Median | Middle of the sorted list | 85 | Skewed data or data with outliers |
| Mode | Most frequent value | 85 (appears twice) | Categories and most-common questions |
| Range | Largest − smallest | 92 − 78 = 14 | A quick measure of spread |
When is the median better than the mean?
The median is better when the data is skewed or has outliers, meaning values far from the rest, because one extreme value can move the mean a long way but barely moves the median. Income and home prices are the usual examples, since a few very high values stretch the upper end.
Worked example: five employees earn $30,000, $35,000, $40,000, $45,000 and $200,000. The mean is $70,000, a figure that four of the five don't come close to. The median is $40,000, which describes the typical employee far better. Remove the top earner and the mean falls by $32,500 while the median falls by only $2,500.
The mean still has advantages. It uses every value, and it connects directly to totals: mean × count = sum. If you need to budget for total payroll, the $70,000 mean is the useful number, since 5 × $70,000 gives the full $350,000. So the question to ask is what you plan to do with the figure.
| Salaries | Mean | Median |
|---|---|---|
| $30,000, $35,000, $40,000, $45,000, $200,000 | $70,000 | $40,000 |
| $30,000, $35,000, $40,000, $45,000 | $37,500 | $37,500 |
| Change after removing $200,000 | −$32,500 | −$2,500 |
What is the formula for a weighted average?
The weighted average formula is Σ(w × x) ÷ Σ(w), where each value x is multiplied by its weight w, the products are added, and the total is divided by the sum of the weights. The symbol Σ means "add them all up." A weight is a number showing how much a value should count. An ordinary mean is the special case where every weight is equal.
Worked example: a course counts homework for 20% of the grade, the midterm for 30% and the final for 50%. A student scores 90, 80 and 70. The table shows the calculation. Our average calculator has no weights field, so we did this arithmetic by hand and then confirmed it by entering 90 twice, 80 three times and 70 five times, which also returns 77.
| Component | Score | Weight | Score × weight |
|---|---|---|---|
| Homework | 90 | 0.20 | 18 |
| Midterm | 80 | 0.30 | 24 |
| Final exam | 70 | 0.50 | 35 |
| Total | 1.00 | 77 |
Why does a weighted average differ from a simple average?
A simple average gives every value the same influence, so it misleads when the values represent different amounts. In the grade example, the simple average of 90, 80 and 70 is 80, but the weighted grade is 77 because the lowest score carries half the weight. The same logic produces a GPA, where courses are weighted by credit hours.
Prices work the same way. If you buy 100 shares at $10 and 50 shares at $16, you spent $1,000 + $800 = $1,800 on 150 shares, so your average cost is $1,800 ÷ 150 = $12 a share. The simple average of the two prices is $13, which overstates your cost because you bought fewer shares at the higher price.
This is also why you shouldn't take a plain average of averages when the groups differ in size. Suppose a class of 20 students averages 80 and a class of 30 averages 90. The combined average is (20 × 80 + 30 × 90) ÷ 50 = 4,300 ÷ 50 = 86, not the 85 you get from averaging 80 and 90.
What is the average rate of change formula?
The average rate of change formula is (f(b) − f(a)) ÷ (b − a), the change in the output divided by the change in the input between two points. Despite the name, this isn't a mean of a list. It is the slope of the straight line joining those two points, which you can compute with the slope calculator or read about in the slope formula guide. These examples are our own arithmetic.
Worked example: a car has traveled 120 miles after 2 hours and 300 miles after 5 hours. Its average rate of change is (300 − 120) ÷ (5 − 2) = 180 ÷ 3 = 60 miles per hour, whatever its speed was at any given moment.
For a function, the same formula applies. For f(x) = x² from x = 1 to x = 4, the result is (16 − 1) ÷ (4 − 1) = 15 ÷ 3 = 5. The units are always output per input, such as miles per hour or dollars per year.
Which average should you use for speeds and returns?
Use total distance ÷ total time for speed and the geometric mean for growth rates, because a plain mean gives the wrong answer in both cases.
Speed: if you drive 120 miles at 60 mph and return at 40 mph, the trip takes 2 + 3 = 5 hours. Your average speed is 240 ÷ 5 = 48 mph, not 50, because you spend more time at the slower speed. For more on this, see the speed, distance and time formula.
Returns: a 50% gain followed by a 50% loss has a plain average of 0%, but $100 becomes $150 and then $75. The geometric mean multiplies the growth factors and takes the root: √(1.5 × 0.5) = 0.866, about a 13.4% loss per period. Entering 1.5 and 0.5 in the average calculator gives a geometric mean of 0.8660254038. This compounding effect is explained further in the compound interest guide.
What mistakes should you avoid when calculating averages?
The common errors come from ignoring what the numbers represent.
- Averaging averages or percentages from groups of different sizes without weighting them.
- Forgetting to sort the data before finding the median.
- Dropping zeros that are real values, which inflates the mean.
- Reporting only a center. Two data sets can share a mean and differ widely in spread, which the standard deviation measures: 79, 80, 81 and 0, 80, 160 both average 80.
- Using weights that don't sum to 1 and then failing to divide by their total.
Calculators in this guide
Frequently asked questions
How do you calculate a weighted average in percent?
Convert each percent weight to a decimal, multiply it by its value and add the products. If the weights total 100%, you don't need to divide, so 90 × 0.20 + 80 × 0.30 + 70 × 0.50 = 77.
Is average the same as mean?
In everyday use, yes: average usually refers to the arithmetic mean. Strictly speaking, average is a broader term that can also refer to the median, the mode or other measures of center.
Can the mean and median be the same?
Yes. In a perfectly symmetric data set they are equal, as with 30,000, 35,000, 40,000 and 45,000, where both are 37,500. A large gap between them is a sign the data is skewed.
What if a data set has no mode?
If every value appears exactly once, there is no mode, and our calculators report none. That is normal for small lists, so use the mean or median instead.
How do you find the average of percentages?
If the percentages share the same base, take a plain mean. If they come from groups of different sizes, weight each by its group size, or add up the underlying totals and calculate the percentage from those.
How do you work out what score you need to reach an average?
Multiply the target average by the total count, then subtract the sum you already have. To average 85 over 4 tests with 80, 85 and 90 so far, you need 85 × 4 − 255 = 85.