An average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.
The average is a fair share: move blocks from the tall stacks into the short ones until all stacks are level. The common height is the average.
Formal definition and notation
For data values x1, x2, …, xn, the average (arithmetic mean) is x1 + x2 + ⋯ + xnn, the sum of the values divided by the number of values. It is the fair-share value: if every value were replaced by it, the total would stay the same. More broadly, any measure of center, including the median and mode, can be called an average.
the spreadsheet function that adds a range and divides by the count
Key facts
The average always lies between the smallest and the largest value.
Zeros count. The average of 0, 4 and 8 is 123 = 4, not 122 = 6.
The average does not have to be one of the data values, or even a whole number: 1, 2, 2, 4 average 2.25.
One extreme value, an outlier, pulls the average toward it. The median barely moves.
Adding the same number to every value adds that number to the average; doubling every value doubles it.
When groups have different sizes, the average of the group averages is not the overall average; use a weighted average.
How to find the average
How to find the average
average = sum of the valuesnumber of values
sum of the values
add every value, including zeros and repeats
number of values
how many values you added
Turned around, sum = average × count. That is the key to “missing score” problems.
Try it: move the average
Interactive
Drag a dot or edit the numbers and watch the mean (the everyday average) change. Add a large value and compare how far the mean moves with how far the median moves.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Average examples, step by step
Example 1: Average of four numbers
Warm-up
Find the average of 12, 15, 9 and 20.
Add the values: 12 + 15 + 9 + 20 = 56.
There are 4 values, so divide by 4: 564 = 14.
Answer: The average is 14.
Example 2: The missing test score
Core
Sam wants an average of 86 on five tests. His first four scores are 82, 90, 79 and 88. What score does he need on the fifth test?
The five scores must add to 5 × 86 = 430.
His first four scores add to 82 + 90 + 79 + 88 = 339.
He needs 430 − 339 = 91.
Check: 339 + 915 = 4305 = 86.
Answer: Sam needs a 91 on the fifth test.
Example 3: A new value joins the group
Test-style
The average score of 20 students on a quiz is 75. A new student takes the quiz and scores 96. What is the new class average?
Total of the 20 scores: 20 × 75 = 1500.
Add the new score: 1500 + 96 = 1596.
Now there are 21 students: 159621 = 76.
Answer: The new average is 76.
Common mistakes
✗ MistakeSkipping a zero: the average of 3, 8, 10, 0, 9 is 304 = 7.5.
✓ InsteadCount all five values: 305 = 6.
A zero adds nothing to the sum but is still a value, so it counts in the divisor.
✗ MistakeAveraging two class averages: 10 students averaged 80 and 30 students averaged 60, so all 40 averaged 80 + 602 = 70.
Average rate of change is the slope between two points of a function, Δ yΔ x, not a measure of center of a data set.
Quick check: 3 questions
Question 1 of 3
What is the average of 3, 8, 10, 0 and 9?
Show answer and explanation
Answer: B. The sum is 30 and there are 5 values, so the average is 305 = 6. 7.5 left out the zero, 8 is the median, and 30 is only the sum.
Question 2 of 3
The average of four numbers is 15. Three of them are 10, 12 and 20. What is the fourth number?
Show answer and explanation
Answer: D. The four numbers add to 4 × 15 = 60. The three known ones add to 42, so the fourth is 60 − 42 = 18. 14 is the average of the three, 42 is their sum, and 15 just repeats the average.
Question 3 of 3
Six numbers have an average of 9. What is their sum?
Show answer and explanation
Answer: A. Sum = average times count = 9 × 6 = 54. 15 adds instead of multiplying, 1.5 divides 9 by 6, and 9 is the average itself.
Average: frequently asked questions
Is average the same as mean?
Usually, yes. When a teacher, a test or a news report says “average” with no other word, it means the arithmetic mean: the sum divided by the count. Statistics books also call the median and the mode averages, because all three describe a center. If a question means the median, it will say median.
Can the average be a number that is not in the data?
Yes. The average of 1, 2, 2 and 4 is 94 = 2.25, which is not one of the values. An average can be a decimal even when every value is a whole number, like 2.5 children per family. It always lies between the smallest and largest values, though.
How do you find the average of percentages?
If every percent comes from the same size total, just average them: 80%, 90% and 70% average 80%. If the totals differ, go back to the counts. 9 of 10 and 30 of 50 is 39 out of 60, or 65%, not the average of 90% and 60%.
MeanThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.Statistics · Grade 5, Grade 6, Grade 7MedianThe median is the middle value when data are listed in order. If there is an even number of values, the median is the average of the two middle values. Half the data lie at or below the median and half at or above it, and outliers barely move it.Statistics · Grade 5, Grade 6, Grade 7ModeThe mode is the value that appears most often in a data set. To find it, count how many times each value occurs and pick the most frequent one. A data set can have one mode, more than one mode, or no mode at all when every value appears only once. The mode works for categories, too.Statistics · Grade 5, Grade 6, Grade 7DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SampleA sample in statistics is a part of a population that is actually observed or measured, such as 60 students surveyed out of a school of 900. You use the sample to estimate facts about the whole population. A random sample, where every member has the same chance to be chosen, tends to represent the population fairly.Statistics · Grade 7, Statistics, Algebra 1PopulationA population in statistics is the whole group of people, animals or objects you want to learn about, such as every student in a school or every bulb a factory makes. Measuring all of them is often too costly, so you study a sample, a part of the population, and use it to draw conclusions about the whole group.Statistics · Grade 7, Statistics, Algebra 1Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5OutlierAn outlier is a data value that is far away from most of the other values in a set. In 4, 5, 6, 7, 50, the value 50 is an outlier. A common test flags any value more than 1.5 times the interquartile range below the first quartile or above the third. Outliers pull the mean strongly.Statistics · Grade 6, Grade 7, Grade 8Weighted AverageA weighted average is an average in which some values count more than others. Multiply each value by its weight, add those products, and divide by the sum of the weights. Course grades, GPAs and combined groups use it, because a plain mean would treat every value as equally important.Statistics · Grade 7, Grade 8, Algebra 1DivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5Average Rate of ChangeThe average rate of change of a function over an interval is the change in output divided by the change in input: f(b) − f(a)b − a. It tells how fast the function changes, on average, from x = a to x = b. On a graph it is the slope of the line through the two endpoints.Functions · Algebra 1, Algebra 2, PrecalculusSlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1SumA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2, Grade 3Standard DeviationStandard deviation is a number that measures how spread out data values are from their mean: roughly, the typical distance of a value from the mean. A small standard deviation means the values cluster near the mean; a large one means they are spread out. It is the square root of the variance.Statistics · Algebra 1, StatisticsRangeThe range of a data set is the difference between the greatest and least values: maximum minus minimum. For 2, 5, 9, the range is 9 − 2 = 7. Range is a simple measure of spread: it tells how far apart the extreme values are, though one outlier can make it misleading about the rest of the data.Statistics · Grade 5, Grade 6, Grade 7