Mathwords logoMathwords

Average

Average

This almost always refers to the arithmetic mean. In general, however, the average could be any single number that represents the center of a set of values.

 

 

See also

Geometric mean, harmonic mean, root mean square, median, mode, weighted average

Key Formula

Average (Arithmetic Mean)=x1+x2+⋯+xnn\text{Average (Arithmetic Mean)} = \frac{x_1 + x_2 + \cdots + x_n}{n}
Where:
  • x1,x2,…,xnx_1, x_2, \ldots, x_n = The individual values in the data set
  • nn = The total number of values

Worked Example

Problem: A student scores 80, 90, 75, 95, and 85 on five math tests. Find the average score.
Step 1: Add all the scores together.
80+90+75+95+85=42580 + 90 + 75 + 95 + 85 = 425
Step 2: Count the number of values. There are 5 test scores.
n=5n = 5
Step 3: Divide the sum by the number of values.
4255=85\frac{425}{5} = 85
Answer: The average test score is 85.

Another Example

Problem: The daily high temperatures for a week are 60°F, 62°F, 58°F, 65°F, 70°F, 68°F, and 63°F. What is the average high temperature?
Step 1: Add all seven temperatures.
60+62+58+65+70+68+63=44660 + 62 + 58 + 65 + 70 + 68 + 63 = 446
Step 2: Divide the sum by 7 (the number of days).
4467≈63.7\frac{446}{7} \approx 63.7
Answer: The average high temperature for the week is approximately 63.7°F.

Frequently Asked Questions

What is the difference between average and mean?
In most contexts, 'average' and 'mean' refer to the same thing: the arithmetic mean. However, 'average' is an informal, everyday word, while 'mean' is the precise mathematical term. Technically, 'average' can also refer to other measures of center like the median or mode, but unless stated otherwise, it means the arithmetic mean.
Can the average be a number not in the data set?
Yes. The average often does not match any individual value in the set. For example, the average of 3 and 8 is 5.5, which is not 3 or 8. The average represents the center of the data, not necessarily an actual data point.

Average (Mean) vs. Median

The average (arithmetic mean) adds all values and divides by the count, so every value affects the result — including outliers. The median is the middle value when the data is sorted, making it resistant to extreme values. For the set {1, 2, 3, 4, 100}, the average is 22, but the median is 3. When data is symmetric, the mean and median are close. When data is skewed or has outliers, the median often gives a better sense of what is 'typical.'

Why It Matters

Average is one of the most widely used concepts in all of mathematics and daily life. Teachers use it to compute grades, scientists use it to summarize experimental data, and economists use it to track indicators like average income or average price. Understanding when the average is a good summary — and when it can be misleading due to outliers — is a critical skill in data literacy.

Common Mistakes

Mistake: Forgetting to divide by the correct number of values, especially when a zero appears in the data.
Correction: A zero is still a data point. If your scores are 90, 80, and 0, you divide by 3 (not 2). The average is (90 + 80 + 0) / 3 ≈ 56.7, not 85.
Mistake: Assuming the average is always a good representation of the data.
Correction: Outliers can pull the average far from most values. For example, if four people earn 30,000andoneearns30,000 and one earns1,000,000, the average salary is $224,000 — far higher than what most of the group earns. In such cases, the median may be more informative.

Related Terms