Average Calculator
Calculate mean, median, mode, standard deviation and more from a list of numbers.
Mean, median and mode: when to use each
Three measures of "average" describe the centre of a data set differently. Choosing the wrong one can seriously misrepresent what the data shows. A single extreme value (outlier) can drag the mean far from the typical value, making the median a better choice for skewed distributions like house prices or salaries.
Standard deviation explained
Standard deviation (Ο) measures how spread out the values are around the mean. A small Ο means values cluster tightly around the mean; a large Ο means they are spread widely. It is calculated as the square root of the variance β the average of the squared distances from the mean.
Data: 2, 4, 4, 4, 5, 5, 7, 9
Mean = (2+4+4+4+5+5+7+9) Γ· 8 = 5
Variance = [(2β5)Β²+(4β5)Β²+(4β5)Β²+(4β5)Β²+(5β5)Β²+(5β5)Β²+(7β5)Β²+(9β5)Β²] Γ· 8
= [9+1+1+1+0+0+4+16] Γ· 8 = 4
Standard deviation Ο = β4 = 2
β Most values fall within 5 Β± 2 (i.e. between 3 and 7)The 68-95-99.7 rule (normal distribution)
In a normal (bell-curve) distribution, standard deviation defines predictable ranges: approximately 68% of values fall within Β±1Ο of the mean, 95% within Β±2Ο, and 99.7% within Β±3Ο. This rule is used in quality control, finance, and scientific research to judge whether an observation is unusual.
Real-world examples
Tips for interpreting averages
- Always report mean and standard deviation together. A mean of 50 tells you the centre; a Ο of 2 vs. Ο of 20 tells you completely different stories about the data.
- Check for skewness. If mean and median differ substantially (e.g. mean $65K vs. median $45K salary), the distribution is skewed. The median is the more honest measure of "typical."
- Sample vs. population standard deviation. This calculator uses population Ο (divide by n). If your data is a sample from a larger population, statistical software divides by nβ1 (Bessel's correction) for a less biased estimate.
Frequently asked questions
What is the difference between mean, median and mode?
The mean is the sum divided by the count, the median is the middle value when sorted, and the mode is the most frequent value. Each describes the data differently.
When should I use the median instead of the mean?
Use the median when data has outliers or is skewed β for example incomes β because a few extreme values can distort the mean.
Can a data set have more than one mode?
Yes. If two values tie for most frequent, the set is bimodal; more than two makes it multimodal. A set with no repeats has no mode.
What is the range?
The range is the difference between the largest and smallest values, giving a quick sense of how spread out the data is.
Mean = sum Γ· count | Median = middle value | Mode = most frequent | Std dev = β(Ξ£(xβΞΌ)Β²Γ·n)