Statistics has a reputation for being difficult, but two ideas carry most of it: how likely something is (probability), and how spread out data is (standard deviation). This guide explains both with worked examples — and the one distinction that quietly changes the answer.
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes, it is simply the number of favourable outcomes divided by the total number of outcomes.
Roll two dice and ask for a total of 7. There are 36 equally likely combinations, and six of them sum to 7. So the probability is 6 ÷ 36 = 1/6, or about 16.7%. Our probability calculator handles these directly.
Two rules cover most cases.
Independent events, both happening — multiply. The chance of three heads in a row is 1/2 × 1/2 × 1/2 = 1/8. Each flip is independent, so the probabilities multiply.
Mutually exclusive events, either happening — add. The chance of rolling a 1 or a 6 on one die is 1/6 + 1/6 = 2/6 = 1/3.
The mean, or average, is the sum of the values divided by how many there are. It is useful, but on its own it hides how spread out the data is.
Two datasets can share a mean of 50 while being completely different — one clustered tightly around 50, the other swinging between 0 and 100. The average alone cannot tell them apart, which is exactly what standard deviation is for. Our statistics calculator reports the mean alongside the spread.
Standard deviation measures how far, on average, the values sit from the mean. A small standard deviation means the data is tightly clustered; a large one means it is widely spread.
The method: find the mean, subtract it from each value and square the result, average those squared differences, then take the square root. For the set 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5 and the standard deviation is 2. Our standard deviation calculator shows each step.
The squaring is deliberate — it stops positive and negative differences from cancelling out, and it gives extra weight to values far from the mean. The final square root returns the answer to the original units.
Here is the subtlety that changes the answer, and that most people never notice.
If your data is the entire population — every value you care about — you divide by n, the number of values. If your data is a sample drawn from a larger population, you divide by n − 1 instead. This adjustment (called Bessel's correction) compensates for the fact that a sample tends to underestimate the true spread of the population it came from.
Before calculating, ask whether your numbers are the whole population or a sample, and choose the matching formula. A good calculator lets you select which, so always check that setting.
Standard deviation becomes far more useful once you know how data is distributed. Many natural measurements — heights, test scores, measurement errors — follow an approximately normal distribution, the familiar symmetric bell curve.
For normally distributed data, standard deviation carries a remarkably consistent meaning, summarised by the 68–95–99.7 rule:
So if a test has a mean of 60 with a standard deviation of 10, roughly 95% of results sit between 40 and 80. A value beyond three standard deviations is genuinely unusual — under 0.3% of the data — which is the basis for spotting outliers.
How do I calculate the probability of an event?
For equally likely outcomes, divide the number of favourable outcomes by the total number of possible outcomes. Rolling a total of 7 with two dice has 6 favourable combinations out of 36, so the probability is 6 ÷ 36 = 1/6, or about 16.7 per cent.
How do I combine two probabilities?
For two independent events both happening, multiply their probabilities — three coin heads in a row is 1/2 × 1/2 × 1/2 = 1/8. For two mutually exclusive events where either can happen, add them — rolling a 1 or a 6 is 1/6 + 1/6 = 1/3.
What is the gambler's fallacy?
It is the mistaken belief that independent events are influenced by past results. After several heads in a row, the next coin flip is still 50/50, because the coin has no memory. Expecting a tail to be more likely to balance things out is a common and costly error.
What does standard deviation measure?
It measures how far, on average, values sit from the mean. A small standard deviation means data is tightly clustered around the average; a large one means it is widely spread. Two datasets can share the same mean but have very different standard deviations.
What's the difference between population and sample standard deviation?
If your data is the entire population, you divide by the number of values (n). If it is a sample from a larger group, you divide by n − 1, which corrects for a sample's tendency to underestimate the true spread. The same data gives different answers, so choose the formula that matches your data.
Why do you square the differences when calculating standard deviation?
Squaring stops positive and negative differences from cancelling each other out, and gives more weight to values far from the mean. Taking the square root at the end returns the result to the original units so it can be interpreted alongside the data.