Centre, spread and shape in one pass
A descriptive summary answers three questions about a dataset. Where is its centre? The mean and median both estimate a typical value — the mean is the balance point, the median the middle value once the data is sorted. How spread out is it? The range, variance and standard deviation all grow as the numbers scatter further from the centre. And how is it distributed across its range? That is the job of the quartiles.
Looking at these together is more revealing than any single number. A mean far above the median, for example, hints that a few large values are pulling the average up.
The five-number summary and IQR
The chart plots the five-number summary: the minimum, first quartile (Q1), median, third quartile (Q3) and maximum. These five points cut the sorted data into four parts that each hold roughly a quarter of the values, and they are the basis of a box-and-whisker plot.
- Q1 is the value below which about 25% of the data falls; Q3 marks the 75% point.
- The interquartile range (IQR = Q3 − Q1) is the spread of the middle half, and unlike the full range it is barely affected by outliers.
- A box that sits low with a long upper whisker indicates right-skewed data.
Population or sample, and a caution
This tool reports both flavours of variance and standard deviation so you can pick the right one. Divide by n (population) when your numbers are the whole group; divide by n−1 (sample) when they are a sample used to estimate a larger group. With only one value the sample variance is undefined and is reported as zero.
Finally, descriptive statistics summarise the data you entered — they do not prove a trend or generalise to a wider population on their own. Treat them as a first look, and use inferential methods such as confidence intervals when you need to draw broader conclusions.
Formula
mean = Σx/n; variance(pop) = Σ(x−mean)²/n; variance(sample) = Σ(x−mean)²/(n−1)Frequently asked questions
- When should I use sample vs population statistics?
- Use sample statistics (dividing by n−1) when your numbers are a sample of a larger population. Use population statistics (dividing by n) when the numbers represent the entire group.