What is the difference between mean, median, and mode?
The mean is the arithmetic average (sum ÷ count). The median is the middle value when sorted — half the data lies above, half below. The mode is the most frequent value. For skewed data like incomes, the median describes "typical" better than the mean, which a few extreme values can drag upward.
Sample vs population standard deviation — which should I use?
Use sample standard deviation (dividing by n−1, Bessel's correction) when your data is a subset of a larger group you're trying to describe — the usual case for surveys and experiments. Use population standard deviation (dividing by n) only when you've measured every member of the group. For large n the difference becomes negligible.
What does standard deviation actually tell me?
How spread out the data is around the mean, in the same units as the data. For roughly bell-shaped data, about 68% of values fall within 1 standard deviation of the mean and 95% within 2. A small SD means consistent values; a large one means high variability.
What if my data has no mode or several modes?
If every value appears exactly once, there is no mode — this calculator reports "None". If two or more values tie for the highest frequency, the data is multimodal and all tied values are listed. Multimodality often hints that your data mixes two distinct groups.
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations from the mean; standard deviation is its square root. They carry identical information, but in different units. If your data is in centimetres, the variance is in square centimetres, which is impossible to interpret directly, while the standard deviation comes back in centimetres and can be compared with the mean. So why keep variance at all? Because it adds. The variance of a sum of independent quantities is the sum of their variances, which is not true of standard deviations. That additivity is why variance is the quantity statistical theory is built on and standard deviation is the one reported.
How large does a sample need to be?
Precision improves with the square root of the sample size, not with the size itself, so the standard error falls as one over the square root of n. Quadrupling your sample halves the uncertainty, and the returns diminish steeply from there — which is why national polls of a few thousand people give margins of a few percent regardless of whether the population is ten million or three hundred million. Far more important than size is whether the sample represents the population. The 1936 Literary Digest poll surveyed over two million people and still called the US election wrongly, because its list of names was drawn from car and telephone owners.