See the distribution the mean can hide.

The mean is the sum divided by the count — the balance point of the data. The median is the middle value once everything is sorted, with the average of the two middle values when the count is even. The mode is whatever value appears most often. They coincide in a perfectly symmetric distribution and separate as soon as it skews, which is why the gap between the mean and the median is itself a useful diagnostic: mean above median means a few large values are pulling the average up.

Whenever the distribution is skewed or contains outliers you do not want to let dominate. Household income is the standard example: a handful of very high incomes drag the mean far above what a typical household earns, so official statistics report the median. The mean is the better choice for symmetric data and whenever you need a figure that adds up correctly across groups, since medians cannot be combined by weighting the way means can.

Because the sample mean is itself estimated from the same data, and it sits by construction slightly closer to the sample points than the true population mean does. Dividing by n would therefore systematically understate the spread. Dividing by n − 1, known as Bessel's correction, makes the variance an unbiased estimator of the population variance. The correction matters most on small samples: with n = 5 it inflates the variance by 25%, with n = 100 by about 1%.

Variance is the average of the squared distances from the mean, so it is expressed in squared units — squared dollars, squared centimeters — which nobody can interpret directly. The standard deviation is its square root, which brings it back into the original units and makes it comparable to the mean. Variance remains the quantity that behaves nicely in algebra: variances of independent variables add, standard deviations do not.

It is the standard deviation divided by the mean, expressed as a percentage, and it lets you compare the spread of data sets that use different units or sit at very different scales. A standard deviation of 5 is enormous around a mean of 10 and negligible around a mean of 10,000. As rough guidance, under 15% counts as tightly clustered, 15–30% as moderate, and above 30% as genuinely scattered. It is only meaningful for ratio data with a positive mean.

Multiply each value by its weight, add up those products, and divide by the sum of the weights. The mistake to avoid is dividing by the number of items instead of by the total weight. For a GPA the values are grade points and the weights are credit hours, so a 3.0 earned in a four-credit course counts twice as much as a 4.0 earned in a two-credit seminar. Weights do not need to sum to one; the formula normalises them for you.

R² is the share of the variation in Y that the fitted line accounts for. An R² of 0.82 means the line explains 82% of how much Y moves, leaving 18% to everything the model does not capture. It equals the square of the correlation coefficient r for a simple linear regression. What it does not tell you is whether the relationship is causal, whether a straight line was the right shape, or whether one outlier is doing all the work.

No. A strong fit is consistent with X causing Y, with Y causing X, with a third variable driving both, and with pure coincidence in a small sample. Ice cream sales and drowning deaths correlate strongly, and the cause of both is summer. Establishing causation needs either an experiment where you control X, or a design that rules out the plausible confounders. The regression tells you about association only.

A permutation counts arrangements where order matters; a combination counts selections where it does not. Picking three people for a committee is a combination — the same three people are the same committee whatever order you name them in. Picking a president, a treasurer and a secretary from the same three is a permutation, because swapping roles gives a different outcome. The two are linked by P(n,k) = C(n,k) × k!, since each selection can be ordered k! ways.

C(49,6) = 49! ÷ (6! × 43!) = 13,983,816. Order does not matter on a lottery ticket, so it is a combination rather than a permutation — the permutation count P(49,6) is 720 times larger, at just over 10 billion. Because there are fourteen million equally likely tickets, buying one of each would guarantee the jackpot, and would also cost far more than most jackpots pay out.

It is the standard deviation divided by the square root of the sample size, and it estimates how much your sample mean would bounce around if you repeated the sampling. It is the quantity behind confidence intervals: roughly 95% of sample means fall within about two standard errors of the true mean. Note the square root — quadrupling your sample size only halves the standard error, which is why precision gets expensive fast.

The arithmetic works from n = 1 for the mean and n = 2 for a sample standard deviation, but working is not the same as being informative. With fewer than about ten values a standard deviation is very unstable, and a regression on three points can show a near-perfect R² purely by chance. Treat small-sample results as a description of the numbers you have, not as an estimate of anything wider.