Median when the data are skewed or contain outliers, mean when they are roughly symmetric. Report both when they disagree noticeably, the gap is informative.

Why divide by n − 1?

Because deviations are measured from the sample mean rather than the true mean, which slightly understates spread. Dividing by n − 1 corrects that bias.

What does a 95% confidence interval mean?

That the method captures the true parameter in 95% of repeated samples. It is a statement about the procedure, not a probability for the specific interval you computed.

Does a small p-value prove an effect?

No. It says the data would be unlikely if the null hypothesis were true. Effect size and study design decide whether the result matters.

Statistics study guide

Statistics moves from describing a sample to making claims about a population. Every formula in the second half exists to quantify how much that leap might be wrong.

Practise statistics & probability

Describing data

Centre and spread must be reported together. The mean and standard deviation suit roughly symmetric data; the median and interquartile range are the honest choice when outliers or skew are present.

Variance uses squared deviations so that distances above and below the mean do not cancel. The sample version divides by n − 1 because using the sample mean costs one degree of freedom.

x ˉ = \frac{\sum x _{i}}{n}

s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

Probability

Start by deciding whether events are mutually exclusive and whether they are independent. Those two answers determine which addition and multiplication rules apply.

Conditional probability re-scales the sample space to the given condition. Bayes' theorem is that idea run in reverse, converting one conditional into the other.

P (A \cup B) = P (A) + P (B) - P (A \cap B)

P (A \cap B) = P (A) P (B ∣ A)

Distributions

The binomial distribution counts successes in a fixed number of independent trials with constant probability. Check all four conditions before using it.

The normal distribution describes continuous data clustered around a mean. Converting to a z-score standardises any normal variable so a single table serves every problem.

P (X = k) =^{n} C_{k} p^{k} (1 - p)^{n - k}

z = \frac{x - μ}{σ}

Inference

The central limit theorem is what makes inference possible: sample means are approximately normal for large samples regardless of the population's shape, with standard error shrinking as the square root of sample size.

A confidence interval reports a range of plausible parameter values; a hypothesis test asks whether the data are surprising under an assumed value. A p-value is that surprise, not the probability the hypothesis is true.

S E = \frac{s}{n}

x ˉ \pm z^{*} \frac{s}{n}

Frequently asked questions

Mean or median?: Median when the data are skewed or contain outliers, mean when they are roughly symmetric. Report both when they disagree noticeably, the gap is informative.
Why divide by n − 1?: Because deviations are measured from the sample mean rather than the true mean, which slightly understates spread. Dividing by n − 1 corrects that bias.
What does a 95% confidence interval mean?: That the method captures the true parameter in 95% of repeated samples. It is a statement about the procedure, not a probability for the specific interval you computed.
Does a small p-value prove an effect?: No. It says the data would be unlikely if the null hypothesis were true. Effect size and study design decide whether the result matters.

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