Statistics Solver with Step-by-Step Answers

• Mean, median, mode, range, quartiles, interquartile range, and outliers by the 1.5 IQR rule.
• Sample and population standard deviation, with the full table of deviations rather than a single number.
• Variance, coefficient of variation, and z-scores for comparing values from different distributions.
• Probability: addition and multiplication rules, conditional probability, independence, Bayes, permutations, and combinations.
• Binomial, Poisson, and normal distributions, including normal approximation and continuity correction.
• Sampling distributions, the central limit theorem, and the standard error that follows from it.
• Confidence intervals for a mean, a proportion, and a difference between two of either.
• Hypothesis tests: z, t, paired t, and chi-square, with p-values and a conclusion in context.
• Correlation and least-squares regression lines, with residuals and the meaning of r squared.

Three facts cannot be inferred from your numbers and change the answer if you leave them out: whether the data is a sample or a whole population, whether the test is one-tailed or two-tailed, and what significance level your course works at. State all three. It is also worth saying whether you are expected to use tables or a calculator, because a table-based answer is rounded at a different point and the two can straddle a critical value.

Three cheap checks catch most statistics errors. The mean has to lie inside the range of the data, and 18 does. The standard deviation should be a sizeable fraction of the range and never larger than it — those six values span 38 and $s$ came out at 13.5, which is reasonable for a set with one value stranded out to the right. Any probability must land between 0 and 1, so a normal-curve answer of 1.4 is a subtraction performed in the wrong direction.

The fourth check is the one used at the end of the third problem. A confidence interval and a two-sided test at the matching level are one calculation viewed from two angles, so they cannot disagree. If your test rejects a value that your interval contains, one of them carries an arithmetic error, and you know that before knowing which.

Most marks lost in an inference question are lost before any arithmetic happens, by running the wrong test on the right data. Three features decide it: what is being compared, whether the observations are paired, and whether the population standard deviation is genuinely known rather than estimated.

Deviations are measured from the sample mean, which sits closer to the sample than the true mean does, so squared deviations come out too small. Dividing by $n-1$ corrects that bias. On six data points the difference is around 10 per cent; on 500 it is negligible, but the rule does not change.

A test that does not reach significance means the evidence was insufficient, which is also what a tiny sample produces when a real effect exists. The correct sentence is that there is not enough evidence to conclude a difference, never that there is no difference.

$P(A\cap B)=P(A)P(B)$ holds only for independent events. For drawing two cards without replacement, the second probability is conditional on the first: $\frac{4}{52}\times \frac{3}{51}$, not $\frac{4}{52}\times \frac{4}{52}$. Ask whether the first event changes the second before multiplying.

A p-value is the probability of data at least this extreme assuming the null is true. It says nothing directly about how likely the null is. Getting this backwards makes 0.04 sound like a 96 per cent certainty, which it is not.

Two of those four are computational and two are interpretive, and the interpretive pair costs more. A test statistic can be recomputed in a minute; a sentence claiming a study proved there was no difference cannot be rescued by better arithmetic. Write the conclusion as a decision about the null hypothesis in the wording of the original question, then reread it and confirm it claims nothing about certainty. The wider set of verification habits sits in the guide to checking your own work.