Memory tricks for statistics

When asked to describe any distribution, always address all four. Missing any one costs points on every stats exam.

Every complete hypothesis test needs all 8 steps. Missing any step costs full points on AP Statistics and college exams.

In any normal distribution: 68% within 1 standard deviation, 95% within 2, 99.7% within 3. These three numbers are on almost every statistics exam.

Type I (α): crying wolf — H₀ IS true but you reject it anyway. Type II (β): missing the wolf — H₀ IS false but you fail to reject it. Power = 1−β = correctly detecting a real effect.

Direction (positive/negative), Unusual points (outliers or influential points), Form (linear or curved), Unusual clusters, Strength (strong/moderate/weak). Always reference the actual variables in context!

CI = point estimate ± (critical value × standard error). A 95% CI means: if you repeated the procedure many times, 95% of intervals would capture the true parameter. Never say "95% probability the parameter is in this interval."

P(A AND B) = P(A)×P(B|A). If independent: P(A)×P(B). P(A OR B) = P(A)+P(B)−P(A AND B). If mutually exclusive: P(A)+P(B). Remember: AND=multiply, OR=add and subtract overlap.

r = direction and strength of linear relationship (−1 to +1). r=1 or −1 = perfect linear, r=0 = no linear relationship. r² = proportion of variation in y explained by x. Always interpret r² as a percentage.

IQR = Q3−Q1. Lower fence = Q1−1.5×IQR. Upper fence = Q3+1.5×IQR. Any value outside these fences is an outlier. This is the standard rule used in box plots and statistical reports.

Random (SRS): every individual equally likely. Stratified: sample randomly from each stratum. Cluster: randomly select whole clusters. Systematic: every kth individual. Voluntary response: self-selected — most prone to bias.

The p-value is the probability of getting results as extreme as observed, ASSUMING H₀ is true. If p<α (usually 0.05): reject H₀. Never say "accept H₀" — only "fail to reject." Low p-value means data is unlikely under H₀.

Mean = sum of all values ÷ n. Sensitive to outliers — one extreme value pulls it dramatically. Median = middle value when sorted. Resistant to outliers. For skewed distributions: report the median. For symmetric: either works.

Permutation nPr: ordered arrangements — passwords, race results, seating. Formula: n factorial divided by (n minus r) factorial. Combination nCr: unordered selections — committees, card hands, pizza toppings. Formula: n factorial divided by (r factorial times (n minus r) factorial). nCr equals nPr divided by r factorial — you divide out the ways to rearrange each group.

BINS: Binary (success or failure only), Independent trials, fixed Number n, Same probability p. Formula: P(X equals k) = nCk times p to the k times (1 minus p) to the (n minus k). Mean equals np. Standard deviation equals square root of np(1 minus p). Use normal approximation when np and n(1 minus p) are both at least 10.

z equals (x minus mu) divided by sigma. Positive z is above mean, negative is below, zero is at the mean. For probability P(X less than x): convert to z then look up left area. P(X greater than x) equals 1 minus P(X less than x). To find x from percentile: x equals mu plus z times sigma. For sample means use sigma divided by square root of n as the denominator.

For random samples from any population with mean mu and standard deviation sigma, the sampling distribution of x-bar is approximately normal with mean mu and standard error sigma divided by square root of n when n is at least 30. Larger n gives smaller standard error. The distribution of x-bar always centers at the true population mean. Doubling n reduces standard error by a factor of square root of 2.

Goodness of Fit tests if counts match a claimed distribution — df equals categories minus 1. Independence test checks association between two categorical variables — df equals (rows minus 1) times (columns minus 1). Expected count equals row total times column total divided by grand total. Always right-tailed. Reject null if chi-square exceeds critical value or p-value is less than alpha.

Use z when population sigma is known (rare). Use t when sigma is unknown — use sample standard deviation s instead. Degrees of freedom equals n minus 1 for one-sample t. The t-distribution is bell-shaped and symmetric but with heavier tails than z. As degrees of freedom increase, t approaches z. t-interval: x-bar plus or minus t-star times s divided by square root of n.