Formula sheet

Every formula, when to use it, and a worked example.

Memory hooks

Steps, Variance and SDMy Desk Shows Some Daily RiskMean, Distance, Square, Sum, Divide, Root

μ vs x, True vs estimateThe real answer vs your best guessGreek is real, bar or hat is a guess

n − 1, Sample correctionA sample gets one taken offOnly the sample gets it

Mean vs median, OutliersAverage salary with the boss in the roomMedian is the typical person

Skew, TailThe mean chases the tailIncome = right skew

SD, SpreadHow bumpy the ride isSame average, smoother wins

CV, SD ÷ meanRisk per unit of rewardLower is better

68 / 95 / 99.7, Empirical ruleDaily noise, monthly jolt, yearly headline1 in 3, 1 a month, 1 in 1.5 years

Or, AdditionTwo guest lists, count overlaps onceDon't count the heart face cards twice

And, MultiplicationCombo lock: every dial must hitMore dials, less likely

Given, ConditionalFilter the list, then countBreak-point conversion

Mutually exclusive, Can't bothHeads and tails on one flipNever independent

Expected value, Weighted averageWhat a bet is worth on averageYou never roll 3.5

Correlation, Move togetherDiversification dialBelow 1 cuts risk

Binomial, Count wins in n triesTiebreaks won out of 3np, np(1−p)

Poisson, Count per windowCalls to a help deskRescale λ first

z, SDs from the mean1.28, 1.645, 1.96, 2.576Memorize four numbers

√n, Noise cancelling4× the data buys half the noiseSE = σ/√n

CI, Estimate ± marginMargin of error around your guess95% of intervals catch μ

Regression, Best-fit lineRise over runNever extrapolate

VIF, RedundancyTwo ingredients that are the same thingAbove 10 is trouble

p-value, Surprise scoreIf the p is low, H₀ must goSmall p: reject. Big p: keep H₀, but it is not proven

Type I vs II, False alarm vs missI cry wolf, II misses the wolfα is the false-alarm rate; power catches the wolf

1. Data and sampling

What statistics isUsing data to make decisionsSurvey, chart, average, decide

Describe vs inferDescribe what you have; infer the bigger groupTeam average age vs election poll

SamplingRandom = fair, stratified = every group, risk-based = riskiest groupOversample the largest claims, then reweight

Data typesQuantitative vs qualitative; discrete vs continuous; univariate vs multivariateZip code is a label

2. Where is the middle

MeanAdd up, divide by how many15, 7, 22, 7, 29, 11, 5, 0 = 12

MedianMiddle once sorted; average the two middles if evenSame data = 9

ModeMost frequent; none or bimodal possible40, 55, 45, 70, 55, 60, 40, 35 = 40 and 55

SkewRight skew: mode < median < meanIncome

3. How spread out is it

RangeMax − min; one outlier sets it75 − 12 = 63

Population varianceΣ(y − μ)² / N4, 7, 10: 18/3 = 6, σ = 2.45

Sample varianceΣ(y − ȳ)² / (n − 1)18/2 = 9, s = 3

Shortcut[Σy² − (Σy)²/n] / (n − 1)165 − 147 = 18, then ÷ 2

Empirical rule68 / 95 / 99.7% within 1 / 2 / 3 SD400 ± 50: 350 to 450

Coefficient of variations / x, risk per unit of returnB: 8.4 / 10.8 = 78%

4. Probability rules

ComplementP(not A) = 1 − P(A)P(not a 1 or 2) = 4/6

OrP(A) + P(B) − P(both)Heart or face card = 22/52

GivenP(A and B) / P(B)P(spa | gym) = 0.12 / 0.60

AndP(A) × P(B | A)Two aces = 4/52 × 3/51

IndependentP(A and B) = P(A)P(B)Two heads = 0.25

TreeMultiply along, add across0.85 × 0.95 = 0.8075

5. Random variables and expected value

Expected valueΣ y P(y)Four-sided die = 2.5

VarianceΣ (y − μ)² P(y) = Σy²P(y) − μ²Y: 2,684

CVσ / μLower is better

ConstantsE(a + bY) = a + bE(Y); Var = b² Var(Y)Y + 10: mean shifts, SD stays

6. Two variables: covariance, correlation, portfolios

CovarianceΣΣ xy P(x, y) − μx μy99 − 86.4 = 12.6

CorrelationCov / (σx σy)12.6 / (6 × 8.4) = 0.25

Portfolio meanbE(X) + cE(Y)0.5(8) + 0.5(10.8) = 9.4

Portfolio varianceb²V(X) + c²V(Y) + 2bc Cov32.94, σ = 5.74

Cov = 0Not the same as independentY = X²

7. Counting and the binomial

Factorialn! = n(n − 1)…1; 0! = 16! = 720

CombinationsnCx = n! / [x!(n − x)!]C(5,2) = 10

BinomialnCy pʸ (1 − p)ⁿ⁻ʸP(2 of 8 at 0.2) = 0.294

Mean, variancenp, np(1 − p)8 × 0.2 = 1.6

HypergeometricC(K, k) C(N − K, n − k) / C(N, n)10 phones, 3 bad, pick 4: P(1 bad) = 0.5

8. Poisson: counts per unit

Poissone⁻λ λʸ / y!λ = 3: P(3) = 0.224

Rescale λrate × window30 an hour → 5 per 10 minutes

Mean = varianceBoth λλ = 6: mean 6, variance 6

9. The normal curve and z

z score(y − μ) / σ(525 − 400) / 50 = 2.5

Less thanΦ(z)P(Y < 525) = 0.9938

Greater than1 − Φ(z)P(Y > 475) = 0.0668

BetweenΦ(z of b) − Φ(z of a)400 to 475: 0.4332

Reversey = μ + zσ10% run out: 400 + 1.28(50) = 464

Key z1.28, 1.645, 1.96, 2.57690%, 95% one-sided, 95%, 99%

Uniform(d − c) / (b − a)0 to 20, P(over 15) = 0.25

10. Sampling distributions and the CLT

Standard errorσ / √n12 / √36 = 2

z for x(x − μ) / (σ / √n)(51 − 50) / 2 = 0.5

Sum of nMean nμ, SD σ√n64 bills: 40,960 and 3,840

ProportionSE = √[p(1 − p) / n]0.3, n = 400: 0.0229

How big is nRule of thumb 30; depends on skewA normal population works at any n

UnbiasedE(estimator) = parameterx, p, s² with n − 1

11. Confidence intervals and sample size

σ knownx ± z σ / √n502.4 ± 1.96(5 / √25)

σ unknownx ± t s / √n, df = n − 172.5 ± 2.131(12 / √16)

Proportionp ± z √[p(1 − p) / n]0.6 ± 1.645(0.0346)

n for μ(zσ / E)²(1.96 × 2,500 / 400)² → 151

n for pz² p(1 − p) / E², p = 0.5 worst case(1.96 / 0.03)² × 0.25 → 1,068

Confidence level1 − α; z: 1.645, 1.96, 2.576Round n up

12. Hypothesis tests

Test statistic(estimate − H₀ value) / SE(247.5 − 250) / 1 = −2.5

p-value, two-sided2 × P(Z > |z|)z = 2.5: 0.0124

DecideReject H₀ if p < α0.012 < 0.05: reject

Proportion SE√[p₀(1 − p₀) / n], with p₀p₀ = 0.5, n = 400: 0.025

ErrorsType I = α, Type II = βPower = 1 − β

13. Simple regression

Slope[nΣxy − ΣxΣy] / [nΣx² − (Σx)²]192 / 168 = 1.143

Interceptȳ − b₁x6 − 1.143(4) = 1.429

Sums of squaresSST = SSR + SSE; R² = SSR / SST36.57 / 42 = 87.1%

Residual SDs = √(SSE / (n − 2))√(5.43 / 4) = 1.16

Slope testt = b₁ / SE, df = n − 25.19 vs 2.776, reject

IntervalsCI for the mean y is narrower than the PI for one yWidest far from x

14. Multiple regression

ANOVAF = MSR / MSE, df (k, n − k − 1)18.2 vs 3.89, reject

Each tbⱼ / SE(bⱼ)SqFt t = 3.40, p = 0.005

Adjusted R²Penalizes extra x'sNever above R²

VIF1 / (1 − Rⱼ²); trouble above 10r = 0.95 gives 10.3

Dummiesk categories → k − 1Corner lot = 1, else 0

Sequential SSOrder changes the split, not the total52.0 + 8.0 = 38.5 + 21.5 = 60.0

Traps

1. Data and samplingA bigger sample fixes randomness, not bias.10,000 app users who answered a survey say nothing about the ones who ignore surveys. That is a convenience sample.

2. Where is the middleDuplicates count.In 15, 7, 22, 7, 29, 11, 5, 0 the 7 appears twice, and both stay in the sorted list.

3. How spread out is itSame range, different spread.10, 50, 50, 50, 90 and 10, 10, 50, 90, 90 both span 80, but SDs of about 28 and 40. Range sees only the ends.

4. Probability rulesMutually exclusive events are not independent.If A happens B cannot, so knowing A changes B completely.

5. Random variablesThe expected value is not the most likely outcome, and may be impossible (2.5 on a four-sided die).It is the long-run average.

6. Two variables and portfoliosIndependent means Cov = 0, but Cov = 0 does not mean independent.Y = X² is the classic catch.

7. Counting and the binomialAt least one is 1 − P(none).Adding P(1) + P(2) + … by hand is slow and invites mistakes.

8. PoissonForgetting to rescale λ.3 calls an hour over 2 hours is λ = 6, not 3.

9. The normal curveSome z tables give the area from 0 to z, not from the far left.For less than a positive z, add 0.5; Excel's NORM.S.DIST already starts from the left.

10. Sampling and the CLTDividing σ by n instead of √n.The standard error shrinks with the square root: four times the sample only halves the noise.

11. Confidence intervalsA 95% CI is not a 95% chance that μ is inside.μ is fixed; the interval moves. For sample size, width is 2E: width 0.06 means E = 0.03.

12. Hypothesis testsA p-value is not the chance H₀ is true.It is the chance of data at least this extreme under H₀. For a proportion, the SE uses p₀, not p.

13. Simple regressionCorrelation is symmetric; regression is not.Swap x and y and you get a different line. Never predict outside the x range you have.

14. Multiple regressionHolding other variables fixed is about the math, not the world.If size and rooms move together in reality, you can't hold one fixed.