Formula sheet

n = number of subjects, i.e. rows of the data table

n

number of subjects (rows)

Taught in What is data?

Round half-up to the reported dp: 0.495 → 0.50

Taught in Fractions, proportions, percentages and rounding·Diagnostic test accuracy (2×2)

Interim values to 4 dp, reported measures to 2 dp

Taught in Fractions, proportions, percentages and rounding

P = favourable ÷ equally likely outcomes

Taught in Chance and probability

Value = position read against the axis scale

Taught in Reading a graph

Share of shuffled gaps ≥ observed ≈ how surprising the gap is if there is no real difference

Taught in Comparing groups: real difference or chance?

Data = the values a variable takes for each subject or material

Taught in What is data?

Proportion = part ÷ whole = x/n

x

number with the characteristic

n

total

Taught in Fractions, proportions, percentages and rounding·Summary measures by data type·Graphs for quantitative data·Surrogate outcomes

P(not A) = 1 − P(A); q = 1 − p; rest = 1 − proportion

p

probability of the event ("success")

q

probability of failure

Taught in Fractions, proportions, percentages and rounding·Chance and probability·Types of distribution

x̄ = Σxᵢ / n (sum of values ÷ number of values)

x̄

sample mean

xᵢ

each value

n

number of values

Taught in Reading the symbols (Σ, x̄, √, ², subscripts, ≥)·Averages and spread, by eye·Summary statistics

Variance = SD²

SD

standard deviation

Taught in Reading the symbols (Σ, x̄, √, ², subscripts, ≥)·Summary statistics

SD = √[Σ(xᵢ − x̄)² / (n − 1)]

xᵢ

each value

x̄

sample mean

n

number of values

Taught in Reading the symbols (Σ, x̄, √, ², subscripts, ≥)·Summary statistics

Median = middle value when the data are sorted

Taught in Averages and spread, by eye

Range = highest − lowest = x_max − x_min

Taught in Averages and spread, by eye·Summary statistics

IQR = Q3 − Q1

Q1

lower quartile (25th percentile)

Q3

upper quartile (75th percentile)

Taught in Summary statistics·Summary measures by data type·Graphs for quantitative data

Mean > median → positive skew; mean < median → negative skew

Taught in Normal vs non-normal

Floor 0: SD > ½x̄ → skew likely; SD ≥ x̄ → highly likely. Otherwise D = x̄ − lowest (or highest − x̄): SD > ½D likely, SD ≥ D highly likely

x̄

mean

D

distance from the mean to the lowest (or highest) possible value

Taught in Normal vs non-normal

Excess kurtosis = kurtosis − 3

Taught in Normal vs non-normal

Sample skewness and kurtosis from the data

Taught in Normal vs non-normal

KS test: low p → non-normal; it can miss non-normality

Taught in Normal vs non-normal

Counted: equal intervals (8 − 6 = 6 − 4 = 2); ordinal: ordered, intervals unequal or undefined

Taught in Data and types of variables

SE = σ/√N

σ

standard deviation

N

sample size

Taught in Reading the symbols (Σ, x̄, √, ², subscripts, ≥)·Samples and populations·Confidence intervals

The sample mean x̄ estimates the population mean μ

Taught in Samples and populations

Sample means differ because of random error

Taught in Samples and populations

0 ≤ P ≤ 1

Taught in Chance and probability·Distribution of data

Probability = frequency ÷ total

Taught in Distribution of data

Σ P(x) = 1

Taught in Distribution of data

Density = proportion ÷ bin width

Taught in Distribution of data

Bins should be of similar width or the shape misleads

Taught in Reading a graph

P(k) = C(n, k)·pᵏ·qⁿ⁻ᵏ

n

number of trials

k

number of successes

p

probability of success

q

1 − p

Taught in Types of distribution

P(k) = λᵏ·e^(−λ)/k!

λ

events per unit time

k

number of events

Taught in Types of distribution

z = (x − μ)/σ

x

a value

μ

mean

σ

standard deviation

Taught in Reading the symbols (Σ, x̄, √, ², subscripts, ≥)·Types of distribution

μ ± 1 / 2 / 3 SD covers 68 / 95 / 99.7% of values

Taught in Types of distribution

% ≥ x = 1 − Φ((x − μ)/σ)

Φ

standard normal cumulative distribution

Taught in Types of distribution

df = n − 1

Taught in Types of distribution·One-sample t-test

z = 1.2816 / 1.6449 / 1.960 / 2.5758 for 80 / 90 / 95 / 99%; power 0.80 → 0.842, 0.90 → 1.282

Taught in Confidence intervals·Continuous outcomes

95% CI = x̄ ± 1.96 × SE (lower x̄ − 1.96 SE, upper x̄ + 1.96 SE)

x̄

sample mean

SE

standard error, SD/√n

Taught in Confidence intervals

Risk = r/n = events ÷ total; in a 2×2, a/(a + b) and c/(c + d)

r

number with the event

n

total in the group

Taught in Counting in a 2×2 table·Summary statistics·Confounding·Missing data·Cohort studies·Risk, odds, RR, OR, RD, NNT

Odds = x/y (with ÷ without the event) = r/(n − r); in a 2×2, a/b and c/d

x

with the event

y

without the event

Taught in Summary statistics·Cohort studies·Summary measures by data type·Risk, odds, RR, OR, RD, NNT

Odds = Risk/(1 − Risk); Risk = Odds/(1 + Odds)

Taught in Summary statistics

P(A and B) = P(A) × P(B) for independent events

Taught in Chance and probability·Alpha error

(1/3)³ = 1/27 ≈ 0.037

Taught in Chance and probability·Alpha error

H0: difference < MCID; H1: difference ≥ MCID (H1 is the opposite of the test hypothesis)

MCID

minimal clinically important difference

Taught in Test, null and alternate hypotheses

False-positive rate ≈ α at any n

Taught in Alpha error

p = P(difference at least as large as observed | test hypothesis and model)

Taught in Tests, p-values, significance

Significant (reject H0, accept H1) if p < the pre-set threshold, usually 0.05

Taught in Tests, p-values, significance·Interpreting results

SE diff = σ·√(2/n)

σ

common SD

n

per group

Taught in Tests, p-values, significance·Beta error and power

t = (observed − test value)/SE, df = 2n − 2; p = 2·P(T > |t|)

Taught in Tests, p-values, significance

P(≥ 1 significant by chance) = 1 − (1 − α)ᵏ

α

significance level per test

k

number of tests

Taught in p-value misconceptions·Selection of the reported result

p does not measure the size or importance of an effect

Taught in p-value misconceptions

CI of the difference excludes the test value ↔ p < 0.05

Taught in CIs and p-values

Continuous outcome: two 95% CIs that don’t overlap → p < 0.05

Taught in CIs and p-values

SE diff = √(SE₁² + SE₂²)

Taught in CIs and p-values

CI diff = d ± 1.96 × SE diff

d

estimated difference

Taught in CIs and p-values

SE = (upper − lower)/(2 × 1.96)

Taught in Interpreting results

z = d/SE; p = 2[1 − Φ(|z|)]

Taught in Interpreting results

T = (μ − x̄)/(s/√n); the usual form (x̄ − μ)/(s/√n) differs only in sign

μ

known population mean

x̄

sample mean

s

sample SD

n

sample size

Taught in One-sample t-test

p = T.DIST.2T(ABS(T), n − 1)

Taught in One-sample t-test

One-sided p = two-sided p / 2, in the hypothesised direction

Taught in One-sample t-test·One- vs two-sided; equivalence

Two-sided p-values are the default

Taught in One- vs two-sided; equivalence

Equivalence: the whole CI lies within ±margin

Taught in One- vs two-sided; equivalence

Power = 1 − β; β = 1 − power

β

beta (type II, false-negative) error

Taught in Beta error and power·Why sample size matters·Continuous outcomes·No previous study·Power calculations

Power = Φ(Δ/(σ√(2/n)) − z₁₋α/₂) [+ Φ(−Δ/(σ√(2/n)) − z₁₋α/₂)]

Δ

difference in means

σ

SD

n

per group

Φ

standard normal cumulative distribution

Taught in Beta error and power·Why sample size matters·Power calculations

n = 2(z₁₋α/₂ + z₁₋β)² σ² / Δ² per group, rounded up

Δ

difference to detect

σ

SD

z₁₋α/₂

z for the two-sided α

z₁₋β

z for the power

Taught in Beta error and power·Continuous outcomes·Minimal important difference

Expected false positives = k × α

Taught in Selection of the reported result

OR = ad/bc

Taught in Classification of interventions·Risk, odds, RR, OR, RD, NNT

MH RR = Σ(aᵢ·n0ᵢ/Nᵢ) / Σ(cᵢ·n1ᵢ/Nᵢ)

i

stratum

n1ᵢ, n0ᵢ

exposed and unexposed totals

Nᵢ

stratum total

Taught in Confounding

Estimated (observed) difference = true difference + bias (systematic error)

Taught in What is systematic error·Measurement of outcomes

More samples narrow the spread but leave the bias

Taught in What is systematic error

Confounder: related to intervention and outcome; mediator: on the pathway from intervention to outcome; moderator: related to the outcome, not to the intervention

Taught in Confounder, mediator, moderator

Bias = analysed difference − all-eligible difference

Taught in Selection bias

Non-differential misclassification: usually biased towards the null; differential: likely to bias

Taught in Classification of interventions

ITT effect = effect of being assigned to (starting) the treatment; per-protocol effect = effect of adhering to it

Taught in Deviations from intended interventions

Re-analyse with missing outcomes as favourable and as unfavourable

Taught in Missing data

Assessors aware of the intervention; different methods by group; measurement error related to the intervention

Taught in Measurement of outcomes

Before start or unrelated to treatment → generalisability; else initial follow-up missing → selection bias; else → missing data

Taught in Decision flowcharts

Excluded because the intervention was not received as planned → deviations; else → missing data (corrected reading)

Taught in Decision flowcharts

Prevalence = r/n (e.g. 2700/10,000 = 27%)

Taught in Cross-sectional studies

IR = r/(n × t)

r

new cases

n

people

t

time followed

Taught in Cohort studies

Odds(cases) = E_case/(N_case − E_case); Odds(controls) = E_control/(N_control − E_control)

E

number exposed

N

number of cases (or controls)

Taught in Case-control studies

RCT measures and calculations are as in cohort studies

Taught in Randomised controlled trials

Compare the trend before vs after the intervention

Taught in Quasi-experimental designs

DiD = (T_P2 − T_P1) − (C_P2 − C_P1)

T

treated group

C

control group

P1, P2

before and after periods

Taught in Quasi-experimental designs

Equivalence: two-sided α and p; zone −Δ to +Δ, e.g. ±2%

Δ

equivalence margin

Taught in Non-inferiority and equivalence

Non-inferiority: one-sided α in the sample size, one-sided p in the analysis

Taught in Non-inferiority and equivalence

Non-inferiority is shown if the CI limit in the harmful direction stays inside Δ

Δ

non-inferiority margin

Taught in Non-inferiority and equivalence

Total n = number of arms × n per arm; factorial total = 4 × 75 = 300

Taught in Parallel and multi-arm trials·Factorial trials and interactions

Pairwise comparisons = k(k − 1)/2

k

number of arms

Taught in Parallel and multi-arm trials

Additive benchmark = sum of the single effects (10 + 5 = 15 pp). Synergistic: combined > sum; antagonistic: largest single < combined < sum; negative: combined < at least one single

Taught in Factorial trials and interactions

outcome ~ β0 + β1·A + β2·B + β3·A×B; test β3

Taught in Factorial trials and interactions

DE = 1 + (m − 1)·ρ

m

mean cluster size

ρ

intra-cluster correlation (ICC)

Taught in Cluster and stepped-wedge

n_cluster = DE × n_individual

Taught in Cluster and stepped-wedge

N_crossover / N_parallel ≈ (1 − ρ)/2 for a continuous outcome

ρ

within-person correlation

Taught in Cross-over

0/3 DLT → escalate; 1/3 → add 3; 2–3/3 → add 3 at the lower level; MTD = 0 or 1 of 6 with DLT

DLT

dose-limiting toxicity

MTD

maximum tolerated dose

Taught in Dose escalation (3+3)

Feasibility: about 24–50 patients usually sufficient; pilot: no special calculation, based on the recruitment period and the main-study sample size

Taught in Pilot vs feasibility

n = (z₁₋α/₂ + z₁₋β)² (σ₁² + σ₂²) / Δ²

Taught in Continuous outcomes

Use the observed Δ only if it exceeds the MID; otherwise Δ = MID

MID

minimal important difference

Taught in Minimal important difference

n = [z₁₋α/₂ √(2p̄(1 − p̄)) + z₁₋β √(p₁(1 − p₁) + p₂(1 − p₂))]² / (p₁ − p₂)², p̄ = (p₁ + p₂)/2

p₁

control proportion

p₂

intervention proportion

p̄

average of p₁ and p₂

Taught in Binary outcomes

n_cc = (n/4)(1 + √(1 + 4/(n|p₁ − p₂|)))²

Taught in Binary outcomes

Variance = p(1 − p)

Taught in Binary outcomes

n_recruit = n/(1 − d)

d

expected dropout proportion

Taught in Binary outcomes

CI half-width = 1.96 × σ × √(2/n)

Taught in Why sample size matters

Percentage = proportion × 100 = count ÷ group total × 100

Taught in Fractions, proportions, percentages and rounding·Tables·Graphs for categorical data

Decimal places = digits right of the point; significant digits ignore the point, leading zeros and some trailing zeros

Taught in Tables

Fences = Q1 − 1.5 × IQR and Q3 + 1.5 × IQR

Q1, Q3

lower and upper quartiles

IQR

Q3 − Q1

Taught in Graphs for quantitative data

Mean paired change = Σ(after − before) / n

Taught in Graphs for quantitative data

Row total = a + b; column total = a + c; grand total = a + b + c + d; a marginal total sums a row or column

Taught in Counting in a 2×2 table·Tables

Intervention (exposure) on the rows, outcome on the columns

Taught in Counting in a 2×2 table

RR = [a/(a + b)] ÷ [c/(c + d)]: risk in one group ÷ risk in the other

Taught in Confounding·Classification of interventions·Missing data·Cohort studies·Risk, odds, RR, OR, RD, NNT

OR = (a/b) ÷ (c/d); in a case-control study OR = odds(cases) ÷ odds(controls)

Taught in Cohort studies·Case-control studies·Risk, odds, RR, OR, RD, NNT

RD = a/(a + b) − c/(c + d)

Taught in Risk, odds, RR, OR, RD, NNT

NNT (or NNH) = 1/RD; use 1/|RD| rounded up

Taught in Risk, odds, RR, OR, RD, NNT

RR = OR / (1 + control-group risk × (OR − 1))

Taught in Risk, odds, RR, OR, RD, NNT

Difference = mean (or median) of group A − mean (or median) of group B, chosen by the distribution

Taught in Comparing groups: real difference or chance?·Other outcome types

SMD = mean difference ÷ standard deviation (pooled SD: ext)

Taught in Other outcome types

Add 0.5 to every cell when a cell is zero

Taught in Other outcome types

RR_k = risk_k / risk_ref; OR_k = odds_k / odds_ref

k

exposure level

ref

reference level

Taught in Other outcome types

r = Σ(x − x̄)(y − ȳ) / √(Σ(x − x̄)² Σ(y − ȳ)²)

Taught in Scatter plots and correlation

ŷ = a + bx

a

intercept

b

slope

Taught in Scatter plots and correlation

Normally distributed → Pearson; not normally distributed → Spearman

Taught in Pearson vs Spearman

Spearman ρ = Pearson r computed on the ranks

Taught in Pearson vs Spearman

ASE = √(p(1 − p)/n); 95% CI = p ± 1.96 × ASE

p

proportion

ASE

asymptotic standard error

Taught in Summary measures by data type

Accuracy = nearness of a measurement to the true value; precision = agreement between replicate or repeat measurements

Taught in Accuracy vs precision

Bias = mean of scans − true value; spread = SD of scans

Taught in Accuracy vs precision

eᵢ = observed − true

Taught in Random vs systematic error

Random: errors sum to zero, average unchanged, variability up; systematic: errors don’t sum to zero, average shifted, biased estimate

Taught in Random vs systematic error

SD_observed = √(SD_true² + SD_error²)

Taught in Random vs systematic error

Sensitivity = TP/(TP + FN)

Taught in Diagnostic test accuracy (2×2)

Specificity = TN/(FP + TN)

Taught in Diagnostic test accuracy (2×2)

PPV = TP/(TP + FP)

Taught in Diagnostic test accuracy (2×2)

NPV = TN/(FN + TN)

Taught in Diagnostic test accuracy (2×2)

Positive if the value ≥ threshold; negative if below

Taught in Thresholds and ROC curves

ROC curve = sensitivity (y) against 1 − specificity (x) across thresholds

Taught in Thresholds and ROC curves

AUC by the trapezoid rule

Taught in Thresholds and ROC curves

% agreement = (both yes + both no) / total observations

Taught in Agreement: kappa

κ < 0 poor; 0–0.20 slight; 0.21–0.40 fair; 0.41–0.60 moderate; 0.61–0.80 substantial; 0.81–1.00 almost perfect

Taught in Agreement: kappa

κ = (p_o − p_e)/(1 − p_e)

p_o

observed agreement

p_e

agreement expected by chance

Taught in Agreement: kappa

p_e = (row₁ × col₁ + row₂ × col₂)/N²

Taught in Agreement: kappa

dᵢ = x₁ᵢ − x₂ᵢ; mᵢ = (x₁ᵢ + x₂ᵢ)/2; plot d against m

dᵢ

difference between methods

mᵢ

mean of the two methods

Taught in Bland–Altman and ICC

Limits = mean difference ± 2 × SD of the differences (±1.96 is the commonly cited value, ext)

Taught in Bland–Altman and ICC

RSD = SD / mean × 100; < 5% indicates minimal error

Taught in Bland–Altman and ICC

ICC and CCC: named only, no formula

Taught in Bland–Altman and ICC