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1Formulas and methods
Evidence-based practice statistics
Sensitivity and specificity
Se = a/(a + c), Sp = d/(b + d)
Predictive values
PPV = a/(a + b), NPV = d/(c + d)
Accuracy
(a + d)/n
Likelihood ratios
LR+ = Se/(1 − Sp), LR− = (1 − Se)/Sp
PPV at prevalence p
Se·p / (Se·p + (1 − Sp)(1 − p))
Odds
odds = p/(1 − p); p = odds/(1 + odds); post-test odds = pre-test odds × LR
Risk measures
RR = EER/CER, ARR = CER − EER, RRR = ARR/CER
Number needed to treat
NNT = 1/ARR, rounded up; NNH = 1/(EER − CER)
Odds ratio
OR = ad/(bc)
Incidence rate
new cases / person-time
Method
Draw the 2 × 2 table and label a, b, c and d before you calculate.
Read sensitivity and specificity down the columns, predictive values along the rows.
For another prevalence, imagine 10 000 people and fill the table with Se, Sp and the prevalence.
For a likelihood ratio, convert the probability to odds, multiply, and convert back.
For a trial, find EER and CER first; then RR = EER/CER, ARR = CER − EER and NNT = 1/ARR rounded up.
Descriptive statistics
Mean
x̄ = Σx/n
Sample standard deviation
s = √(Σ(x − x̄)²/(n − 1))
Median
middle value of the sorted data
z-score
z = (x − μ)/σ
Normal distribution
68% within ±1σ, 95% within ±2σ
Confidence intervals
σ known
x̄ ± z·σ/√n
σ unknown
x̄ ± t·s/√n, df = n − 1
Critical z
90%: 1.645, 95%: 1.96, 99%: 2.576
Sample size
n = (z·σ/E)², rounded up
Method
Decide between z (population σ known) and t (only the sample s, with n − 1 degrees of freedom).
Find the critical value for the confidence level.
Margin of error: E = critical value × standard deviation/√n.
Interval: x̄ ± E.
Hypothesis tests & confidence intervals
z-test for a mean
z = (x̄ − μ0)/(σ/√n)
t-test for a mean
t = (x̄ − μ0)/(s/√n), df = n − 1
Two-sided at 5%
reject H0 if |z| > 1.96
Confidence interval
x̄ ± z·σ/√n
Method
State H₀ and H₁ and decide whether the test is one- or two-sided.
Calculate the test statistic: (x̄ − μ₀)/(σ/√n), or with s and t.
Compare with the critical value, or find the p-value.
Reject H₀ if the statistic is beyond the critical value (p below α), and state the conclusion in words.
2Practice problems
1.Evidence-based practice statisticsMedium
Practice exercise (made-up values, not clinical guidance). In a cohort study, exposed people contributed 4200 person-years with 107 new cases, and unexposed people 7000 person-years with 119 new cases. Find the incidence rate ratio IRR (exposed versus unexposed). Round to two decimal places.
exposed: 107/4200 PYnot exposed: 119/7000 PY
IRR =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). 10 000 people are screened with a test of sensitivity 88% and specificity 90%. The prevalence of the disease is 21%. How many false-positive results FP are expected?
N = 10 000p = 21%Se = 88%Sp = 90%
FP =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 15, 59, 36, 15, 30.
n = 5
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 50 and standard deviation 15. What proportion P of values is below 56?
μ = 50σ = 15x = 56
P =
5.Confidence intervalsMedium
A sample of 21 items has mean 111 and standard deviation s = 3.5. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.
n = 21x̄ = 111s = 3.599%t(20) = 2.845
E =
6.Confidence intervalsHard
Two fertilisers are tested. Plot group 1: 15 plots, mean yield 10 kg, standard deviation 8 kg. Plot group 2: 6 plots, mean 19 kg, standard deviation 6 kg. Assume equal variances (pooled t, df = n₁ + n₂ − 2). Find the margin of error E of the 99% confidence interval for μ₁ − μ₂.
Two teaching methods are compared. Group 1: 5 students, mean score 29, standard deviation 10. Group 2: 7 students, mean score 42.7, standard deviation 11. Assume equal population variances and use the pooled two-sample t test. Find the test statistic t for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (group 1 minus group 2).
n₁ = 5x̄₁ = 29s₁ = 10n₂ = 7x̄₂ = 42.7s₂ = 11
t =
8.Hypothesis tests & confidence intervalsHard
A sample of 64 measurements has mean 196.4. The population standard deviation is 10. Find the lower limit L of the 99% confidence interval for the mean.
x̄ = 196.4σ = 10n = 6499%
L =
3Solutions
1.Evidence-based practice statistics
Answer: IRR = 1.5
Likelihood ratios and risk measures: LR+ = sensitivity/(1 − specificity), LR− = (1 − sensitivity)/specificity. In a trial, EER and CER are the event rates with and without treatment: RR = EER/CER, ARR = CER − EER, RRR = ARR/CER and NNT = 1/ARR, rounded up. In a case-control study only the odds ratio OR = ad/(bc) can be used. An incidence rate = new cases/person-time.
Incidence rates
exposed: 107/4200 = 25.48 per 1000 person-years
not exposed: 119/7000 = 17 per 1000 person-years
Rate ratio
IRR = 25.48/17 = 1.499 ≈ 1.5
Answer
IRR = 1.5
2.Evidence-based practice statistics
Answer: FP = 790
Predictive values depend on prevalence: PPV = Se·p/(Se·p + (1 − Sp)(1 − p)) and NPV = Sp(1 − p)/(Sp(1 − p) + (1 − Se)p). With likelihood ratios: pre-test odds = p/(1 − p), post-test odds = pre-test odds × LR, probability = odds/(1 + odds). For a population of N, true positives = N·p·Se and false positives = N·(1 − p)·(1 − Sp). NNT = 1/(CER × RRR), rounded up.
With and without the disease
10 000·0.21 = 2100
10 000 − 2100 = 7900
False positives
FP = 7900·0.1 = 790
Answer
FP = 790
Of the 2638 positives only 1848 have the disease (PPV 70.1%).
3.Descriptive statistics
Answer: s = 18.18
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Standardise
z = 0.4
Standard normal table
Φ(0.4) = 0.6554
P = 0.6554
Answer
P = 0.6554
5.Confidence intervals
Answer: E = 2.173
Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.
Critical value
df = n − 1 = 20, t = 2.845
Margin of error
E = 2.845·3.5/√21 = 2.173
Answer
E = 2.173
6.Confidence intervals
Answer: E = 10.4 kg
Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.
Pooled standard deviation
sp² = (14·8² + 5·6²)/19 = 56.63
sp = 7.525
Margin of error
E = 2.861·7.525·√(1/15 + 1/6) = 10.4
Answer
E = 10.4 kg
7.Hypothesis tests & confidence intervals
Answer: t = −2.205
Hypothesis test: the test statistic = (estimate − value under H₀)/(standard error under H₀). The p-value is the probability, assuming H₀ is true, of a statistic at least as extreme as the one observed: the area in one tail for a one-sided H₁, twice the tail area for a two-sided H₁. Reject H₀ when p ≤ α.
Pooled variance
sp² = (4·10² + 6·11²)/10 = 112.6
sp = 10.61
Standard error
SE = 10.61·√(1/5 + 1/7) = 6.213
Test statistic
t = (29 − 42.7)/6.213 = −2.205
Answer
t = −2.205
With df = 10, the two-sided 5% critical value is 2.228; |t| is smaller, so H₀ is not rejected.
8.Hypothesis tests & confidence intervals
Answer: L = 193.2
Testing a mean: the test statistic measures how many standard errors the sample mean lies from the value claimed in H₀: z = (x̄ − μ₀)/(σ/√n) when σ is known, t = (x̄ − μ₀)/(s/√n) when it is estimated from the sample. A confidence interval is x̄ ± (critical value)·(standard error).