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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). A test has a sensitivity of 85% and a specificity of 60%. Find the positive likelihood ratio LR+. Round to two decimal places.
Se = 85%Sp = 60%
LR+ =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 10% for a disease. A test with sensitivity 70% and specificity 91% comes back positive. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 10%Se = 70%Sp = 91%positive
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 24, 31, 43, 55, 27, 49, 43.
n = 7
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 70 and standard deviation 10. Find the z-score z of a score of 80.
μ = 70σ = 10x = 80
z =
5.Confidence intervalsMedium
The battery life of two phone models is compared. Model 1: 60 phones, mean 72 h, known standard deviation 18 h. Model 2: 30 phones, mean 60.5 h, known standard deviation 4 h. Find the margin of error E of the 95% confidence interval for μ₁ − μ₂.
Two fertilisers are tested. Plot group 1: 13 plots, mean yield 14 kg, standard deviation 4 kg. Plot group 2: 8 plots, mean 11.5 kg, standard deviation 10 kg. Assume equal variances (pooled t, df = n₁ + n₂ − 2). Find the margin of error E of the 90% confidence interval for μ₁ − μ₂.
Two production lines make the same part. Line 1: a sample of 50 parts has mean length 59 mm, and the line's known standard deviation is 3 mm. Line 2: a sample of 100 parts has mean 59.7 mm, known standard deviation 9 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
A one-sided z test of H₀: μ = 100 against H₁: μ > 100 is done at α = 0.10 with a sample of 36; the standard deviation is known to be 6. In reality the mean is 102.3. Find the power 1 − β of the test in percent.
μ₀ = 100μ = 102.3σ = 6n = 36α = 0.10
1 − β = %
3Solutions
1.Evidence-based practice statistics
Answer: LR+ = 2.13
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.
Positive likelihood ratio
LR+ = 0.85/(1 − 0.6) = 2.125 ≈ 2.13
Answer
LR+ = 2.13
2.Evidence-based practice statistics
Answer: p′ = 46.4 %
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.
Likelihood ratio
LR+ = 0.7/(1 − 0.91) = 7.778
Pre-test odds
0.1/(1 − 0.1) = 0.1111
Post-test odds
0.1111·7.778 = 0.8642
Post-test probability
p′ = 0.8642/(1 + 0.8642) = 0.4636 ≈ 46.4%
Answer
p′ = 46.4%
3.Descriptive statistics
Answer: s = 11.7
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 = (80 − 70)/10 = 1
Answer
z = 1
5.Confidence intervals
Answer: E = 4.774 h
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.
Point estimate
x̄₁ − x̄₂ = 72 − 60.5 = 11.5
Standard error
SE = √(18²/60 + 4²/30) = 2.436
Margin of error
E = 1.96·2.436 = 4.774
Answer
E = 4.774 h
The interval does not contain 0, so the means differ significantly.
6.Confidence intervals
Answer: E = 5.323 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 = (12·4² + 7·10²)/19 = 46.95
sp = 6.852
Margin of error
E = 1.729·6.852·√(1/13 + 1/8) = 5.323
Answer
E = 5.323 kg
7.Hypothesis tests & confidence intervals
Answer: z = −0.7035
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.
Standard error of the difference
SE = √(3²/50 + 9²/100) = √0.99 = 0.995
Test statistic
z = (59 − 59.7)/0.995 = −0.7035
Answer
z = −0.7035
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: 1 − β = 84.57 %
Power of a test: power = 1 − β is the probability of rejecting H₀ when it is false. Find the cutoff x̄c from μ₀ and z_α, then compute the probability of landing beyond x̄c using the true mean: z = (x̄c − μ)/(σ/√n).