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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 randomised trial, 500 patients received a new treatment and 710 received the control treatment. The table shows how many in each group had a side effect. Find the relative risk RR of the event with treatment. Round to two decimal places.
treatment: 63/500control: 36/710
RR =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 30% for a disease. A test with sensitivity 65% and specificity 80% comes back positive. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 30%Se = 65%Sp = 80%positive
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 43, 24, 17, 31, 46.
n = 5
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 100 and standard deviation 15. What proportion P of values is above 88?
μ = 100σ = 15x = 88
P =
5.Confidence intervalsMedium
A sample of 20 items has mean 52 and standard deviation s = 11. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.
n = 20x̄ = 52s = 1199%t(19) = 2.861
E =
6.Confidence intervalsHard
Two fertilisers are tested. Plot group 1: 9 plots, mean yield 37.5 kg, standard deviation 8.5 kg. Plot group 2: 12 plots, mean 48 kg, standard deviation 5 kg. Assume equal variances (pooled t, df = n₁ + n₂ − 2). Find the upper limit of the 95% confidence interval for μ₁ − μ₂.
12 employees take a typing test before and after a training course. For each person the difference d = after − before (words per minute) is recorded. The mean difference is −2.7 and the standard deviation of the differences is 7.2. Find the paired test statistic t for H₀: μd = 0.
n = 12d̄ = −2.7sd = 7.2
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 45 of 100 patients on treatment A recover, and 72 of 200 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 45n₁ = 100x₂ = 72n₂ = 200
z =
3Solutions
1.Evidence-based practice statistics
Answer: RR = 2.49
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.
Event rates
EER = 63/500 = 0.126
CER = 36/710 = 0.0507
Relative risk
RR = 0.126/0.0507 = 2.485 ≈ 2.49
Answer
RR = 2.49
2.Evidence-based practice statistics
Answer: p′ = 58.2 %
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+ = 0.65/(1 − 0.8) = 3.25
Pre-test odds
0.3/(1 − 0.3) = 0.4286
Post-test odds
0.4286·3.25 = 1.393
Post-test probability
p′ = 1.393/(1 + 1.393) = 0.5821 ≈ 58.2%
Answer
p′ = 58.2%
3.Descriptive statistics
Answer: s = 12.32
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 = −0.8
Standard normal table
Φ(−0.8) = 0.2119
P = 1 − 0.2119 = 0.7881
Answer
P = 0.7881
5.Confidence intervals
Answer: E = 7.037
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 = 19, t = 2.861
Margin of error
E = 2.861·11/√20 = 7.037
Answer
E = 7.037
6.Confidence intervals
Answer: upper limit = −4.316 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 = (8·8.5² + 11·5²)/19 = 44.89
sp = 6.7
Margin of error
E = 2.093·6.7·√(1/9 + 1/12) = 6.184
Interval
−10.5 ± 6.184 → [−16.68, −4.316]
Answer
upper limit = −4.316 kg
7.Hypothesis tests & confidence intervals
Answer: t = −1.299
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).
Standard error of the mean difference
SE = 7.2/√12 = 2.078
Test statistic
t = −2.7/2.078 = −1.299
Answer
t = −1.299
With df = 11, the two-sided 5% critical value is 2.201; |t| is smaller, so H₀ is not rejected.
8.Hypothesis tests & confidence intervals
Answer: z = 1.507
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.
Sample proportions
p̂₁ = 45/100 = 0.45
p̂₂ = 72/200 = 0.36
p̂ = 117/300 = 0.39
Standard error under H₀
SE = √(0.39·0.61·(1/100 + 1/200)) = 0.05974
Test statistic
z = (0.45 − 0.36)/0.05974 = 1.507
Answer
z = 1.507
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).