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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 case-control study compares 130 patients with a disease (cases) and 360 people without it (controls). The table shows how many in each group were exposed to a risk factor. Find the odds ratio OR for the exposure. Round to two decimal places.
a = 69b = 130c = 61d = 230
OR =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 60% for a disease. A test with sensitivity 93% and specificity 72% comes back negative. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 60%Se = 93%Sp = 72%negative
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 28, 17, 39, 48, 11.
n = 5
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 500 and standard deviation 100. What proportion P of values is below 300?
μ = 500σ = 100x = 300
P =
5.Confidence intervalsMedium
The battery life of two phone models is compared. Model 1: 80 phones, mean 74.5 h, known standard deviation 9 h. Model 2: 60 phones, mean 89 h, known standard deviation 14 h. Find the upper limit of the 99% confidence interval for μ₁ − μ₂.
The population standard deviation is 19. How large a sample n do you need so that a 99% confidence interval for the mean has a margin of error of at most 1?
σ = 19E = 199%
n =
7.Hypothesis tests & confidence intervalsMedium
A sample of 81 students has a mean score of 499.2 and a sample standard deviation of 4. Find the test statistic t for H₀: μ = 500.
μ₀ = 500x̄ = 499.2s = 4n = 81
t =
8.Hypothesis tests & confidence intervalsHard
A sample of 100 measurements has mean 102.6. The population standard deviation is 10. Find the lower limit L of the 99% confidence interval for the mean.
x̄ = 102.6σ = 10n = 10099%
L =
3Solutions
1.Evidence-based practice statistics
Answer: OR = 2
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.
Odds of exposure
cases: 69/61 = 1.131
controls: 130/230 = 0.5652
Odds ratio
OR = (69·230)/(130·61) = 2.001 ≈ 2
Answer
OR = 2
OR above 1: exposure is more common among the cases.
2.Evidence-based practice statistics
Answer: p′ = 12.7 %
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− = (1 − 0.93)/0.72 = 0.09722
Pre-test odds
0.6/(1 − 0.6) = 1.5
Post-test odds
1.5·0.09722 = 0.1458
Post-test probability
p′ = 0.1458/(1 + 0.1458) = 0.1273 ≈ 12.7%
Answer
p′ = 12.7%
3.Descriptive statistics
Answer: s = 15.24
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 = −2
Standard normal table
Φ(−2) = 0.02275
P = 0.02275
Answer
P = 0.02275
5.Confidence intervals
Answer: upper limit = −9.171 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 = 74.5 − 89 = −14.5
Standard error
SE = √(9²/80 + 14²/60) = 2.069
Margin of error
E = 2.576·2.069 = 5.329
Interval
−14.5 ± 5.329 → [−19.83, −9.171]
Answer
upper limit = −9.171 h
The interval does not contain 0, so the means differ significantly.
6.Confidence intervals
Answer: n = 2396
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.
Solve for n
n = (2.576·19/1)² = 2396
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 2396
Answer
n = 2396
7.Hypothesis tests & confidence intervals
Answer: t = −1.8
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
SE = 4/√81 = 4/9 = 0.4444
Test statistic
t = (499.2 − 500)/0.4444 = −1.8
Answer
t = −1.8
t0.025; df = 80 = 1.99
|t| < 1.99, so H₀ is not rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: L = 100
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).