In the print window, choose “Save as PDF” to keep a copy on your device.
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, 410 patients received a new treatment and 500 received the control treatment. The table shows how many in each group had a side effect. Find the number needed to harm NNH: how many patients must be treated for one extra patient to have the side effect? Round up to a whole number of patients.
treatment: 207/410control: 140/500
NNH =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A screening test has a sensitivity of 81% and a specificity of 94%. It is used in a population where the prevalence is 10%. Find the positive predictive value PPV as a percentage. Round to one decimal place.
Se = 81%Sp = 94%p = 10%
PPV = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 45, 25, 19, 36, 52.
n = 5
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 170 and standard deviation 20. Find the z-score z of a score of 200.
μ = 170σ = 20x = 200
z =
5.Confidence intervalsMedium
The battery life of two phone models is compared. Model 1: 100 phones, mean 115 h, known standard deviation 19 h. Model 2: 60 phones, mean 125 h, known standard deviation 4 h. Find the margin of error E of the 90% confidence interval for μ₁ − μ₂.
Nothing is known in advance about the proportion of defective products. How many products n must be inspected so that a 95% confidence interval for the proportion has a margin of error of at most 0.03?
p = 0.5E = 0.0395%
n =
7.Hypothesis tests & confidence intervalsMedium
13 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 3.88 and the standard deviation of the differences is 5.6. Find the paired test statistic t for H₀: μd = 0.
n = 13d̄ = 3.88sd = 5.6
t =
8.Hypothesis tests & confidence intervalsHard
A sample of 16 measurements has mean 125.6. The population standard deviation is 15. Find the upper limit U of the 90% confidence interval for the mean.
x̄ = 125.6σ = 15n = 1690%
U =
3Solutions
1.Evidence-based practice statistics
Answer: NNH = 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.
Event rates
EER = 207/410 = 0.5049
CER = 140/500 = 0.28
Number needed to harm
ARI = 0.5049 − 0.28 = 0.2249
NNH = 1/0.2249 = 4.447 → 5
Answer
NNH = 5
About one in every 5 treated patients has the side effect because of the treatment.
2.Evidence-based practice statistics
Answer: PPV = 60 %
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.
True and false positives (per person)
Se·p = 0.81·0.1 = 0.081
(1 − Sp)·(1 − p) = 0.06·0.9 = 0.054
Positive predictive value
PPV = 0.081/(0.081 + 0.054) = 0.6 = 60%
Answer
PPV = 60%
The lower the prevalence, the lower the PPV of the same test.
3.Descriptive statistics
Answer: s = 13.65
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 = (200 − 170)/20 = 1.5
Answer
z = 1.5
5.Confidence intervals
Answer: E = 3.239 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 = 115 − 125 = −10
Standard error
SE = √(19²/100 + 4²/60) = 1.969
Margin of error
E = 1.645·1.969 = 3.239
Answer
E = 3.239 h
The interval does not contain 0, so the means differ significantly.
6.Confidence intervals
Answer: n = 1068
Confidence interval for a proportion: p̂ ± z·√(p̂(1 − p̂)/n), where p̂ = x/n. To reach a margin E you need n ≥ z²·p(1 − p)/E², rounded up; without a prior estimate use p = 0.5, which gives the largest n.
Solve for n
n = 1.96²·0.5·0.5/0.03² = 1067
Round up
n = 1068
Answer
n = 1068
7.Hypothesis tests & confidence intervals
Answer: t = 2.498
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 = 5.6/√13 = 1.553
Test statistic
t = 3.88/1.553 = 2.498
Answer
t = 2.498
With df = 12, the two-sided 5% critical value is 2.179; |t| is larger, so H₀ is rejected.
8.Hypothesis tests & confidence intervals
Answer: U = 131.8
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).