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). A new test was compared with the reference standard in a study (see the 2 × 2 table). Find the negative likelihood ratio LR−. Round to two decimal places.
a = 353b = 381c = 27d = 849
LR− =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A screening test has a sensitivity of 97% and a specificity of 97%. It is used in a population where the prevalence is 20%. Find the negative predictive value NPV as a percentage. Round to one decimal place.
Se = 97%Sp = 97%p = 20%
NPV = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 32, 56, 29, 30, 28, 13.
n = 6
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 60 and standard deviation 12. Find the z-score z of a score of 67.2.
μ = 60σ = 12x = 67.2
z =
5.Confidence intervalsMedium
A sample of 21 items has mean 28.5 and standard deviation s = 15. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.
n = 21x̄ = 28.5s = 1599%t(20) = 2.845
E =
6.Confidence intervalsHard
The population standard deviation is 18. How large a sample n do you need so that a 95% confidence interval for the mean has a margin of error of at most 4?
σ = 18E = 495%
n =
7.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 120; the population standard deviation is known to be 3. A random sample of 36 observations has a mean of 120.54. Test H₀: μ = 120 against H₁: μ > 120. Find the p-value p.
μ₀ = 120σ = 3n = 36x̄ = 120.54H₁: μ > μ₀
p =
8.Hypothesis tests & confidence intervalsHard
A one-sided z test at α = 0.05 must detect a shift of the mean of 1.5 units with a power of 80%. The standard deviation is 4. Find the minimum sample size n.
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.
Sensitivity and specificity
Se = 353/380 = 0.9289
Sp = 849/1230 = 0.6902
Negative likelihood ratio
LR− = (1 − 0.9289)/0.6902 = 0.1029 ≈ 0.1
Answer
LR− = 0.1
2.Evidence-based practice statistics
Answer: NPV = 99.2 %
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.
True and false negatives (per person)
Sp·(1 − p) = 0.97·0.8 = 0.776
(1 − Se)·p = 0.03·0.2 = 0.006
Negative predictive value
NPV = 0.776/(0.776 + 0.006) = 0.9923 ≈ 99.2%
Answer
NPV = 99.2%
3.Descriptive statistics
Answer: s = 13.88
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 = (67.2 − 60)/12 = 0.6
Answer
z = 0.6
5.Confidence intervals
Answer: E = 9.312
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 = 2.845·15/√21 = 9.312
Answer
E = 9.312
6.Confidence intervals
Answer: n = 78
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 = (1.96·18/4)² = 77.79
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 78
Answer
n = 78
7.Hypothesis tests & confidence intervals
Answer: p = 0.1401
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.
Test statistic
SE = σ/√n = 3/√36 = 0.5
z = (120.54 − 120)/0.5 = 1.08
Tail area
Φ(1.08) = 0.8599
1 − Φ(1.08) = 0.1401
p-value
p = 0.1401
Answer
p = 0.1401
p > 0.05, so H₀ is not rejected at the 5% level.
8.Hypothesis tests & confidence intervals
Answer: n = 44
Sample size for a test: to detect a true shift δ with power 1 − β in a one-sided z test at level α, the standard error must satisfy δ = (z_α + z_β)·σ/√n, so n = ((z_α + z_β)·σ/δ)², rounded up.