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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 200 patients with a disease (cases) and 190 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 = 77b = 46c = 123d = 144
OR =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A screening test has a sensitivity of 73% and a specificity of 96%. It is used in a population where the prevalence is 5%. Find the positive predictive value PPV as a percentage. Round to one decimal place.
Se = 73%Sp = 96%p = 5%
PPV = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 40, 41, 42, 39, 28, 44.
n = 6
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 50 and standard deviation 5. What proportion P of values is below 44?
μ = 50σ = 5x = 44
P =
5.Confidence intervalsMedium
A 90% confidence interval for a mean, based on 50 observations and a known σ, has a margin of error of 10.5. What is the margin of error E₂ if the same study uses 100 observations instead?
90%n₁ = 50E₁ = 10.5n₂ = 100
E₂ =
6.Confidence intervalsHard
The diameters of 15 machined shafts have a sample standard deviation of 6.4 µm. Assuming the diameters are normally distributed, find the lower limit σ of the 95% confidence interval for the population standard deviation. With df = 14, 2.5% of the χ² distribution lies below 5.629 and 2.5% lies above 26.119.
n = 15s = 6.4 µmχ² = 5.629, 26.119
σ = µm
7.Hypothesis tests & confidence intervalsMedium
Two teaching methods are compared. Group 1: 6 students, mean score 36, standard deviation 12. Group 2: 6 students, mean score 24.9, standard deviation 10.5. Assume equal population variances and use the pooled two-sample t test. Find the test statistic t for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (group 1 minus group 2).
n₁ = 6x̄₁ = 36s₁ = 12n₂ = 6x̄₂ = 24.9s₂ = 10.5
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 45 of 150 patients on treatment A recover, and 45 of 250 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 45n₁ = 150x₂ = 45n₂ = 250
z =
3Solutions
1.Evidence-based practice statistics
Answer: OR = 1.96
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: 77/123 = 0.626
controls: 46/144 = 0.3194
Odds ratio
OR = (77·144)/(46·123) = 1.96
Answer
OR = 1.96
OR above 1: exposure is more common among the cases.
2.Evidence-based practice statistics
Answer: PPV = 49 %
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 = 0.73·0.05 = 0.0365
(1 − Sp)·(1 − p) = 0.04·0.95 = 0.038
Positive predictive value
PPV = 0.0365/(0.0365 + 0.038) = 0.4899 ≈ 49%
Answer
PPV = 49%
The lower the prevalence, the lower the PPV of the same test.
3.Descriptive statistics
Answer: s = 5.657
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 = −1.2
Standard normal table
Φ(−1.2) = 0.1151
P = 0.1151
Answer
P = 0.1151
5.Confidence intervals
Answer: E₂ = 7.425
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.
Ratio of margins
E₂/E₁ = (1.645/1.645)·√(50/100) = 0.7071
New margin
E₂ = 10.5·0.7071 = 7.425
Answer
E₂ = 7.425
6.Confidence intervals
Answer: σ = 4.686 µm
Interval for a variance: (n − 1)s²/σ² has a χ² distribution with n − 1 degrees of freedom. The 95% interval for σ² runs from (n − 1)s² divided by the upper χ² value (2.5% above it) to (n − 1)s² divided by the lower χ² value (2.5% below it); the interval for σ is the square root of both limits. It is not symmetric around s.
Sum of squares
(n − 1)s² = 14·6.4² = 573.4
Limits for σ²
573.4/26.119 = 21.95
573.4/5.629 = 101.9
Limit for σ
σ = √21.95 = 4.686 µm
Answer
σ = 4.686 µm
95% interval for σ: [4.686, 10.09] µm.
7.Hypothesis tests & confidence intervals
Answer: t = 1.705
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.
Pooled variance
sp² = (5·12² + 5·10.5²)/10 = 127.1
sp = 11.27
Standard error
SE = 11.27·√(1/6 + 1/6) = 6.51
Test statistic
t = (36 − 24.9)/6.51 = 1.705
Answer
t = 1.705
With df = 10, the two-sided 5% critical value is 2.228; |t| is smaller, so H₀ is not rejected.
8.Hypothesis tests & confidence intervals
Answer: z = 2.782
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 = 45/150 = 0.3
p̂₂ = 45/250 = 0.18
p̂ = 90/400 = 0.225
Standard error under H₀
SE = √(0.225·0.775·(1/150 + 1/250)) = 0.04313
Test statistic
z = (0.3 − 0.18)/0.04313 = 2.782
Answer
z = 2.782
|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).