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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 250 patients with a disease (cases) and 550 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 = 37b = 165c = 213d = 385
OR =
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 98% and specificity 65% comes back positive. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 30%Se = 98%Sp = 65%positive
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 16, 37, 20, 48, 43, 13.
n = 6
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 170 and standard deviation 10. Find the z-score z of a score of 185.
μ = 170σ = 10x = 185
z =
5.Confidence intervalsMedium
A sample of 7 items has mean 54 and standard deviation s = 19.5. The population is roughly normal. Find the lower limit of the 99% confidence interval for the mean.
n = 7x̄ = 54s = 19.599%t(6) = 3.707
lower limit =
6.Confidence intervalsHard
The population standard deviation is 2. 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 0.5?
σ = 2E = 0.599%
n =
7.Hypothesis tests & confidence intervalsMedium
8 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 8.3 and the standard deviation of the differences is 8.7. Find the paired test statistic t for H₀: μd = 0.
n = 8d̄ = 8.3sd = 8.7
t =
8.Hypothesis tests & confidence intervalsHard
A sample of 49 measurements has mean 201.3. The population standard deviation is 8. Find the upper limit U of the 99% confidence interval for the mean.
x̄ = 201.3σ = 8n = 4999%
U =
3Solutions
1.Evidence-based practice statistics
Answer: OR = 0.41
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: 37/213 = 0.1737
controls: 165/385 = 0.4286
Odds ratio
OR = (37·385)/(165·213) = 0.4053 ≈ 0.41
Answer
OR = 0.41
OR below 1: exposure is less common among the cases, which suggests a protective effect.
2.Evidence-based practice statistics
Answer: p′ = 54.5 %
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+ = 0.98/(1 − 0.65) = 2.8
Pre-test odds
0.3/(1 − 0.3) = 0.4286
Post-test odds
0.4286·2.8 = 1.2
Post-test probability
p′ = 1.2/(1 + 1.2) = 0.5455 ≈ 54.5%
Answer
p′ = 54.5%
3.Descriptive statistics
Answer: s = 15
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 = (185 − 170)/10 = 1.5
Answer
z = 1.5
5.Confidence intervals
Answer: lower limit = 26.68
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 = 6, t = 3.707
Margin of error
E = 3.707·19.5/√7 = 27.32
Interval
54 ± 27.32 → [26.68, 81.32]
Answer
lower limit = 26.68
6.Confidence intervals
Answer: n = 107
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 = (2.576·2/0.5)² = 106.2
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 107
Answer
n = 107
7.Hypothesis tests & confidence intervals
Answer: t = 2.698
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 = 8.7/√8 = 3.076
Test statistic
t = 8.3/3.076 = 2.698
Answer
t = 2.698
With df = 7, the two-sided 5% critical value is 2.365; |t| is larger, so H₀ is rejected.
8.Hypothesis tests & confidence intervals
Answer: U = 204.2
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).