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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 270 patients with a disease (cases) and 520 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 = 123b = 130c = 147d = 390
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 67% and specificity 61% comes back positive. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 30%Se = 67%Sp = 61%positive
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 38, 56, 45, 51, 33, 36, 55.
n = 7
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 500 and standard deviation 50. What proportion P of values is above 460?
μ = 500σ = 50x = 460
P =
5.Confidence intervalsMedium
A 95% confidence interval for a mean, based on 100 observations and a known σ, has a margin of error of 9.9. What is the margin of error E₂ of a 90% interval based on 200 observations?
95%n₁ = 100E₁ = 9.990%n₂ = 200
E₂ =
6.Confidence intervalsHard
The diameters of 20 machined shafts have a sample standard deviation of 7.5 µm. Assuming the diameters are normally distributed, find the lower limit σ of the 95% confidence interval for the population standard deviation. With df = 19, 2.5% of the χ² distribution lies below 8.907 and 2.5% lies above 32.852.
n = 20s = 7.5 µmχ² = 8.907, 32.852
σ = µm
7.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 100; the population standard deviation is known to be 24. A random sample of 64 observations has a mean of 96.64. Test H₀: μ = 100 against H₁: μ ≠ 100. Find the p-value p.
μ₀ = 100σ = 24n = 64x̄ = 96.64H₁: μ ≠ μ₀
p =
8.Hypothesis tests & confidence intervalsHard
A sample of 49 measurements has mean 74.7. The population standard deviation is 5. Find the upper limit U of the 99% confidence interval for the mean.
x̄ = 74.7σ = 5n = 4999%
U =
3Solutions
1.Evidence-based practice statistics
Answer: OR = 2.51
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: 123/147 = 0.8367
controls: 130/390 = 0.3333
Odds ratio
OR = (123·390)/(130·147) = 2.51
Answer
OR = 2.51
OR above 1: exposure is more common among the cases.
2.Evidence-based practice statistics
Answer: p′ = 42.4 %
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+ = 0.67/(1 − 0.61) = 1.718
Pre-test odds
0.3/(1 − 0.3) = 0.4286
Post-test odds
0.4286·1.718 = 0.7363
Post-test probability
p′ = 0.7363/(1 + 0.7363) = 0.4241 ≈ 42.4%
Answer
p′ = 42.4%
3.Descriptive statistics
Answer: s = 9.406
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 = −0.8
Standard normal table
Φ(−0.8) = 0.2119
P = 1 − 0.2119 = 0.7881
Answer
P = 0.7881
5.Confidence intervals
Answer: E₂ = 5.875
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 = (1.645/1.96)·√(100/200) = 0.5935
New margin
E₂ = 9.9·0.5935 = 5.875
Answer
E₂ = 5.875
6.Confidence intervals
Answer: σ = 5.704 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 = 19·7.5² = 1069
Limits for σ²
1069/32.852 = 32.53
1069/8.907 = 120
Limit for σ
σ = √32.53 = 5.704 µm
Answer
σ = 5.704 µm
95% interval for σ: [5.704, 10.95] µm.
7.Hypothesis tests & confidence intervals
Answer: p = 0.2627
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 = 24/√64 = 3
z = (96.64 − 100)/3 = −1.12
Tail area
Φ(1.12) = 0.8686
1 − Φ(1.12) = 0.1314
p-value
p = 2·0.1314 = 0.2627
Answer
p = 0.2627
p > 0.05, so H₀ is not rejected at the 5% level.
8.Hypothesis tests & confidence intervals
Answer: U = 76.54
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).