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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). In a randomised trial, 590 patients received a new treatment and 270 received the control treatment. The table shows how many in each group had the outcome (an unwanted event such as a stroke). Find the number needed to treat NNT to prevent one event. Round up to a whole number of patients.
treatment: 161/590control: 105/270
NNT =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). 10 000 people are screened with a test of sensitivity 76% and specificity 85%. The prevalence of the disease is 12%. How many positive results n₊ are expected in total?
N = 10 000p = 12%Se = 76%Sp = 85%
n₊ =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 46, 33, 18, 30, 29.
n = 5
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 60 and standard deviation 8. Find the z-score z of a score of 72.
μ = 60σ = 8x = 72
z =
5.Confidence intervalsMedium
A sample of 7 items has mean 114.5 and standard deviation s = 5. The population is roughly normal. Find the lower limit of the 99% confidence interval for the mean.
n = 7x̄ = 114.5s = 599%t(6) = 3.707
lower limit =
6.Confidence intervalsHard
The diameters of 16 machined shafts have a sample standard deviation of 10.9 µm. Assuming the diameters are normally distributed, find the lower limit σ of the 95% confidence interval for the population standard deviation. With df = 15, 2.5% of the χ² distribution lies below 6.262 and 2.5% lies above 27.488.
n = 16s = 10.9 µmχ² = 6.262, 27.488
σ = µm
7.Hypothesis tests & confidence intervalsMedium
A sample of 36 students has a mean score of 198.5 and a sample standard deviation of 5. Find the test statistic t for H₀: μ = 200.
μ₀ = 200x̄ = 198.5s = 5n = 36
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 30 of 150 patients on treatment A recover, and 8 of 150 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 30n₁ = 150x₂ = 8n₂ = 150
z =
3Solutions
1.Evidence-based practice statistics
Answer: NNT = 9
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 = 161/590 = 0.2729
CER = 105/270 = 0.3889
Number needed to treat
ARR = 0.3889 − 0.2729 = 0.116
NNT = 1/0.116 = 8.62 → 9
Answer
NNT = 9
Treating 9 patients prevents about one event.
2.Evidence-based practice statistics
Answer: n₊ = 2232
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.
With and without the disease
10 000·0.12 = 1200
10 000 − 1200 = 8800
All positive results
TP = 1200·0.76 = 912
FP = 8800·0.15 = 1320
n₊ = 912 + 1320 = 2232
Answer
n₊ = 2232
Of the 2232 positives only 912 have the disease (PPV 40.9%).
3.Descriptive statistics
Answer: s = 10.03
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 = (72 − 60)/8 = 1.5
Answer
z = 1.5
5.Confidence intervals
Answer: lower limit = 107.5
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 = 3.707·5/√7 = 7.006
Interval
114.5 ± 7.006 → [107.5, 121.5]
Answer
lower limit = 107.5
6.Confidence intervals
Answer: σ = 8.052 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 = 15·10.9² = 1782
Limits for σ²
1782/27.488 = 64.83
1782/6.262 = 284.6
Limit for σ
σ = √64.83 = 8.052 µm
Answer
σ = 8.052 µm
95% interval for σ: [8.052, 16.87] µm.
7.Hypothesis tests & confidence intervals
Answer: t = −1.8
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).
Standard error
SE = 5/√36 = 5/6 = 0.8333
Test statistic
t = (198.5 − 200)/0.8333 = −1.8
Answer
t = −1.8
t0.025; df = 35 = 2.03
|t| < 2.03, so H₀ is not rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: z = 3.819
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 = 30/150 = 0.2
p̂₂ = 8/150 = 0.05333
p̂ = 38/300 = 0.1267
Standard error under H₀
SE = √(0.1267·0.8733·(1/150 + 1/150)) = 0.03841
Test statistic
z = (0.2 − 0.05333)/0.03841 = 3.819
Answer
z = 3.819
|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).