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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 cohort study, exposed people contributed 4000 person-years with 48 new cases, and unexposed people 1100 person-years with 4 new cases. Find the incidence rate ratio IRR (exposed versus unexposed). Round to two decimal places.
exposed: 48/4000 PYnot exposed: 4/1100 PY
IRR =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). 10 000 people are screened with a test of sensitivity 93% and specificity 80%. The prevalence of the disease is 16%. How many positive results n₊ are expected in total?
N = 10 000p = 16%Se = 93%Sp = 80%
n₊ =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 26, 17, 17, 31, 34.
n = 5
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 70 and standard deviation 20. What proportion P of values is below 110?
μ = 70σ = 20x = 110
P =
5.Confidence intervalsMedium
The battery life of two phone models is compared. Model 1: 40 phones, mean 88.5 h, known standard deviation 8 h. Model 2: 40 phones, mean 80.5 h, known standard deviation 15 h. Find the lower limit of the 95% confidence interval for μ₁ − μ₂.
The diameters of 16 machined shafts have a sample standard deviation of 9.1 µm. Assuming the diameters are normally distributed, find the upper 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 = 9.1 µmχ² = 6.262, 27.488
σ = µm
7.Hypothesis tests & confidence intervalsMedium
Two production lines make the same part. Line 1: a sample of 30 parts has mean length 113 mm, and the line's known standard deviation is 10 mm. Line 2: a sample of 30 parts has mean 104 mm, known standard deviation 13 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
A one-sided z test of H₀: μ = 500 against H₁: μ < 500 is done at α = 0.10 with a sample of 25; the standard deviation is known to be 2.5. In reality the mean is 498.7. Find the power 1 − β of the test in percent.
μ₀ = 500μ = 498.7σ = 2.5n = 25α = 0.10
1 − β = %
3Solutions
1.Evidence-based practice statistics
Answer: IRR = 3.3
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.
Incidence rates
exposed: 48/4000 = 12 per 1000 person-years
not exposed: 4/1100 = 3.636 per 1000 person-years
Rate ratio
IRR = 12/3.636 = 3.3
Answer
IRR = 3.3
2.Evidence-based practice statistics
Answer: n₊ = 3168
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
10 000·0.16 = 1600
10 000 − 1600 = 8400
All positive results
TP = 1600·0.93 = 1488
FP = 8400·0.2 = 1680
n₊ = 1488 + 1680 = 3168
Answer
n₊ = 3168
Of the 3168 positives only 1488 have the disease (PPV 47%).
3.Descriptive statistics
Answer: s = 7.842
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.
Mean
x̄ = 125/5 = 25
Squared deviations
Σ(x − x̄)² = 1² + (−8)² + (−8)² + 6² + 9² = 246
Standard deviation
s = √(246/4) = 7.842
Answer
s = 7.842
4.Descriptive statistics
Answer: P = 0.9772
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 = 2
Standard normal table
Φ(2) = 0.9772
P = 0.9772
Answer
P = 0.9772
5.Confidence intervals
Answer: lower limit = 2.732 h
Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.
Point estimate
x̄₁ − x̄₂ = 88.5 − 80.5 = 8
Standard error
SE = √(8²/40 + 15²/40) = 2.688
Margin of error
E = 1.96·2.688 = 5.268
Interval
8 ± 5.268 → [2.732, 13.27]
Answer
lower limit = 2.732 h
The interval does not contain 0, so the means differ significantly.
6.Confidence intervals
Answer: σ = 14.08 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 = 15·9.1² = 1242
Limits for σ²
1242/27.488 = 45.19
1242/6.262 = 198.4
Limit for σ
σ = √198.4 = 14.08 µm
Answer
σ = 14.08 µm
95% interval for σ: [6.722, 14.08] µm.
7.Hypothesis tests & confidence intervals
Answer: z = 3.006
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.
Standard error of the difference
SE = √(10²/30 + 13²/30) = √8.967 = 2.994
Test statistic
z = (113 − 104)/2.994 = 3.006
Answer
z = 3.006
|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: 1 − β = 90.62 %
Power of a test: power = 1 − β is the probability of rejecting H₀ when it is false. Find the cutoff x̄c from μ₀ and z_α, then compute the probability of landing beyond x̄c using the true mean: z = (x̄c − μ)/(σ/√n).