In one sentence Estimation means using data from a sample to make an approximate statement about an unknown characteristic of the whole population.
Estimation uses a sample to describe a population you cannot measure
Estimation is the process of using information obtained from a sample to make an approximate statement about an unknown characteristic of a population. A population parameter — a mean, a proportion, a variance or a standard deviation — is usually unknown, because studying every individual is expensive, slow or simply impossible. Suppose a researcher wants the average systolic blood pressure of every adult in a city. Measuring all of them is not realistic, so the researcher selects a sample of 500 adults and calculates their average. That sample average is then used as the estimate of the population average. The logic is empirical: the evidence comes from observed data rather than from a theory. The estimate is honest only if the sample was selected so that it represents the population, which is why the sampling methods of the earlier sections matter here.
A hospital wants the mean waiting time in its outpatient clinic. It times 120 consecutive patients, obtains a mean of 48 minutes, and reports that mean as its estimate of the waiting time for all outpatients.
A researcher measures the haemoglobin of 200 students and reports a mean of 13.5 g/dL. What exactly is being estimated, and why is the true value unknown?
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The sample mean of 13.5 g/dL is estimating the population mean haemoglobin of all students in the group being studied. The true value is unknown because not every student was measured; only a sample of 200 was, and each different sample would give a slightly different mean.
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- What a confidence interval adds to a point estimate
A point estimate is a single value, and a single value hides how much it could be wrong. A confidence interval supplies the missing information by giving a range of plausible values for the parameter. The recipe uses three quantities: the estimate itself, the standard error of the estimate, and a multiplier that depends on the chosen confidence level. The standard error of a sample mean is s ÷ √n, where s is the sample standard deviation and n is the sample size; it measures how much the sample mean would wobble from one sample to the next. For 95% confidence the multiplier is about 1.96, so the interval runs from estimate − 1.96 × standard error to estimate + 1.96 × standard error. Suppose a sample of 100 patients has a mean systolic pressure of 128 mmHg and a standard deviation of 15 mmHg. The standard error is 15 ÷ √100 = 15 ÷ 10 = 1.5 mmHg, and the margin of error is 1.96 × 1.5 ≈ 2.9 mmHg, so the 95% interval is roughly 125.1 to 130.9 mmHg. Note carefully what the interval does and does not say. It does not say that 95% of patients have pressures in this range; that is the empirical rule for individual values, not for means. It says that if the study were repeated many times, 95% of the intervals so built would contain the true population mean. The width responds to sample size through the square root: to halve the margin of error you must quadruple n, from 100 to 400. That square-root rule is why very large studies are expensive, and why a narrow interval is the real currency of estimation. In nursing research, an interval that crosses a difference of zero — or a value that matters clinically — tells the reader that the data are compatible with no effect, even when the point estimate looks impressive.
- The two mistakes a test can make: false positives, false negatives and power
A decision to reject H0 is a judgement made under uncertainty, so it can be wrong in two directions, and naming those directions is what lets you read a test result honestly. The first mistake is a Type I error, also called a false positive: H0 is true, but the data look extreme enough to reject it. The significance level α is exactly the long-run rate of these errors you are willing to accept; choosing α = 0.05 means that when H0 is genuinely true, about 5 in 100 identical studies would still produce p < 0.05. The second mistake is a Type II error, a false negative: H0 is false, a real effect exists, but the data are too weak to detect it. Its rate is written β, and the ability to avoid it is the power of the test, power = 1 − β. Conventionally researchers aim for power of 0.80 or more, and power is raised by increasing the sample size, reducing measurement error, or choosing a design with less variability. Power is calculated before a study begins, which is the real reason the sample size matters: an underpowered study can miss a genuine and important effect and then be reported as “no difference”. Two further details change the arithmetic. A two-sided test splits α across both directions of the effect, while a one-sided test puts it all in one direction and is easier on the same data; a one-sided test is only legitimate when a change in the opposite direction would be uninteresting, which is rare in clinical work. Finally, because a researcher could compute a p-value and then choose a threshold to fit it, α must be declared in advance. Fixing the threshold before the data are opened, and reporting the test used and the effect size with its units, is what keeps the procedure honest rather than merely arithmetical.
- Matching the statistical test to the research situation
No single test fits every question. The appropriate test depends on the research question, the study design, the type of variable measured and the assumptions the data must satisfy. The common matches are these. When one sample mean is compared with a fixed reference value — a ward’s mean waiting time against a national target — the one-sample t-test is used. When the means of two separate groups are compared — patients on a new drug against patients on a standard drug — the independent-samples t-test is used, and the two groups must be genuinely independent. When the same people are measured twice — blood pressure before and after a treatment, or a matched pair of twins — the paired t-test is used, because each person acts as their own control and the two measurements are linked. When two categorical variables are cross-tabulated — sex against whether a patient has hypertension — the chi-square test is used, and it works on the counts in the cells rather than on means. When the association between two quantitative variables is examined — height and weight, or dose and blood level — correlation is used. When one quantitative variable is used to predict another — a regression of blood pressure on age — regression is used. When more than two group means are compared in one analysis — three diets, or four wards — analysis of variance, shortened to ANOVA, is used, because running several t-tests instead would inflate the chance of a false positive. The choice must be justified by the design and not by familiarity. It also carries assumptions worth knowing: t-tests and ANOVA assume roughly normal data and independent observations, chi-square assumes adequate expected counts in every cell, and correlation assumes a straight-line relationship. Checking those assumptions is part of the analysis, and a wrong test produces a p-value that answers a question nobody asked.