Key takeaways
– In statistics, a population is the complete set of items, events, or people you want to study (e.g., all daisies in a country, every share price for a given index). (Investopedia)
– A sample is a subset of a population used to make inferences about population characteristics (parameters). Random sampling reduces bias. (Investopedia; CUEMATH)
– Population parameters (µ, σ) describe the full population; sample statistics (x̄, s) estimate those parameters. Confidence intervals and hypothesis tests quantify uncertainty. (Investopedia; CUEMATH)
– In finance, some datasets (historic prices for all traded days) can be treated as a population; survivorship bias and nonrandom exclusions still matter. (Investopedia)
What “population” means in statistics
– Definition: The population is the entire set of entities (people, animals, transactions, time points, etc.) about which you want to draw conclusions. It is not limited to living organisms—an “individual” can be a trade, a firm, or a measured unit. (Investopedia)
– Examples:
– All great white sharks in the world’s oceans (biological population).
– All doctors in a country who treat a condition (population targeted by an ad).
– All daily closing prices of a publicly traded stock since listing (investment population).
Population vs. sample, parameter vs. statistic
– Population parameter: a number that summarizes a characteristic of the population (e.g., population mean µ, population standard deviation σ). These are often denoted with Greek letters. (Investopedia)
– Sample statistic: the analogous number computed from a sample (e.g., sample mean x̄, sample standard deviation s). Sample statistics are used to estimate parameters and are subject to sampling error. (Investopedia; CUEMATH)
– When you can avoid inference: If you truly observe every member of the population, you can compute parameters directly and do not need sampling-based inference. In practice this is rare except for well-recorded datasets (e.g., complete historical price records). (Investopedia)
Statistics commonly used to describe populations
– Size (N), density
– Measures of central tendency: mean (µ), median, mode
– Measures of spread: variance, standard deviation (σ), interquartile range
– Shape: skewness, kurtosis, distribution type
– Position: percentiles, quantiles
– Relationships: covariance, correlation, regression coefficients
(Investopedia; CUEMATH)
Why sampling is usually necessary
– Practical constraints (time, cost, accessibility) make measuring every member infeasible for most populations (e.g., all brown‑eyed people worldwide or all great white sharks).
– A properly selected random sample allows valid inference about population parameters and reduces bias. (Investopedia)
Common sources of bias to avoid
– Selection bias: sample systematically differs from population.
– Nonresponse bias: certain types don’t respond.
– Survivorship bias: excluding failed/removed items (common in finance).
– Measurement error: inaccurate or inconsistent measurement.
– Convenience sampling bias: using easily available units instead of random selection.
Practical steps for studying a population (step-by-step)
1. Define the population precisely
– Specify inclusion/exclusion criteria and the target universe (who/what, time period, geographic scope).
– Example: “All U.S. board-certified oncologists actively treating adult patients as of 2024.”
2. Choose or construct a sampling frame
– The sampling frame is the list or mechanism by which units are selected (registries, customer lists, trading databases).
– Ensure the frame covers the population; document gaps.
3. Pick a sampling method
– Probability (random) sampling: simple random sampling, stratified sampling, cluster sampling, systematic sampling — preferred for unbiased inference.
– Nonprobability sampling: convenience, quota, snowball — use only when necessary and report limitations.
4. Determine sample size
– For proportions: n = (Z^2 * p*(1-p)) / E^2 (use p≈0.5 if unknown to maximize required n); apply finite population correction if population is small.
– For means: n = (Z * σ / E)^2 where E is desired margin of error and σ is an estimate of population SD.
– Choose confidence level (commonly 95%) and margin of error appropriate for decision needs.
5. Collect the data carefully
– Use standardized instruments, train data collectors, pilot test surveys.
– Track response rates and reasons for nonresponse.
6. Clean and weight the sample if needed
– Correct entry errors, handle missing data with transparent rules.
– Apply weights to correct for sampling design or nonresponse when appropriate.
7. Compute sample statistics and estimate parameters
– Calculate point estimates (x̄, p̂, s) and accompanying measures of uncertainty (standard errors).
– Construct confidence intervals (e.g., x̄ ± Z*SE) and perform hypothesis tests where relevant.
8. Report results and limitations
– State sampling method, sample size, response rate, potential biases, and assumptions.
– Present both estimates and measures of uncertainty (confidence intervals, p-values).
Worked examples
– Great white sharks (ecology)
– Population: all great whites globally (infeasible to enumerate).
– Approach: tag a random sample from accessible areas, collect data on size/age/movement, use mark-recapture or model-based estimators to infer population-level parameters.
– Doctors recommending a drug (survey claim)
– Population: all doctors treating the condition.
– Risk: ad may report the statistic for respondents only, not the full population (nonresponse bias). Valid inference requires probability sampling and adjustment for nonresponse.
– Stock prices (finance)
– Population: all recorded daily closing prices for a stock (this can be treated as a population if you have all days).
– Analysts can compute parameters directly, but must be careful of survivorship bias (excluding delisted stocks) and structural changes over time.
Population mean and formulas (quick reference)
– Population mean µ = Σ xi / N (where N is population size).
– Sample mean x̄ = Σ xi / n (n is sample size).
– Population variance σ^2 = Σ (xi − µ)^2 / N.
– Sample variance s^2 = Σ (xi − x̄)^2 / (n − 1) (unbiased estimator).
(See CUEMATH for inferential formula details.)
Inference: how a sample informs a population
– Sampling distribution: repeated random sampling yields a distribution of sample statistics around the true parameter; its standard deviation is the standard error.
– Confidence intervals quantify plausible ranges for parameters given observed sample statistics.
– Hypothesis testing evaluates evidence against a null claim about a parameter.
Special considerations for investment analysis
– Some financial datasets (prices, volumes) are complete records and can be treated as populations, permitting direct calculation of parameters (e.g., historical mean return). (Investopedia)
– Still watch out for:
– Survivorship bias: backtests using only surviving firms can overstate performance.
– Nonstationarity: distributions can change over time (regime shifts).
– Data snooping: extensive searching for patterns inflates false positives.
Checklist before publishing or acting on results
– Is the population definition clear and appropriate?
– Does the sampling frame adequately cover the population?
– Was sampling random or are there known selection issues?
– Is sample size sufficient for the desired precision?
– Have you quantified uncertainty (SEs, CIs)?
– Are potential biases and limitations disclosed?
– For finance, have you accounted for survivorship and nonstationarity?
Common pitfalls and how to avoid them
– Treating a convenience sample as representative — use probability sampling or clearly qualify conclusions.
– Ignoring nonresponse — implement follow-ups and weighting adjustments.
– Misinterpreting statistics from respondents as population values — report as sample-based estimates.
– Forgetting to use n−1 in sample variance when estimating population variance — use correct formula.
Fast fact
– Parameter notation: population mean = µ, population standard deviation = σ; sample mean = x̄, sample standard deviation = s. (Investopedia; CUEMATH)
The bottom line
A population in statistics is simply the full set of items you want to study. Because measuring entire populations is often impractical, thoughtful sampling and transparent analysis are essential. Define your population precisely, choose an appropriate sampling strategy, compute sample statistics and uncertainty, and always disclose limitations and potential biases. In finance, some datasets function as full populations, but methodological caution (survivorship, nonstationarity) remains crucial. (Investopedia; CUEMATH; CliffsNotes)
Sources
– Investopedia, “Population” (Matthew Collins) — https://www.investopedia.com/terms/p/population.asp
– CUEMATH, “Inferential Statistics” — https://www.cuemath.com/statistics/inferential-statistics/
– CliffsNotes, “Populations, Samples, Parameters, and Statistics” (overview guide)
(Continuation of article on population in statistics)
Additional Sections
Types of Populations
– Finite vs. infinite: A finite population has a fixed number of elements (e.g., all 500 stocks in an index today); an infinite population is conceptual or effectively unbounded (e.g., future customer arrivals, theoretical repeated trials). For many practical problems you treat the population as finite if you can enumerate it.
– Target population vs. sampling frame: The target population is the full set you want to study (e.g., “all adults in a country”). The sampling frame is the list or mechanism from which you actually draw the sample (e.g., voter registration lists, phone numbers). Mismatch between the two creates coverage bias.
– Cross-sectional vs. longitudinal populations: Cross-sectional populations are measured at one point in time (e.g., survey of current employees). Longitudinal populations are tracked over time (e.g., cohort of patients followed for 10 years).
Population Parameters and Sample Statistics — Notation and Meaning
– Population size: N (total number of individuals or units)
– Sample size: n (number selected from N)
– Population mean: μ (mu) = (sum of all values in population) / N
– Sample mean: x̄ (x-bar) = (sum of sampled values) / n
– Population variance: σ^2 and standard deviation: σ
– Sample variance: s^2 and sample standard deviation: s
– Proportion (population): P; sample proportion: p̂
Why Distinguish Parameter vs Statistic?
– Parameters describe the entire population and are often unknown.
– Statistics are computed from samples and are used to estimate parameters.
– Notation differences (Greek for parameters, Roman for statistics) remind us whether a value is known exactly or estimated.
Common Sampling Methods (with when to use them)
– Simple random sampling: every unit has equal chance. Good baseline; minimizes selection bias if frame is accurate.
– Systematic sampling: pick every k-th element. Simple and efficient, but beware of periodicity.
– Stratified sampling: divide population into strata (e.g., age groups) and sample within each. Increases precision when strata differ.
– Cluster sampling: randomly select clusters (e.g., schools) and sample all or some units within. Useful when population is geographically dispersed.
– Convenience and voluntary response sampling: easy but prone to bias; avoid for inference.
– Purposive (judgmental) sampling: used in qualitative research; not for estimating population parameters statistically.
Bias and Error: Types and Mitigation
– Sampling error: natural variability because you observe only a sample; reduced by increasing n.
– Nonresponse bias: causal when respondents differ systematically from nonrespondents; mitigate with follow-ups, incentives, weighting.
– Selection/coverage bias: occurs when sampling frame omits parts of population; use an inclusive frame or adjust with weighting.
– Measurement error: poor question wording or faulty instruments; pilot tests and instrument calibration help.
– Response bias: social desirability, leading questions; use neutral wording, anonymity where appropriate.
Practical Steps for Designing a Study (step-by-step)
1. Define the research question and the target population precisely.
– Example: “What proportion of U.S. primary-care physicians recommend Drug XYZ to eligible patients?”
2. Specify inclusion and exclusion criteria for individuals/units.
3. Construct or choose an appropriate sampling frame.
4. Choose a sampling method appropriate for the frame and cost constraints.
5. Determine required sample size for the desired precision and confidence.
– For estimating a proportion: n = (Z^2 * p*(1−p)) / E^2
– Z = Z-score for desired confidence (1.96 for 95% CI)
– p = estimated proportion (use 0.5 for conservative max sample size)
– E = desired margin of error (in proportion units)
– Example: To estimate a proportion with ±5% margin at 95% confidence, n ≈ (1.96^2 * 0.25) / 0.05^2 ≈ 385.
– For estimating a mean: n = (Z^2 * σ^2) / E^2 (use prior σ or pilot study)
6. Collect data, monitoring response rates and data quality.
7. Compute sample statistics and construct confidence intervals.
– Proportion CI: p̂ ± Z * sqrt(p̂(1−p̂)/n)
– Mean CI (unknown σ): x̄ ± t_(n−1) * s/√n
8. Evaluate assumptions and potential biases; adjust or report limitations.
Confidence Intervals and Margin of Error (practical)
– Confidence interval gives a range likely to contain the population parameter at a specified confidence level.
– Margin of error depends on variability and sample size; halving the margin of error requires quadrupling sample size.
– When a full census (N observed) is available, no inference is required; parameters are computed directly.
Examples and Walkthroughs
Example 1 — The “62% of doctors” ad
– Claim: 62% of doctors recommend Drug XYZ.
– Practical questions: What was the sampling method? What was response rate? Margin of error? If the advertiser surveyed 400 doctors and 62% responded “yes,” the 95% CI is:
– p̂ = 0.62; SE = sqrt(0.62*0.38/400) ≈ 0.024; CI ≈ 0.62 ± 1.96*0.024 ≈ 0.62 ± 0.047 → (0.573, 0.667).
– Interpretation: If sample was random and unbiased, we’re 95% confident the true proportion is between ~57.3% and ~66.7%.
– Caveat: If the survey had low response or nonrandom recruitment, this CI is misleading.
Example 2 — Great white shark tagging (ecology)
– Target population: all great white sharks in a region.
– Practical approach: capture–mark–recapture to estimate population size. Random encounters are unlikely, so use stratified sampling by habitat, season, tagging, and recapture probabilities.
– Inference requires models that account for detection probability.
Example 3 — Investing and market populations
– Populations can be concrete: all daily closing prices for a listed stock over 20 years — you can compute exact historical mean and standard deviation (a population parameter for that historical interval).
– But when inferring future returns, treat historical data as a sample from a larger stochastic population; use confidence intervals and models.
– Key financial parameters vs. statistical terms:
– Finance “alpha”: excess return over benchmark. (Different from statistical alpha.)
– Statistical α: probability of Type I error in hypothesis testing.
– Finance “beta”: sensitivity to market returns. (Different from β in hypothesis testing.)
Advanced Topics (brief)
– Finite population correction (FPC): When sample is a large fraction of a finite population (say, n/N > 0.05), adjust variance: multiply by sqrt((N−n)/(N−1)).
– Bootstrapping: resampling technique to estimate sampling distribution when analytic forms are complex.
– Weighted sampling and post-stratification: correction for unequal probabilities or to align sample with known population margins.
– Power analysis: for hypothesis testing, calculate sample size to achieve desired power (probability of detecting an effect if it exists).
Common Pitfalls and How to Avoid Them
– Confusing population with sample — always clarify which set your measures describe.
– Over-generalizing results from convenience samples — limit inferences to populations represented by the sample.
– Ignoring nonresponse bias — track and model differences between respondents and nonrespondents.
– Misinterpreting statistical significance as practical importance — report effect sizes and confidence intervals.
Practical Checklist for Analysts and Students
– Have I clearly defined the target population and sampling frame?
– Which sampling method best balances bias control and cost?
– Is my planned sample size sufficient for the desired precision?
– Are there known sources of bias and how will I address them?
– Are the formulas and statistical methods I plan to use appropriate given assumptions (normality, independence, known/unknown variance)?
– Do I understand the distinction between population parameters and sample statistics, in notation and interpretation?
Real-World Applications
– Public health: estimating disease prevalence in a country (requires representative sampling and often stratification by region and age).
– Market research: estimating customer satisfaction (careful sampling and weighting often required).
– Ecology: estimating animal populations using capture–recapture and spatial stratification.
– Finance: measuring historical volatility (often uses the full recorded population of past prices) versus forecasting (treats history as sample of possible outcomes).
Concluding Summary
A population in statistics is the complete set of items or events that you want to study. Because it is usually impractical to measure every element, samples are drawn to estimate population parameters. Understanding the distinction between population parameters (μ, σ) and sample statistics (x̄, s) is fundamental. Choosing an appropriate sampling method, calculating an adequate sample size, and guarding against sources of bias are essential steps to make valid inferences. In many areas — from ecology and medicine to finance — clear definition of the population and careful sampling design determine whether conclusions are trustworthy. When full population data are available (as with many historical financial series), parameters can be computed directly; otherwise, inference methods, confidence intervals, and proper reporting of assumptions guide reliable conclusions.
Sources
– Investopedia, “Population” (Matthew Collins).
– CliffsNotes, “Populations, Samples, Parameters, and Statistics.”
– CUEMATH, “Inferential Statistics.”
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