Written and maintained by CASRAI Editorial Board
Last updated
A parameter in statistics is a number that describes a whole population, such as the true mean or the true proportion of everyone you want to learn about. A statistic is the matching number computed from a sample. Parameters are usually unknown and fixed; statistics are known and change from sample to sample. This guide gives the definition, the notation, a worked example with every step of the arithmetic, a rule for telling the two apart (including whether 3.7% or 50% is a statistic or a parameter), and what a “parameter of interest” is.
What is a parameter in statistics?
A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. A statistic is a number that represents a property of the sample (OpenStax, Introductory Statistics 2e, section 1.1). The NIST/SEMATECH e-Handbook of Statistical Methods calls the population value “the population parameter or true value of the parameter” and its sample counterpart “a point estimate or a sample estimate”.
The University of Illinois Data Science Exploration site describes the parameter as the true but often unknown value we would ideally like to know. The practical consequence: you almost never get to compute a parameter directly, so you compute a statistic from a sample and use it to estimate the parameter. See population vs sample for the population side of that distinction.
Notation: which symbol is the parameter?
Parameters are conventionally written with Greek letters and statistics with Latin letters. The Illinois resource lists the pairs below.
| Quantity | Statistic (sample) | Parameter (population) |
|---|---|---|
| Proportion | p̂ (p-hat) | p (or π) |
| Mean | x̄ (x-bar) | μ |
| Standard deviation | s | σ |
| Variance | s² | σ² |
| Median | m | M |
What is an example of a parameter?
Any summary of an entire population is a parameter. The Illinois resource gives a real case: the population is all Airbnb listings in Chicago in March 2023. For that population, the mean number of guests the listings accommodate (μ) was 4.31, the proportion with a local host (p) was 0.7862, and the median price (M) was $124.
A sample of 700 of those listings gave x̄ = 4.42, p̂ = 0.8042 and a median (m) of $126. Each sample statistic is close to its parameter but not equal to it: the sample mean overshoots μ by 0.11 (4.42 − 4.31), and the sample proportion overshoots p by 0.018 (0.8042 − 0.7862). That gap is sampling error, and it is the reason a statistic is described as an estimate of a parameter rather than as the parameter itself.
Formula: the parameter and its estimate, with every symbol explained
For a mean, the population parameter and its sample estimate use the same arithmetic on different data.
- Population mean: μ = (Σ xᵢ) / N, where xᵢ is each value, Σ means “sum over all values” and N is the number of values in the whole population.
- Sample mean: x̄ = (Σ xᵢ) / n, where n is the number of values in the sample.
- Population standard deviation: σ = √( Σ (xᵢ − μ)² / N ).
- Sample standard deviation: s = √( Σ (xᵢ − x̄)² / (n − 1) ). Dividing by n − 1 rather than n is the usual correction that makes s² an unbiased estimator of σ²; the reason is explained in what is an unbiased estimator.
- Proportion: p̂ = x / n, where x is the number of sample members with the characteristic.
Worked example: how to calculate parameter stats
Hypothetical data for illustration. Suppose a study of one university’s 5,000 undergraduates asks how many hours each sleeps per night. Nobody measures all 5,000, so a random sample of n = 8 students is taken: 6.5, 7.0, 8.0, 6.0, 7.5, 7.0, 5.5 and 8.5 hours.
- Sum: 6.5 + 7.0 + 8.0 + 6.0 + 7.5 + 7.0 + 5.5 + 8.5 = 56.0.
- Sample mean: x̄ = 56.0 / 8 = 7.0 hours. This is the statistic. It estimates μ, the mean for all 5,000 students, which stays unknown.
- Deviations from x̄: −0.5, 0, 1.0, −1.0, 0.5, 0, −1.5, 1.5.
- Squared deviations: 0.25, 0, 1.0, 1.0, 0.25, 0, 2.25, 2.25, which sum to 7.0.
- Sample variance: s² = 7.0 / (8 − 1) = 1.0, so s = 1.0 hour.
If you divided by n instead (7.0 / 8 = 0.875, giving 0.94), you would be using the population formula on sample data, which tends to understate σ. Only if the eight students were the entire population would 0.94 be the parameter σ.
Is 3.7% a statistic or a parameter? Is 50% a statistic or a parameter?
The value of the number does not decide it. What decides it is whether it was computed from a sample or from every member of the population. A percentage is a statistic when it describes a sample and a parameter when it describes the whole population.
| The number | How it was obtained | Reading |
|---|---|---|
| 3.7% | 37 of 1,000 randomly sampled employees (37 / 1,000 = 0.037) report a condition | Statistic (p̂) |
| 3.7% | Records for all employees in the company show it | Parameter (p) |
| 50% | 25 of 50 sampled participants chose option A (25 / 50 = 0.50) | Statistic (p̂) |
| 50% | A hypothesised or theoretical value, such as a fair coin’s probability of heads | Parameter (p) |
The quick test: ask “who is this number about?” If the answer is a subset you drew, it is a statistic. If the answer is everyone in the group the question is about, it is a parameter. Check how the group was defined with sampling methods before deciding what counts as the population.
What is a parameter of interest?
The parameter of interest is the specific population quantity a study is designed to estimate or test: the one number that answers the research question. The same population has many parameters (mean, proportion, variance); the research question picks one.
How to find the parameter of interest
- State the question in plain words, for example “Do students sleep enough?” or “Does the new treatment lower blood pressure?”
- Identify the population the question is about.
- Decide what kind of quantity answers it: a mean for a continuous measure, a proportion for a yes/no outcome, or a difference between two means or proportions for a comparison.
- Name it with its symbol before collecting data (for example μ, or μ₁ − μ₂), so the sample size, the measure and the analysis all target the same quantity.
The statistic that estimates it (x̄, p̂ or x̄₁ − x̄₂) then carries a margin of uncertainty, which a confidence interval expresses.
Interpretation: reading the two numbers
| If you see | It means |
|---|---|
| Greek letter (μ, σ, π) or “population” | A parameter: a fixed property of the whole population, usually unknown. |
| Latin letter (x̄, s, p̂) or “sample” | A statistic: computed from data, varies from sample to sample. |
| Statistic close to a known parameter | A small sampling error, as in the Chicago listings example (0.11 on a mean near 4.3). |
| Different samples give different statistics | Expected: the parameter did not change, the sample did. |
Assumptions and common mistakes
- Treating a statistic as if it were the parameter. A sample mean of 7.0 does not mean μ = 7.0; it is an estimate with sampling error.
- Deciding by value. Neither 3.7% nor 50% is a parameter or a statistic by itself.
- Using σ’s formula on sample data. For a sample, divide by n − 1 (Excel STDEV.S, R sd, Python ddof=1).
- Assuming any sample estimates the parameter well. The estimate is only as representative as the sampling process behind it.
- Mixing up the population. If the “population” is redefined (all students versus one class), the parameter changes with it.
How to report it (APA example)
The APA Style numbers and statistics guide (7th edition, last updated September 11, 2024) says to use the symbol with an operator for statistics (for example M = 7.7), to italicise Latin letters used as statistical symbols but not Greek letters, to report means and standard deviations on integer scales to one decimal and other means and standard deviations to two, and to put no zero before the decimal for proportions. It also says not to define symbols such as M, SD and n.
Applied to the worked example: “Sleep duration in the sample (n = 8) averaged M = 7.00 hours (SD = 1.00).” Hours are not an integer-scale survey item, so two decimals apply. The wording says “in the sample”, which signals that these are statistics, not parameters.
How to calculate it in Excel, R or Python
These compute sample statistics for the eight values above. For the pairing of each function with the population or sample formula, see Excel statistics functions and standard deviation in R.
Excel (values in A1:A8): =AVERAGE(A1:A8) returns 7; =STDEV.S(A1:A8) returns 1 (sample, n − 1); =STDEV.P(A1:A8) returns 0.935 (population formula, n).
R: x = c(6.5, 7.0, 8.0, 6.0, 7.5, 7.0, 5.5, 8.5); mean(x); sd(x) gives 7 and 1; sd uses n − 1.
Python: import numpy as np; x = np.array([6.5, 7.0, 8.0, 6.0, 7.5, 7.0, 5.5, 8.5]); np.mean(x); np.std(x, ddof=1) gives 7.0 and 1.0; np.std(x) without ddof=1 gives 0.935, the population formula.
FAQ
What is an example of a parameter?
The mean income of all households in a country, the proportion of all voters who support a measure, or the 4.31 mean number of guests across all Chicago listings in March 2023 in the Illinois example. Each describes a whole population.
How to calculate parameter stats?
If you have every member of the population, apply the population formulas (divide by N). If you have a sample, calculate the statistic (divide by n, or n − 1 for the standard deviation) and treat it as an estimate of the parameter.
Is the value of 3.7% a statistic or parameter?
It is a statistic if it came from a sample and a parameter if it describes the entire population. The value itself does not tell you which.
Is 50% a statistic or parameter?
The same rule applies: a sample result such as 25 of 50 is a statistic; a theoretical or population-wide 50% is a parameter.
What is the parameter of interest in statistics?
The population quantity the study is built to estimate or test, such as a mean, a proportion or a difference between groups.
Sources
- OpenStax. Introductory Statistics 2e, section 1.1, Definitions of Statistics, Probability, and Key Terms. openstax.org (checked 2026-10-29).
- NIST/SEMATECH e-Handbook of Statistical Methods. itl.nist.gov (checked 2026-10-29).
- University of Illinois Urbana-Champaign, Data Science Exploration: Population Parameters and Sample Statistics. exploration.stat.illinois.edu (checked 2026-10-29).
- American Psychological Association (2024). APA Style numbers and statistics guide. apastyle.apa.org (checked 2026-10-29).
