Parameters are numbers that summarize data for an entire population. Statistics are numbers that summarize data from a sample, i.e. some subset of the entire population. Show
Problems (1) through (6) below each present a statistical study*. For each study, identify both the parameter and the statistic in the study. 1) A researcher wants to estimate the average height of women aged 20 years or older. From a simple random sample of 45 women, the researcher obtains a sample mean height of 63.9 inches. 2) A nutritionist wants to estimate the mean amount of sodium consumed by children under the age of 10. From a random sample of 75 children under the age of 10, the nutritionist obtains a sample mean of 2993 milligrams of sodium consumed. 3) Nexium is a drug that can be used to reduce the acid produced by the body and heal damage to the esophagus. A researcher wants to estimate the proportion of patients taking Nexium that are healed within 8 weeks. A random sample of 224 patients suffering from acid reflux disease is obtained, and 213 of those patients were healed after 8 weeks. 4) A researcher wants to estimate the average farm size inKansas. From a simple random sample of 40 farms, the researcher obtains a sample mean farm size of 731 acres. 5) An energy official wants to estimate the average oil output per well in theUnited States. From a random sample of 50 wells throughout theUnited States, the official obtains a sample mean of 10.7 barrels per day. 6) An education official wants to estimate the proportion of adults aged 18 or older who had read at least one book during the previous year. A random sample of 1006 adults aged 18 or older is obtained, and 835 of those adults had read at least one book during the previous year. 7) The International Dairy Foods Association (IDFA) wants to estimate the average amount of calcium male teenagers consume. From a random sample of 50 male teenagers, the IDFA obtained a sample mean of 1081 milligrams of calcium consumed. 8) A sociologist wants to the proportion of adults with children under the age of 18 that eat dinner together 7 nights a week. A simple random sample of 1122 adults with children under the age of 18 was obtained, and 337 of those adults reported eating dinner together with their families 7 nights a week. 9) A school administrator wants to estimate the mean score on the verbal portion of the SAT for students whose first language is not English. From a simple random sample of 20 students whose first language is not English, the administrator obtains a sample mean SAT verbal score of 458. * These research objectives were adapted from problems in Michael Sullivan, Fundamentals of Statistics, 2nd edition, Pearson Education 2008. A parameter is a number describing a whole population (e.g., population mean), while a statistic is a number describing a sample (e.g., sample mean). The goal of quantitative research is to understand characteristics of populations by finding parameters. In practice, it’s often too difficult, time-consuming or unfeasible to collect data from every member of a population. Instead, data is collected from samples. With inferential statistics, we can use sample statistics to make educated guesses about population parameters. Table of contents
Population vs sample A sample is a smaller group taken from the population. The sample is the group of elements that you will actually collect data from. Population vs sampleYou want to identify the level of support for the death penalty among US residents. Since the population you’re interested in is all US residents, it’s not practical to collect data from the whole population. Instead, you use random sampling to survey a sample of 2000 participants.What kinds of numbers are parameters and statistics?Statistics and parameters are numbers that summarize any measurable characteristic of a sample or a population. For categorical variables (e.g., political affiliation), the most common statistic or parameter is a proportion. For numerical variables (e.g., height), the mean or standard deviation are commonly reported statistics or parameters. Examples of statistics vs parametersSample statisticPopulation parameterProportion of 2000 randomly sampled participants that support the death penalty.Proportion of all US residents that support the death penalty.Median income of 850 college students in Boston and Wellesley.Median income of all college students in Massachusetts.Standard deviation of weights of avocados from one farm.Standard deviation of weights of all avocados in the region.Mean screen time of 3000 high school students in India.Mean screen time of all high school students in India.Statistical notationDifferent symbols are used for statistics versus parameters to show whether a sample or a population is being referred to. Greek letters and capital letters usually refer to populations, whereas Latin letters and lower-case letters refer to samples. Symbols for statistics vs parametersSample statisticPopulation parameterProportionp̂ (called “p-hat”)PMeanx̄ (called “x-bar”)μ (Greek letter “mu”)Standard deviations (Latin letter “s”)σ (Greek letter “sigma”)Variances2σ2Receive feedback on language, structure and formattingProfessional editors proofread and edit your paper by focusing on:
See an example Telling the difference between a parameter and a statisticIn news and research reports, it’s not always clear whether a number is a parameter or statistic. To figure out which type of number you’re dealing with, ask yourself the following:
If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters. If the answer is no to either of the questions, then the number is more likely to be a statistic. Sampling is used to collect data from large populations and generalize the statistics to the broader population in an externally valid way. Quiz: Statistic or parameter?Estimating parameters from statisticsUsing inferential statistics, you can estimate population parameters from sample statistics. To make unbiased estimates, your sample should ideally be representative of your population and/or randomly selected. There are two important types of estimates you can make about the population parameter: point estimates and interval estimates.
Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie. Estimating a population parameter from a sample statisticIn your study on support for the death penalty among US residents, you find that 61% of participants in your sample support the death penalty. To estimate the population parameter, you calculate a point estimate and an interval estimate from your sample statistic.Your point estimate is your sample statistic – you estimate that 61% of all US residents support the death penalty. To find the interval estimate, you construct a 95% confidence interval that tells you where the population parameter is expected to lie most of the time. With random sampling, there is a 0.95 probability that the true population parameter for support for the death penalty among US residents lies between 57% and 65%. Frequently asked questions about parameters and statisticsWhat’s the difference between a statistic and a parameter? A statistic refers to measures about the sample, while a parameter refers to measures about the population. How do you know whether a number is a parameter or a statistic? To figure out whether a given number is a parameter or a statistic, ask yourself the following:
If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters. If the answer is no to either of the questions, then the number is more likely to be a statistic. Why are samples used in research? Samples are used to make inferences about populations. Samples are easier to collect data from because they are practical, cost-effective, convenient, and manageable. When are populations used in research? Populations are used when a research question requires data from every member of the population. This is usually only feasible when the population is small and easily accessible. What’s the difference between descriptive and inferential statistics? Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population. Cite this Scribbr articleIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.
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Is the value a parameter or a statistic?
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