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More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, Z-scores, and T-scores.

Statistics is a branch of mathematics working with data collection, organization, analysis, interpretation and presentation. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.

Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.

A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is falsely rejected giving a "false positive") and Type II errors (null hypothesis fails to be rejected and an actual difference between populations is missed giving a "false negative"). Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.

Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.

Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more heavily from calculus and probability theory. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.

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Margin of error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the "true" figures; that is, the figures for the whole population.

The margin of error is usually defined as the radius of a confidence interval for a particular statistic from a survey. One example is the percent of people who prefer product A versus product B. When a single, global margin of error is reported for a survey, it refers to the maximum margin of error for all reported percentages using the full sample from the survey. If the statistic is a percentage, this maximum margin of error can be calculated as the radius of the confidence interval for a reported percentage of 50%.

The margin of error has been described as an "absolute" quantity, equal to a confidence interval radius for the statistic. For example, if the true value is 50 percentage points, and the statistic has a confidence interval radius of 5 percentage points, then we say the margin of error is 5 percentage points. As another example, if the true value is 50 people, and the statistic has a confidence interval radius of 5 people, then we might say the margin of error is 5 people.

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William Gosset

William Sealy Gosset (1876–1937) is better known by his pen name Student and gave this name to Student's t-test and Student's t-distribution. He joined the Dublin brewery of Arthur Guinness & Son in 1899, where he applied his statistical knowledge both in the brewery and on the farm to the selection of the best yielding varieties of barley. Gosset's key 1908 papers addressed the brewer's concern with small samples. To prevent further disclosure of confidential information, Guinness prohibited its employees from publishing any papers regardless of the contained information, so Gosset used the pseudonym Student for his publications to avoid their detection by his employer.

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