In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.
Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s.
An average of a collection or group is a value that is most central or most common in some sense, and represents its overall position.
In mathematics, especially in colloquial usage, it most commonly refers to the arithmetic mean, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5. In situations where the data is skewed or has outliers, and it is desired to focus on the main part of the group rather than the long tail, "average" often instead refers to the median; for example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people. In certain real-world scenarios, such computing the average speed from multiple measurements taken over the same distance, the average used is the harmonic mean. In situations where a histogram or probability density function is being referenced, the "average" could instead refer to the mode. Other statistics that can be used as an average include the mid-range and geometric mean, but they would rarely, if ever, be colloquially referred to as "the average".
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics. Each attempts to summarize or typify a given group of data, illustrating the magnitude and sign of the data set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose.
The arithmetic mean, also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar, . If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the sample mean () to distinguish it from the group mean (or expected value) of the underlying distribution, denoted or .
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered. On the other hand, when the variance is small, the data in the set is clustered.
Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and analysing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics) by its aim to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of probability theory, and are frequently nonparametric statistics. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups (e.g., for each treatment or exposure group), and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, the proportion of subjects with related co-morbidities, etc.
Some measures that are commonly used to describe a data set are measures of central tendency and measures of variability or dispersion. Measures of central tendency include the mean, median and mode, while measures of variability include the standard deviation (or variance), the minimum and maximum values of the variables, kurtosis and skewness.
All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias) are frequently used. When a biased estimator is used, bounds of the bias are calculated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population; because an estimator is difficult to compute (as in unbiased estimation of standard deviation); because a biased estimator may be unbiased with respect to different measures of central tendency; because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators (notably in shrinkage estimators); or because in some cases being unbiased is too strong a condition, and the only unbiased estimators are not useful.
Raw data, also known as primary data, are data (e.g., numbers, instrument readings, figures, etc.) collected from a source. In the context of examinations, the raw data might be described as a raw score (after test scores).
If a scientist sets up a computerized thermometer which records the temperature of a chemical mixture in a test tube every minute, the list of temperature readings for every minute, as printed out on a spreadsheet or viewed on a computer screen are "raw data". Raw data have not been subjected to processing, "cleaning" by researchers to remove outliers, obvious instrument reading errors or data entry errors, or any analysis (e.g., determining central tendency aspects such as the average or median result). As well, raw data have not been subject to any other manipulation by a software program or a human researcher, analyst or technician. They are also referred to as primary data. Raw data is a relative term (see data), because even once raw data have been "cleaned" and processed by one team of researchers, another team may consider these processed data to be "raw data" for another stage of research. Raw data can be inputted to a computer program or used in manual procedures such as analyzing statistics from a survey. The term "raw data" can refer to the binary data on electronic storage devices, such as hard disk drives (also referred to as "low-level data").
In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). The geometric mean of numbers is the nth root of their product, i.e., for a collection of numbers a1, a2, ..., an, the geometric mean is defined as
When the collection of numbers and their geometric mean are plotted in logarithmic scale, the geometric mean is transformed into an arithmetic mean, so the geometric mean can equivalently be calculated by taking the natural logarithm of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale using the exponential function ,
The average absolute deviation (AAD) of a data set is the average of the absolute deviations from a central point. It is a summary statistic of statistical dispersion or variability. In the general form, the central point can be a mean, median, mode, or the result of any other measure of central tendency or any reference value related to the given data set. AAD includes the mean absolute deviation and the median absolute deviation (both abbreviated as MAD).
In statistics, the mode is the value that appears most often in a set of data values. If X is a discrete random variable, the mode is the value x at which the probability mass function P(X) takes its maximum value, i.e., x = argmaxxi P(X = xi). In other words, it is the value that is most likely to be sampled.
Like the statistical mean and median, the mode is a summary statistic about the central tendency of a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, but it may be very different in highly skewed distributions.
