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T-statistic
In statistics, the ''t''-statistic is the ratio of the difference in a number’s estimated value from its assumed value to its standard error. It is used in hypothesis testing via Student's ''t''-test. The ''t''-statistic is used in a ''t''-test to determine whether to support or reject the null hypothesis. It is very similar to the z-score but with the difference that ''t''-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the ''t''-statistic is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown. It is also used along with p-value when running hypothesis tests where the p-value tells us what the odds are of the results to have happened. Definition and features Let \hat\beta be an estimator of parameter ''β'' in some statistical model. Then a ''t''-statistic for this parameter is any quantity of the form : t_ = \frac, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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P-value
In null-hypothesis significance testing, the ''p''-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small ''p''-value means that such an extreme observed outcome would be very unlikely ''under the null hypothesis''. Even though reporting ''p''-values of statistical tests is common practice in academic publications of many quantitative fields, misinterpretation and misuse of p-values is widespread and has been a major topic in mathematics and metascience. In 2016, the American Statistical Association (ASA) made a formal statement that "''p''-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone" and that "a ''p''-value, or statistical significance, does not measure the size of an effect or the importance of a result" or "evidence regarding a model or hypothesis". That ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ordinary Least Squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ... model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable. Some sources consider OLS to be linear regression. Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Errors And Residuals In Statistics
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the '' estimated'' value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances. Introduction Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case, the errors a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Z-score
In statistics, the standard score or ''z''-score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores. It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This process of converting a raw score into a standard score is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see ''Normalization'' for more). Standard scores are most commonly called ''z''-scores; the two terms may be used interchangeably, as they are in this article. Other equivalent terms in use include z-value, z-statistic, normal score, standardized variable and pull in high energy physics. Computing a z-score requires knowledge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pivotal Quantity
In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot need not be a statistic — the function and its can depend on the parameters of the model, but its must not. If it is a statistic, then it is known as an ''ancillary statistic''. More formally, let X = (X_1,X_2,\ldots,X_n) be a random sample from a distribution that depends on a parameter (or vector of parameters) \theta . Let g(X,\theta) be a random variable whose distribution is the same for all \theta . Then g is called a ''pivotal quantity'' (or simply a ''pivot''). Pivotal quantities are commonly used for Normalization (statistics), normalization to allow data from different data sets to be compared. It is relatively easy to construct pivots for location and scale parameters: for the former we form differences so that location cancels, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Student's T-distribution
In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped. However, t_\nu has Heavy-tailed distribution, heavier tails, and the amount of probability mass in the tails is controlled by the parameter \nu. For \nu = 1 the Student's distribution t_\nu becomes the standard Cauchy distribution, which has very fat-tailed distribution, "fat" tails; whereas for \nu \to \infty it becomes the standard normal distribution \mathcal(0, 1), which has very "thin" tails. The name "Student" is a pseudonym used by William Sealy Gosset in his scientific paper publications during his work at the Guinness Brewery in Dublin, Ireland. The Student's distribution plays a role in a number of widely used statistical analyses, including Student's t- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Standard Score
In statistics, the standard score or ''z''-score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores. It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the Statistical population, population standard deviation. This process of converting a raw score into a standard score is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see ''Normalization (statistics), Normalization'' for more). Standard scores are most commonly called ''z''-scores; the two terms may be used interchangeably, as they are in this article. Other equivalent terms in use include z-value, z-statistic, normal score, standardized variable and pull in high energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Student's T-test
Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known (typically, the scaling term is unknown and is therefore a nuisance parameter). When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's ''t'' distribution. The ''t''-test's most common application is to test whether the means of two populations are significantly different. In many cases, a ''Z''-test will yield very similar results to a ''t''-test because the latter converges to the former as the size of the dataset increases. History The term "''t''-statistic" is abbreviated from " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Standard Normal
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution conver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Student's T Test
Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known (typically, the scaling term is unknown and is therefore a nuisance parameter). When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's ''t'' distribution. The ''t''-test's most common application is to test whether the means of two populations are significantly different. In many cases, a ''Z''-test will yield very similar results to a ''t''-test because the latter converges to the former as the size of the dataset increases. History The term "''t''-statistic" is abbreviated from "hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prediction Interval
In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval (statistics), interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis. A simple example is given by a six-sided die with face values ranging from 1 to 6. The confidence interval for the estimated expected value of the face value will be around 3.5 and will become narrower with a larger sample size. However, the prediction interval for the next roll will approximately range from 1 to 6, even with any number of samples seen so far. Prediction intervals are used in both frequentist statistics and Bayesian statistics: a prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter: prediction intervals predict the distribution of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Unit Root
In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary but does not always have a trend. If the other roots of the characteristic equation lie inside the unit circle—that is, have a modulus (absolute value) less than one—then the first difference of the process will be stationary; otherwise, the process will need to be differenced multiple times to become stationary. If there are ''d'' unit roots, the process will have to be differenced ''d'' times in order to make it stationary. Due to this characteristic, unit root processes are also called difference stationary. Unit root processes may sometimes be confused with trend-stationary processes; while they share many properties, they are different in m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |