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68–95–99.7 Rule
In statistics, the 68–95–99.7 rule, also known as the empirical rule, and sometimes abbreviated 3sr or 3, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: approximately 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: \begin \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\ \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\ \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\% \end The usefulness of this heuristic especially depends on the question under consideration. In the empirical sciences, the so-called three-sigma rule of thumb (or 3 rule) expresses a convention ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Rule Histogram
Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how the terms ''evidence'' and ''empirical'' are to be defined. Often different fields work with quite different conceptions. In epistemology, evidence is what justifies beliefs or what determines whether holding a certain belief is rational. This is only possible if the evidence is possessed by the person, which has prompted various epistemologists to conceive evidence as private mental states like experiences or other beliefs. In philosophy of science, on the other hand, evidence is understood as that which '' confirms'' or ''disconfirms'' scientific hypotheses and arbitrates between competing theories. For this role, evidence must be public and uncontroversial, like observable physical objects or events and unlike private mental states, so th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discovery (observation)
Discovery is the act of detecting something new, or something previously unrecognized as meaningful, "portal". In sciences and academic disciplines, discovery is the observation of new phenomena, new actions, or new events and involves providing new reasoning to explain the knowledge gathered through such observations, using knowledge previously acquired through abstract thought and from everyday experiences. Some discoveries represent a radical breakthrough in knowledge or technology. Others are based on earlier discoveries, collaborations or ideas. In such cases, the process of discovery requires at least the awareness that an existing concept or method could be modified or transformed. New discoveries are made using various senses, and are usually added to pre-existing knowledge. Questioning plays a key role in discovery; discoveries are often made due to questions. Some discoveries lead to the invention of objects, processes, or techniques. Science Within scientific dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Deviation (statistics)
In mathematics and statistics, deviation serves as a measure to quantify the disparity between an observed value of a variable and another designated value, frequently the mean of that variable. Deviations with respect to the sample mean and the population mean (or "true value") are called errors and residuals, ''errors'' and ''residuals'', respectively. The Sign (mathematics), sign of the deviation reports the direction of that difference: the deviation is positive when the observed value exceeds the reference value. The absolute value of the deviation indicates the size or magnitude of the difference. In a given sample (statistics), sample, there are as many deviations as sample points. Summary statistics can be derived from a set of deviations, such as the ''standard deviation'' and the ''mean absolute deviation'', measures of statistical dispersion, dispersion, and the ''mean signed deviation'', a measure of bias of an estimator, bias. The deviation of each data point is calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normality Test
In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution 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 ... and to compute how likely it is for a random variable underlying the data set to be normally distributed. More precisely, the tests are a form of model selection, and can be interpreted several ways, depending on one's interpretations of probability: * In descriptive statistics terms, one measures a goodness of fit of a normal model to the data – if the fit is poor then the data are not well modeled in that respect by a normal distribution, without making a judgment on any underlying variable. * In frequentist statistics statistical hypothesis testing, data are tested against the null hypothesis that it is normally distribute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outliers
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the data set. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set, measurement error, or that the population has a heavy-tailed distribution. In the case of measurement error, one wishes to discard them or use statistics that are robust to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high skewness and that one should be very cautious in using tools or intuitions that assume a normal distribution. A frequent cause of outliers is a mixture of two distributions, which may be two d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Normal Distribution
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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Integration By Substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards." This involves differential forms. Substitution for a single variable Introduction (indefinite integrals) Before stating the result rigorously, consider a simple case using indefinite integrals. Compute \int(2x^3+1)^7(x^2)\,dx. Set u=2x^3+1. This means \frac=6x^2, or as a differential form, du=6x^2\,dx. Now: \begin \int(2x^3 +1)^7(x^2)\,dx &= \frac\int\underbrace_\underbrace_ \\ &= \frac\int u^\,du \\ &= \frac\left(\fracu^\right)+C \\ &= \frac(2x^3+1)^+C, \end where C is an arbitrary constant of integration. This procedure is frequently used, but not all integrals are of a form that permits its use. In any event, the result should be verified by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
