query: what percent of data with a normal distribution is within one standard deviation of the mean

pos: 68–95–99.7 rule 68–95–99.7 rule In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively; more accurately, 68.27%, 95.45% and 99.73% 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 Χ is an observation from a normally distributed random variable, μ is the mean of the distribution, and σ is its standard

neg: Standard deviation of the mean is defined by: where formula_59 is the error function. The proportion that is less than or equal to a number, "x", is given by the cumulative distribution function: If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, "μ" ± "σ", where "μ" is the arithmetic mean), about 95 percent are within two standard deviations ("μ" ± 2"σ"), and about 99.7 percent lie within three standard deviations ("μ" ± 3"σ"). This is known as the "68-95-99.7 rule", or "the empirical rule". For various

neg: Standard normal table the standard normal table to find probabilities. Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. The "standard" normal distribution, represented by the letter Z, is the normal distribution having a mean of 0 and a standard deviation of 1. If "X" is a random variable from a normal distribution with mean μ and standard deviation σ, its Z-score may be calculated from X by subtracting μ and dividing by the standard deviation: For the average of a sample of size n from some population in which the mean is μ and the standard deviation is

neg: Reference range this example is estimated to be 4.4 to 6.4. An alternative method of establishing a reference range with the assumption of log-normal distribution is to use the arithmetic mean and arithmetic value of standard deviation. This is somewhat more tedious to perform, but may be useful for example in cases where a study that establishes a reference range presents only the arithmetic mean and standard deviation, leaving out the source data. If the original assumption of arithmetically normal distribution is shown to be less appropriate than the log-normal one, then, using the arithmetic mean and standard deviation may be the

neg: Calibration of radiocarbon dates In the example CALIB output shown at left, the input data is 1270 BP, with a standard deviation of 10 radiocarbon years. The curve selected is the northern hemisphere INTCAL13 curve, part of which is shown in the output; the vertical width of the curve corresponds to the width of the standard error in the calibration curve at that point. A normal distribution is shown at left; this is the input data, in radiocarbon years. The central darker part of the normal curve is the range within one standard deviation of the mean; the lighter grey area shows the range

neg: Contour boxplot than one) times the cBD of the 50% ranked samples. The following example is an ensemble of data from 2D incompressible Navier–Stokes simulation consisting of 40 members, where each ensemble member is a simulation with Reynolds number and inlet velocity chosen randomly. The inlet velocity values are randomly drawn from a normal distribution with mean value of 1 and standard deviation of ±0.01 (in non-dimensionalized units); likewise, Reynolds numbers are generated from a normal distribution with mean value of 130 and standard deviation of ±3. The example below is from an ensemble of publicly available data from the National Oceanic

neg: Average absolute deviation than or equal to the mean absolute deviation from any other fixed number. The mean absolute deviation from the mean is less than or equal to the standard deviation; one way of proving this relies on Jensen's inequality. For the normal distribution, the ratio of mean absolute deviation to standard deviation is formula_3. Thus if "X" is a normally distributed random variable with expected value 0 then, see Geary (1935): In other words, for a normal distribution, mean absolute deviation is about 0.8 times the standard deviation. However, in-sample measurements deliver values of the ratio of mean average deviation /

neg: Probability box easy to obtain by enveloping extreme distributions given the possible parameters. For instance, if a quantity is known to be normal with mean somewhere in the interval [7,8] and standard deviation within the interval [1,2], the left and right edges of the p-box can be found by enveloping the distribution functions of four probability distributions, namely, normal(7,1), normal(8,1), normal(7,2), and normal(8,2), where normal(μ,σ) represents a normal distribution with mean μ and standard deviation σ. All probability distributions that are normal and have means and standard deviations inside these respective intervals will have distribution functions that fall entirely within this p-box.

neg: Log-normal distribution mean, and formula_39 the geometric standard deviation of the random variable formula_13, and let formula_41 and formula_42 be the arithmetic mean, or expected value, and the arithmetic standard deviation as usual. The geometric mean of the log-normal distribution is formula_43, and the geometric standard deviation is formula_44. By analogy with the arithmetic statistics, one can define a geometric variance, formula_45, and a geometric coefficient of variation, formula_46. Because the log-transformed variable formula_47 is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median, formula_48. Note that the geometric mean is

neg: Tikhonov regularization normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation formula_78. The data are also subject to errors, and the errors in formula_28 are also assumed to be independent with zero mean and standard deviation formula_80. Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the "a priori" distribution of formula_22, according to Bayes' theorem. If the assumption of normality is replaced by assumptions of homoscedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the

neg: Standard deviation average height for adult men in the United States is about 70 inches (177.8 cm), with a standard deviation of around 3 inches (7.62 cm). This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches (7.62 cm) of the mean (67–73 inches (170.18–185.42 cm))one standard deviationand almost all men (about 95%) have a height within 6 inches (15.24 cm) of the mean (64–76 inches (162.56–193.04 cm))two standard deviations. If the standard deviation were zero, then all men would be exactly 70 inches (177.8 cm) tall. If the standard deviation were 20 inches (50.8

neg: Standard deviation The reported margin of error of a poll is computed from the standard error of the mean (or alternatively from the product of the standard deviation of the population and the inverse of the square root of the sample size, which is the same thing) and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval. In science, many researchers report the standard deviation of experimental data, and only effects that fall much farther than two standard deviations away from what would have been expected are considered statistically significant—normal random error or variation in the measurements

neg: Standard deviation The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of where "μ" is the expected value of the random variables, "σ" equals their distribution's standard deviation divided by "n", and "n" is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the normalizing constant. If a data distribution is approximately normal, then the proportion of data values within "z" standard deviations

neg: Coefficient of variation data sets with different units or widely different means, one should use the coefficient of variation instead of the standard deviation. The coefficient of variation is also common in applied probability fields such as renewal theory, queueing theory, and reliability theory. In these fields, the exponential distribution is often more important than the normal distribution. The standard deviation of an exponential distribution is equal to its mean, so its coefficient of variation is equal to 1. Distributions with CV < 1 (such as an Erlang distribution) are considered low-variance, while those with CV > 1 (such as a hyper-exponential distribution)

neg: Z-factor the definition of the Z-factor is motivated by the normal distribution, for which more than 99% of values occur within 3 standard deviations of the mean. If the data follow a strongly non-normal distribution, the reference points (e.g. the meaning of a negative value) may be misleading. Another issue is that the usual estimates of the mean and standard deviation are not robust; accordingly many users in the high-throughput screening community prefer "Robust Z-prime". Extreme values (outliers) in either the positive or negative controls can adversely affect the Z-factor, potentially leading to an apparently unfavorable Z-factor even when the assay

neg: Student's t-distribution population means, and in linear regression analysis. The Student's "t"-distribution also arises in the Bayesian analysis of data from a normal family. If we take a sample of "n" observations from a normal distribution, then the "t"-distribution with formula_14 degrees of freedom can be defined as the distribution of the location of the sample mean relative to the true mean, divided by the sample standard deviation, after multiplying by the standardizing term formula_15. In this way, the "t"-distribution can be used to construct a confidence interval for the true mean. The "t"-distribution is symmetric and bell-shaped, like the normal distribution,

neg: Prediction interval normal distribution with known mean 0 but unknown variance formula_25, the sample variance formula_26 of the observations formula_14 has, up to scale, a formula_28 distribution; more precisely: while the future observation formula_16 has distribution formula_31 Taking the ratio of the future observation and the sample standard deviation cancels the "σ," yielding a Student's t-distribution with "n" – 1 degrees of freedom: Solving for formula_16 gives the prediction distribution formula_34 from which one can compute intervals as before. Notice that this prediction distribution is more conservative than using a normal distribution with the estimated standard deviation formula_35 and known mean 0,

neg: Student's t-distribution in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the "t"-distribution is often used to account for the extra uncertainty that results from this estimation. In most such problems, if the standard deviation of the errors were known, a normal distribution would be used instead of the "t"-distribution. Confidence intervals and hypothesis

neg: Multimodal distribution commonly used example of how summary statistics such as the mean, median, and standard deviation can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value. The standard deviation is also larger than deviation of each normal distribution. Although several have been suggested, there is no presently generally agreed summary statistic (or set of statistics) to quantify the parameters of a general bimodal distribution. For a mixture of two normal distributions the means and standard deviations along with

neg: Cell Population Data Volume Mean and Standard deviation 2) Conductivity Mean and Standard deviation 3) Scatter Mean and Standard deviation. It has been said that when the Leukocytes are normal, these data are normal, but in the event of diseases, the number of Cells increases or decreases and frequently so with different properties, as size, internal Cell content etc. In addition, where Blood cells are not seen under normal circumstances, as myelocytes (Immature Neutrophil), results in these data frequently become abnormal. When Looking for changes in the value of these data, it is possible to flag (message in the results) the presence of

neg: Normal distribution deviate, then formula_37 will have a normal distribution with expected value formula_3 and standard deviation formula_16. Conversely, if formula_40 is a normal deviate with parameters formula_3 and formula_6, then formula_43 will have a standard normal distribution. This variate is called the standardized form of formula_40 Every normal distribution is the exponential of a quadratic function: where formula_46 and formula_47. In this form, the mean value is formula_48, and the variance is formula_49. For the standard normal distribution, formula_50, formula_51, and formula_52. The probability density of the standard Gaussian distribution (standard normal distribution) (with zero mean and unit variance) is often

neg: Maxwell–Boltzmann distribution velocity ["v", "v", "v"] is the product of the distributions for each of the three directions: where the distribution for a single direction is Each component of the velocity vector has a normal distribution with mean formula_89 and standard deviation formula_90, so the vector has a 3-dimensional normal distribution, a particular kind of multivariate normal distribution, with mean formula_91 and standard deviation formula_92. The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above. Note that the speed is and the volume element in spherical coordinates where formula_95 and formula_96 are the spherical coordinate angles

neg: Normal distribution denotes the double factorial. An asymptotic expansion of the CDF for large "x" can also be derived using integration by parts; see Error function#Asymptotic expansion. About 68% of values drawn from a normal distribution are within one standard deviation "σ" away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68-95-99.7 (empirical) rule, or the "3-sigma rule". More precisely, the probability that a normal deviate lies in the range between formula_171 and formula_172 is given by To 12 significant figures, the values

neg: Normal distribution that this distribution was seen as typical, common – and thus "normal". Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what "would", in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term "normal" as a designation for this distribution. Also, it was Pearson who first wrote the distribution in terms of the standard deviation "σ" as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the

neg: Parametric statistics means that if the mean and standard deviation are known and if the distribution is normal, the probability of any future observation lying in a given range is known. Suppose we have a sample of 99 test scores with a mean of 100 and a standard deviation of 1. If we assume all 99 test scores are random observations from a normal distribution, then we predict there is a 1% chance that the 100th test score will be higher than 102.365 (that is, the mean plus 2.365 standard deviations) assuming that the 100th test score comes from the same distribution

neg: Market profile to 'Improve Performance'. It is described as "the only variable-cost ticker service in the commodities industry." The promotional material says the Profile is to be the link between the CBOT data and the market. A Profile graphic is to be used to tell "what the market is doing"; the LDB data is for finding the market's 'condition'. As a part of the data-display connection in CBOTMP1, the price of the peak cleared volume is identified as the Point of Control (POC). Following the normal distribution analogy, the central seventy percent of trading activity about POC (+/- one standard deviation) is

neg: Normal distribution maximum value formula_21 and has inflection points at formula_26 and formula_27. Authors may differ also on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance formula_28, that is Stigler goes even further, defining the standard normal with variance formula_30 : Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor formula_16 (the standard deviation) and then translated by formula_3 (the mean value): The probability density must be scaled by formula_35 so that the integral is still 1. If formula_36 is a standard normal

neg: Probit model is positive: The use of the standard normal distribution causes no loss of generality compared with using an arbitrary mean and standard deviation because adding a fixed amount to the mean can be compensated by subtracting the same amount from the intercept, and multiplying the standard deviation by a fixed amount can be compensated by multiplying the weights by the same amount. To see that the two models are equivalent, note that Suppose data set formula_5 contains "n" independent statistical units corresponding to the model above. Then their joint log-likelihood function is The estimator formula_7 which maximizes this function will

neg: Standard score normalization for more). Standard scores are also called z-values, "z"-scores, normal scores, and standardized variables. They are most frequently used to compare an observation to a standard normal deviate, though they can be defined without assumptions of normality. Computing a z-score requires knowing the mean and standard deviation of the complete population to which a data point belongs; if one only has a sample of observations from the population, then the analogous computation with sample mean and sample standard deviation yields the "t"-statistic. If the population mean and population standard deviation are known, the standard score of a raw score

neg: Mode (statistics) the log-normal distribution, we find: Indeed, the median is about one third on the way from mean to mode. When "X" has a larger standard deviation, σ = 1, the distribution of "Y" is strongly skewed. Now Here, Pearson's rule of thumb fails. Van Zwet derived an inequality which provides sufficient conditions for this inequality to hold. The inequality holds if for all "x" where F() is the cumulative distribution function of the distribution. It can be shown for a unimodal distribution that the median formula_4 and the mean formula_5 lie within (3/5) ≈ 0.7746 standard deviations of each other.

neg: Outlier variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end (King effect). There is no rigid mathematical definition of what constitutes an outlier; determining whether or not an observation is an outlier is ultimately a subjective exercise. There are various methods of outlier detection. Some are graphical such as normal probability plots. Others are model-based. Box plots are a hybrid. Model-based methods which are commonly used for identification assume that the data are from a normal distribution, and identify observations which are deemed "unlikely" based on mean and standard deviation: It is proposed

neg: Normal distribution deviation formula_16 is where formula_129 is the imaginary unit. If the mean formula_18, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain, with mean 0 and standard deviation formula_35. In particular, the standard normal distribution formula_54 is an eigenfunction of the Fourier transform. In probability theory, the Fourier transform of the probability distribution of a real-valued random variable formula_40 is closely connected to the characteristic function formula_134 of that variable, which is defined as the expected value of formula_135, as a function of the real variable formula_136

neg: Normal score Normal score The term normal score is used with two different meanings in statistics. One of them relates to creating a single value which can be treated as if it had arisen from a standard (zero mean, unit variance) normal distribution. The second relates to assigning alternative values to data points within a dataset, with the broad intention of creating data values than can be interpreted as being approximations for values that might have been observed had the data arisen from a standard normal distribution. The first meaning is as an alternative name for the standard score or z score,

neg: Robust measures of scale this case is 1, whereas the population variance does not exist. These robust estimators typically have inferior statistical efficiency compared to conventional estimators for data drawn from a distribution without outliers (such as a normal distribution), but have superior efficiency for data drawn from a mixture distribution or from a heavy-tailed distribution, for which non-robust measures such as the standard deviation should not be used. For example, for data drawn from the normal distribution, the MAD is 37% as efficient as the sample standard deviation, while the Rousseeuw–Croux estimator "Q" is 88% as efficient as the sample standard deviation. Rousseeuw

neg: Normal distribution standard deviation "σ", then the variable , for any real numbers "a" and "b", is also normally distributed, with mean "aμ" + "b" and standard deviation |"a"|"σ". Also if "X" and "X" are two independent normal random variables, with means "μ", "μ" and standard deviations "σ", "σ", then their sum will also be normally distributed, with mean "μ" + "μ" and variance formula_254. In particular, if "X" and "Y" are independent normal deviates with zero mean and variance "σ", then and are also independent and normally distributed, with zero mean and variance 2"σ". This is a special case of the

neg: Chi-squared distribution of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman's analysis of variance by ranks. If "Z", ..., "Z" are independent, standard normal random variables, then the sum of their squares, is distributed according to the chi-squared distribution with "k" degrees of freedom. This is usually denoted as The chi-squared distribution has one parameter: "k", a positive integer that specifies

neg: Stock trader into an index fund. In 1963 Benoit Mandelbrot analyzed the variations of cotton prices on a time series starting in 1900. There were two important findings. First, price movements had very little to do with a normal distribution in which the bulk of the observations lies close to the mean (68% of the data are within one standard deviation). Instead, the data showed a great frequency of extreme variations. Second, price variations followed patterns that were indifferent to scale: the curve described by price changes for a single day was similar to a month’s curve. Surprisingly, these patterns of self-similarity

neg: Median the median is useful in statistical data-analysis, for example, in "k"-medians clustering. It can be shown for a unimodal distribution that the median formula_10 and the mean formula_11 lie within (3/5) ≈ 0.7746 standard deviations of each other. In symbols, where |·| is the absolute value. A similar relation holds between the median and the mode: they lie within 3 ≈ 1.732 standard deviations of each other: If the distribution has finite variance, then the distance between the median and the mean is bounded by one standard deviation. This bound was proved by Mallows, who used Jensen's inequality twice, as

neg: Robust confidence intervals same techniques can be used in other spreadsheet programs such as in OpenOffice.org Calc and gnumeric. After removing obvious outliers, one could subtract the median from the other two values for each object, and examine the distribution of the 200 resulting numbers. It should be normal with mean near zero and standard deviation a little larger than σ. A simple Monte Carlo spreadsheet calculation would reveal typical values for the standard deviation (around 105 to 115% of σ). Or, one could subtract the mean of each triplet from the values, and examine the distribution of 300 values. The mean is

neg: Chauvenet's criterion Chauvenet's criterion In statistical theory, Chauvenet's criterion (named for William Chauvenet) is a means of assessing whether one piece of experimental data — an outlier — from a set of observations, is likely to be spurious. The idea behind Chauvenet's criterion is to find a probability band, centered on the mean of a normal distribution, that should reasonably contain all n samples of a data set. By doing this, any data points from the n samples that lie outside this probability band can be considered to be outliers, removed from the data set, and a new mean and standard deviation

neg: Coefficient of variation the sample standard deviation formula_9 to the sample mean formula_10: But this estimator, when applied to a small or moderately sized sample, tends to be too low: it is a biased estimator. For normally distributed data, an unbiased estimator for a sample of size n is: In many applications, it can be assumed that data are log-normally distributed (evidenced by the presence of skewness in the sampled data). In such cases, a more accurate estimate, derived from the properties of the log-normal distribution, is defined as: where formula_14 is the sample standard deviation of the data after a natural log

neg: Standard error 1.96 is the 0.975 quantile of the normal distribution: In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the error in the process by which it was generated. In other words, it is the actual or estimated standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of SE, SEM (for standard error of "measurement" or "mean"), or S. Standard errors provide simple measures of uncertainty in a value and are often used because: In scientific and technical literature,

neg: Process capability process mean (average) and standard deviation are calculated. With a normal distribution, the "tails" can extend well beyond plus and minus three standard deviations, but this interval should contain about 99.73% of production output. Therefore, for a normal distribution of data the process capability is often described as the relationship between six standard deviations and the required specification. The output of a process is expected to meet customer requirements, specifications, or engineering tolerances. Engineers can conduct a process capability study to determine the extent to which the process can meet these expectations. The ability of a process to meet specifications

neg: Particle-size distribution by the size of the particles. ρ: Actual particle density (g/cm) ρ: Gas or sample matrix density (g/cm) r: Least-squares coefficient of determination. The closer this value is to 1.0, the better the data fit to a hyperplane representing the relationship between the response variable and a set of covariate variables. A value equal to 1.0 indicates all data fit perfectly within the hyperplane. λ: Gas mean free path (cm) D: Mass-median-diameter (MMD). The log-normal distribution mass median diameter. The MMD is considered to be the average particle diameter by mass. σ: Geometric standard deviation. This value is determined mathematically

neg: Population proportion sample size, and formula_16 is the upper formula_17 critical value of the standard normal distribution for a level of confidence, formula_18. In order to derive the formula for the one-sample proportion in the Z-interval, a sampling distribution of sample proportions needs to be taken into consideration. The mean of the sampling distribution of sample proportions is usually denoted as formula_19 and its standard deviation is denoted as formula_20. Since the value of formula_1 is unknown, an unbiased statistic formula_3 will be used for formula_1. The mean and standard deviation are rewritten as formula_24 and formula_25 respectively. Invoking the Central Limit

neg: Outlier is possible to test if the number of outliers deviate significantly from what can be expected: for a given cutoff (so samples fall beyond the cutoff with probability "p") of a given distribution, the number of outliers will follow a binomial distribution with parameter "p", which can generally be well-approximated by the Poisson distribution with λ = "pn". Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, "p" is approximately 0.3%, and thus for 1000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ

neg: Standard normal table probability that a statistic is between 0 (the mean) and Z. Note that for z = 1, 2, 3, one obtains (after multiplying by 2 to account for the [-z,z] interval) the results f(z) = 0.6827, 0.9545, 0.9974, characteristic of the 68–95–99.7 rule. This table gives a probability that a statistic is less than Z (i.e. between negative infinity and Z). The values are calculated using the cumulative distribution function of a standard normal distribution with mean of zero and standard deviation of one, usually denoted with the capital Greek letter formula_4 (phi), is the integral formula_4(z) is related to

neg: Standard deviation Standard deviation In statistics, the standard deviation (SD, also represented by the lower case Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values. The standard deviation of a random variable, statistical population, data set, or probability distribution

neg: Exponentially modified Gaussian distribution / "λ"). The shape "K" = "τ" / "σ" is also sometimes used to characterise the distribution. Depending on the values of the parameters, the distribution may vary in shape from almost normal to almost exponential. The parameters of the distribution can be estimated from the sample data with the method of moments as follows: where "m" is the sample mean, "s" is the sample standard deviation, and "γ" is the skewness. Solving these for the parameters gives: Ratcliff has suggested that there be at least 100 data points in the sample before the parameter estimates should be regarded as

neg: Experimental uncertainty analysis time measurements is increased to 0.15 s, or about ten percent. The PDF for the estimated "g" values is also graphed, as it was in Figure 2; note that the PDF for the larger-time-variation case is skewed, and now the biased mean is clearly seen. The approximated (biased) mean and the mean observed directly from the data agree well. The dashed curve is a Normal PDF with mean and variance from the approximations; it does not represent the data particularly well. Rather than the variance, often a more useful measure is the standard deviation "σ", and when this is divided

neg: Standard normal table the error function, or erf("z"). This table gives a probability that a statistic is greater than Z. This table gives a probability that a statistic is greater than Z, for large integer Z values. A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5. Only a "cumulative from mean" table is available. Standard normal table A standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ, which are the values of the cumulative distribution function of the normal distribution. It is used to find

neg: Normal score where values are standardised by subtracting the sample or estimated mean and dividing by the sample or other estimate of the standard deviation. Particularly in applications where the name "normal score" is used, there is usually a presumption that the value can be referred to a table of standard normal probabilities as a means of providing a significance test of some hypothesis, such as a difference in means. The second meaning of normal score is associated with data values derived from the ranks of the observations within the dataset. A given data point is assigned a value which is either

neg: Normality (behavior) a normal reaction would be within one standard deviation, or the most average 68.3%. However, this mathematical model only holds for one particular trait at a time, since, for example, the probability of a single individual being within one standard deviation for 36 independent variables would be one in a million. In statistics, normal is often arbitrarily considered anything that falls within about 1.96 standard deviations of the mean, or the most average 95% (see 1.96). The probability of an individual being within 1.96 standard deviations for 269 independent variables is approximately one in a million. For only 59 independent

neg: Normal probability plot for rankits. The formula used by the "qqnorm" function in the basic "stats" package in R (programming language) is as follows: for , where and is the standard normal quantile function. If the data are consistent with a sample from a normal distribution, the points should lie close to a straight line. As a reference, a straight line can be fit to the points. The further the points vary from this line, the greater the indication of departure from normality. If the sample has mean 0, standard deviation 1 then a line through 0 with slope 1 could be used.

neg: Reference range interval, is often estimated by assuming a normal distribution of the measured parameter, in which case it can be defined as the interval limited by 1.96 (often rounded up to 2) population standard deviations from either side of the population mean (also called the expected value). However, in the real world, neither the population mean nor the population standard deviation are known. They both need to be estimated from a sample, whose size can be designated "n". The population standard deviation is estimated by the sample standard deviation and the population mean is estimated by the sample mean (also called

neg: Chauvenet's criterion a z-score table. "D" is equal to the z-score for "P". Using this method "D" can be determined for any sample size. In Excel, "D" can be found with the following formula: =ABS(NORM.S.INV(1/(4"n"))). To apply Chauvenet's criterion, first calculate the mean and standard deviation of the observed data. Based on how much the suspect datum differs from the mean, use the normal distribution function (or a table thereof) to determine the probability that a given data point will be at the value of the suspect data point. Multiply this probability by the number of data points taken. If the result

neg: Seismic array then the above operation would reproduce the signal S(t) accurately. Real data w(t) are the sum of background noise n(t) plus the signal of interest S(t), i.e. w(t) = S(t) + n(t). Assuming that the signal is coherent and not attenuated, calculating the sum of M observations and including noise we get Assuming that the noise n(t) has a normal amplitude distribution with zero mean and variance σ at all sites, then the variance of the noise after summation is formula_11 and the standard deviation is formula_12. That means the standard deviation of the noise is multiplied by formula_13 while

neg: Frequency distribution the body of the table reports the joint frequencies. Managing and Operating on frequency tabulated data is much simpler than operation on raw data. There are simple algorithms to calculate median, mean, standard deviation etc. from these tables. Statistical hypothesis testing is founded on the assessment of differences and similarities between frequency distributions. This assessment involves measures of central tendency or averages, such as the mean and median, and measures of variability or statistical dispersion, such as the standard deviation or variance. A frequency distribution is said to be skewed when its mean and median are different, or more generally

neg: Medium-power talker talker volume distribution in volume units (vu), and σ is the variance of the distribution. Medium-power talker In telecommunication, a medium-power talker is a hypothetical talker, within a log-normal distribution of talkers, whose volume lies at the medium power of all talkers determining the volume distribution at the point of interest. When the distribution follows a log-normal curve (values expressed in decibels), the mean and standard deviation can be used to compute the medium-power talker. The talker volume distribution follows a log-normal curve and the medium-power talker is uniquely determined by the average talker volume. The medium-power talker volume, "V",

neg: Normal distribution polarization identity. Also, if "X", "X" are two independent normal deviates with mean "μ" and deviation "σ", and "a", "b" are arbitrary real numbers, then the variable is also normally distributed with mean "μ" and deviation "σ". It follows that the normal distribution is stable (with exponent "α" = 2). More generally, any linear combination of independent normal deviates is a normal deviate. For any positive integer "n", any normal distribution with mean "μ" and variance "σ" is the distribution of the sum of "n" independent normal deviates, each with mean "μ/n" and variance "σ""/n". This property is called infinite

neg: Normal distribution and the variance formula_6. The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter formula_150 (phi), is the integral The related error function formula_152 gives the probability of a random variable with normal distribution of mean 0 and variance 1/2 falling in the range formula_153; that is These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions. However, many numerical approximations are known; see below. The two functions are closely related, namely For a generic normal distribution with density formula_78, mean formula_3 and deviation formula_16,

neg: Prediction interval Student's t-statistic, which can be derived by this method and is used in the sequel. A prediction interval ["ℓ","u"] for a future observation "X" in a normal distribution "N"("µ","σ") with known mean and variance may be calculated from Hence or with "z" the quantile in the standard normal distribution for which: or equivalently; The prediction interval is conventionally written as: For example, to calculate the 95% prediction interval for a normal distribution with a mean ("µ") of 5 and a standard deviation ("σ") of 1, then "z" is approximately 2. Therefore, the lower limit of the prediction interval is approximately

neg: Statistical interference shaft must be a little smaller than the hole. (Traditional tolerances may suggest that all dimensions fall within those intended tolerances. A process capability study of actual production, however, may reveal normal distributions with long tails.) Both the shaft and hole sizes will usually form normal distributions with some average (arithmetic mean) and standard deviation. With two such normal distributions, a distribution of interference can be calculated. The derived distribution will also be normal, and its average will be equal to the difference between the means of the two base distributions. The variance of the derived distribution will be the

neg: Medium-power talker Medium-power talker In telecommunication, a medium-power talker is a hypothetical talker, within a log-normal distribution of talkers, whose volume lies at the medium power of all talkers determining the volume distribution at the point of interest. When the distribution follows a log-normal curve (values expressed in decibels), the mean and standard deviation can be used to compute the medium-power talker. The talker volume distribution follows a log-normal curve and the medium-power talker is uniquely determined by the average talker volume. The medium-power talker volume, "V", is given by "V" = "V" + 0.115σ, where "V" is the average of the

neg: Chebyshev's inequality derived from the normal distribution (approximately ). A distribution function "F" is unimodal at "ν" if its cumulative distribution function is convex on (−∞, "ν") and concave on ("ν",∞) An empirical distribution can be tested for unimodality with the dip test. In 1823 Gauss showed that for a unimodal distribution with a mode of zero If the mode is not zero and the mean ("μ") and standard deviation ("σ") are both finite, then denoting the median as "ν" and the root mean square deviation from the mode by "ω", we have and Winkler in 1866 extended Gauss' inequality to "r"

neg: Wrapped normal distribution resultant vector: The mean angle is and the length of the mean resultant is The circular standard deviation, which is a useful measure of dispersion for the wrapped normal distribution and its close relative, the von Mises distribution is given by: A series of "N" measurements "z" = "e" drawn from a wrapped normal distribution may be used to estimate certain parameters of the distribution. The average of the series is defined as and its expectation value will be just the first moment: In other words, is an unbiased estimator of the first moment. If we assume that the mean

neg: Box–Muller transform generates values from any normal distribution with mean formula_24 and variance formula_25. If formula_26 is a standard normal deviate, then formula_27 will have a normal distribution with mean formula_24 and standard deviation formula_29. Note that because the random number generator codice_1 has not been seeded, the same series of values will always be returned from the codice_2 function. Box–Muller transform The Box–Muller transform, by George Edward Pelham Box and Mervin Edgar Muller, is a pseudo-random number sampling method for generating pairs of independent, standard, normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers.

neg: Niche apportionment models More recently statistical tests of the fit of niche apportionment models to empirical data have been developed. The later method (Etienne and Ollf 2005) uses a Bayesian simulation of the models to test their fit to empirical data. The former method, which is still commonly used, simulates the expected relative abundances, from a normal distribution, of each model given the same number of species as the empirical data. Each model is simulated multiple times, and mean and standard deviation can be calculated to assign confidence intervals around each relative abundance distribution. The confidence around each rank can be tested against

neg: Linear belief function on the sales plus a residual ("R") that represents insignificant sales returns and cash discounts. Thus, we can represent the logical relation as a linear equation: Furthermore, if the auditor believes "E" and "B" are 100 thousand dollars on the average with a standard deviation 5 and the covariance 15, we can represent the belief as a multivariate normal distribution. If historical data indicate that the residual R is zero on the average with a standard deviation of 0.5 thousand dollars, we can summarize the historical data by normal distribution "R" ~ N(0, 0.5). If there is a direct observation

neg: Normal distribution analytically, the others being the Cauchy distribution and the Lévy distribution. The normal distribution with density formula_70 (mean formula_3 and standard deviation formula_72) has the following properties: Furthermore, the density formula_54 of the standard normal distribution (i.e. formula_18 and formula_83) also has the following properties: The plain and absolute moments of a variable formula_40 are the expected values of formula_93 and formula_94, respectively. If the expected value formula_3 of formula_40 is zero, these parameters are called "central moments". Usually we are interested only in moments with integer order formula_97. If formula_40 has a normal distribution, these moments exist and are

neg: Outlier In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number – see Poisson distribution – and not indicate an anomaly. If the sample size is only 100, however, just three such outliers are already reason for concern, being more than 11 times the expected number. In general, if the nature of the population distribution is known "a priori", it

neg: Sex differences in human physiology result for the X-chromosome follows, either a double or a single X. Therefore, direct sex differences are usually binary in expression (although the deviations in complex biological processes produce a menagerie of exceptions). These include, most conspicuously, male (vs female) gonads. Indirect sex differences are general differences as quantified by empirical data and statistical analysis. Most differing characteristics will conform to a bell-curve (i.e. normal) distribution which can be broadly described by the mean (peak distribution) and standard deviation (indicator of size of range). Often only the mean or mean difference between sexes is given. This may or may not

neg: Scale parameter statistic. Note that the scale factor depends on the distribution in question. For instance, in order to use the median absolute deviation (MAD) to estimate the standard deviation of the normal distribution, one must multiply it by the factor where Φ is the quantile function (inverse of the cumulative distribution function) for the standard normal distribution. (See MAD for details.) That is, the MAD is not a consistent estimator for the standard deviation of a normal distribution, but 1.4826... MAD is a consistent estimator. Similarly, the average absolute deviation needs to be multiplied by approximately 1.2533 to be a consistent

neg: Robust statistics in the speed-of-light data have more than just an adverse effect on the mean; the usual estimate of scale is the standard deviation, and this quantity is even more badly affected by outliers because the squares of the deviations from the mean go into the calculation, so the outliers' effects are exacerbated. The plots below show the bootstrap distributions of the standard deviation, median absolute deviation (MAD) and Qn estimator of scale. The plots are based on 10,000 bootstrap samples for each estimator, with some Gaussian noise added to the resampled data (smoothed bootstrap). Panel (a) shows the distribution of

neg: Rice distribution and σ, from a sample of data. This can be done using the method of moments, e.g., the sample mean and the sample standard deviation. The sample mean is an estimate of μ and the sample standard deviation is an estimate of μ. The following is an efficient method, known as the "Koay inversion technique". for solving the estimating equations, based on the sample mean and the sample standard deviation, simultaneously . This inversion technique is also known as the fixed point formula of SNR. Earlier works on the method of moments usually use a root-finding method to solve the

neg: Pearson distribution the distribution exists and is equal to σ. Now the parameter "m" only controls the kurtosis of the distribution. If "m" approaches infinity as "λ" and "σ" are held constant, the normal distribution arises as a special case: This is the density of a normal distribution with mean "λ" and standard deviation "σ". It is convenient to require that "m" > 5/2 and to let This is another specialization, and it guarantees that the first four moments of the distribution exist. More specifically, the Pearson type VII distribution parameterized in terms of (λ, σ, γ) has a mean of "λ",

neg: Mann–Whitney U test mean and standard deviation of "U", is approximately a standard normal deviate whose significance can be checked in tables of the normal distribution. "m" and "σ" are given by The formula for the standard deviation is more complicated in the presence of tied ranks. If there are ties in ranks, "σ" should be corrected as follows: Mann–Whitney U test In statistics, the Mann–Whitney "U" test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric test of the null hypothesis that it is equally likely that a randomly selected value from one sample will be less

neg: Beta distribution only −3.5% for "α" = "β" = 1, and it decreases to zero as "α" → ∞, "β" → ∞): At the limit α → ∞, β → ∞, the ratio of the mean absolute deviation to the standard deviation (for the beta distribution) becomes equal to the ratio of the same measures for the normal distribution: formula_76. For α = β = 1 this ratio equals formula_77, so that from α = β = 1 to α, β → ∞ the ratio decreases by 8.5%. For α = β = 0 the standard deviation is exactly equal to the

neg: Standard deviation amount of bias decreases. We obtain more information and the difference between formula_25 and formula_26 becomes smaller. For unbiased estimation of standard deviation, there is no formula that works across all distributions, unlike for mean and variance. Instead, "s" is used as a basis, and is scaled by a correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by "s"/"c", where the correction factor (which depends on "N") is given in terms of the Gamma function, and equals: This arises because the sampling distribution of the sample standard deviation follows a (scaled) chi

neg: Standard error experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the

neg: Sampling distribution sample sizes). The mean of a sample from a population having a normal distribution is an example of a simple statistic taken from one of the simplest statistical populations. For other statistics and other populations the formulas are more complicated, and often they don't exist in closed-form. In such cases the sampling distributions may be approximated through Monte-Carlo simulations, bootstrap methods, or asymptotic distribution theory. The standard deviation of the sampling distribution of a statistic is referred to as the standard error of that quantity. For the case where the statistic is the sample mean, and samples are uncorrelated, the

neg: Process capability parts of process capability are: 1) measure the variability of the output of a process, and 2) compare that variability with a proposed specification or product tolerance. The input of a process usually has at least one or more measurable characteristics that are used to specify outputs. These can be analyzed statistically; where the output data shows a normal distribution the process can be described by the process mean (average) and the standard deviation. A process needs to be established with appropriate process controls in place. A control chart analysis is used to determine whether the process is "in statistical

neg: Paired difference test of age will lead to positive correlations between the cholesterol levels measured within subjects, as long as the duration of the study is small relative to the variation in ages in the sample. Suppose we are using a Z-test to analyze the data, where the variances of the pre-treatment and post-treatment data and are known (the situation with a t-test is similar). The unpaired Z-test statistic is The power of the unpaired, one-sided test carried out at level can be calculated as follows: where "S" is the standard deviation of D, Φ is the standard normal cumulative distribution function, and

neg: Kernel density estimation basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for "h" (that is, the bandwidth that minimises the mean integrated squared error) is where formula_9 is the standard deviation of the samples. This approximation is termed the "normal distribution approximation", Gaussian approximation, or "Silverman's (1986) rule of thumb". While this rule of thumb is easy to compute, it should be used with caution as it can yield widely inaccurate estimates when the density is not close to being normal. For example, consider estimating the bimodal Gaussian mixture: from a sample

neg: Normality test the ratio of the mean absolute deviation to the standard deviation and of the range to the standard deviation. More recent tests of normality include the energy test (Székely and Rizzo) and the tests based on the empirical characteristic function (ECF) (e.g. Epps and Pulley, Henze–Zirkler, BHEP test). The energy and the ECF tests are powerful tests that apply for testing univariate or multivariate normality and are statistically consistent against general alternatives. The normal distribution has the highest entropy of any distribution for a given standard deviation. There are a number of normality tests based on this property, the first

neg: Log-normal distribution Cobb–Douglas. A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain. The log-normal distribution is the maximum entropy probability distribution for a random variate for which the mean and variance of are specified. Given a log-normally distributed random variable formula_13 and two parameters formula_14 and formula_15 that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of formula_13 is normally distributed, and we can write formula_13 as with formula_19 a

neg: Box–Muller transform can be to 0. If the generator uses 32 bits per output value, the smallest non-zero number that can be generated is formula_19. When formula_20 and formula_21 are equal to this the Box–Muller transform produces a normal random variable equal to formula_22 This means that the algorithm will not produce random variables more than 6.66 standard deviations from the mean. This corresponds to a proportion of formula_23 lost due to the truncation. The standard Box–Muller transform generates values from the standard normal distribution ("i.e." standard normal deviates) with mean "0" and standard deviation "1". The implementation below in standard C++

neg: Chi distribution of the molecular speeds in an ideal gas (chi distribution with three degrees of freedom). If formula_1 are independent, normally distributed random variables with means formula_2 and standard deviations formula_3, then the statistic is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution (scaled by the square root of "n" − 1) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution. The chi distribution has one parameter: formula_5 which specifies the number of degrees of freedom (i.e. the number of formula_1). The probability density function (pdf) of

neg: Wrapped normal distribution Wrapped normal distribution In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics. The probability density function of the wrapped normal distribution is where "μ" and "σ" are the mean and standard deviation of the

neg: Normal distribution with two Lagrange multipliers is defined: where formula_70 is, for now, regarded as some density function with mean formula_3 and standard deviation formula_16. At maximum entropy, a small variation formula_243 about formula_70 will produce a variation formula_245 about formula_246 which is equal to 0: Since this must hold for any small formula_243, the term in brackets must be zero, and solving for formula_70 yields: Using the constraint equations to solve for formula_251 and formula_220 yields the density of the normal distribution: The family of normal distributions is closed under linear transformations: if "X" is normally distributed with mean "μ" and

neg: Standard deviation are better in other respects: the uncorrected estimator (using "N") yields lower mean squared error, while using "N" − 1.5 (for the normal distribution) almost completely eliminates bias. The formula for the "population" standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by "s", is known as the uncorrected sample standard deviation, or sometimes the standard deviation of the sample (considered as the entire population), and is

neg: Normality (behavior) behaviour, it may be difficult to measure its normality. This is applicable when trying to diagnose a pathology and is addressed in the DSM. What is viewed as normal can change dependent on both timeframe and environment. Normality can be viewed as "an endless process of man's self-creation and his reshaping of the world". Within this idea, it is possible to surmise that normality is not an all-encompassing term, but simply a relative term based around a current trend in time. With statistics, this is likened to the thought that if the data gathered provides a mean and standard deviation,

neg: Standard deviation cm), then men would have much more variable heights, with a typical range of about 50–90 inches (127–228.6 cm). Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal (bell-shaped). (See the 68-95-99.7 rule, or the "empirical rule," for more information.) Let "X" be a random variable with mean value "μ": Here the operator E denotes the average or expected value of "X". Then the standard deviation of "X" is the quantity (derived using the properties of expected value). In other words, the standard deviation "σ" (sigma) is the square root of the

neg: Risk of ruin can lose significantly more than the price they paid to buy the product.) The probability of ruin is approximately where for a random walk with a starting value of "s", and at every iterative step, is moved by a normal distribution having mean "μ" and standard deviation "σ" and failure occurs if it reaches 0 or a negative value. For example, with a starting value of 10, at each iteration, a Gaussian random variable having mean 0.1 and standard deviation 1 is added to the value from the previous iteration. In this formula, "s" is 10, "σ" is 1, "μ"

neg: Variance his 1918 paper "The Correlation Between Relatives on the Supposition of Mendelian Inheritance": The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations formula_170 and formula_171, it is found that the distribution, when both causes act together, has a standard deviation

neg: Non-local means weighting function, formula_9, is to determine how closely related the image at the point formula_2 is to the image at the point formula_3. It can take many forms. The Gaussian weighting function sets up a normal distribution with a mean, formula_16 and a variable standard deviation: where formula_18 is the filtering parameter (i.e., standard deviation) and formula_19 is the local mean value of the image point values surrounding formula_2. For an image, formula_1, with discrete pixels, a discrete algorithm is required. where formula_11 is given by: Then, for a Gaussian weighting function, where formula_19 is given by: where formula_28 and

neg: Robust confidence intervals intervals calculated from σ, it is necessary to test for and remove a reasonable number of outliers (testing the assumption that the operator is careful and correcting for the fact that he is not perfect), and to test the assumption that the data really have a normal distribution with standard deviation σ. The theoretical analysis of such an experiment is complicated, but it is easy to set up a spreadsheet which draws random numbers from a normal distribution with standard deviation σ to simulate the situation; this can be done in Microsoft Excel using =NORMINV(RAND(),0,σ)), as discussed in and the

neg: Logit-normal distribution by zero and one, and where values of zero and one never occur. The probability density function (PDF) of a logit-normal distribution, for 0 ≤ "x" ≤ 1, is: where "μ" and "σ" are the mean and standard deviation of the variable’s logit (by definition, the variable’s logit is normally distributed). The density obtained by changing the sign of "μ" is symmetrical, in that it is equal to f(1-x;-"μ","σ"), shifting the mode to the other side of 0.5 (the midpoint of the (0,1) interval). The moments of the logit-normal distribution have no analytic solution. The moments can be estimated by

neg: Q–Q plot important for the normal distribution (the location and scale are estimated by the mean and standard deviation, respectively), it can be useful for many other distributions. However, this requires calculating the expected values of the order statistic, which may be difficult if the distribution is not normal. Alternatively, one may use estimates of the "median" of the order statistics, which one can compute based on estimates of the median of the order statistics of a uniform distribution and the quantile function of the distribution; this was suggested by . This can be easily generated for any distribution for which the

neg: Virtual valuation buyer's valuation has a continuous uniform distribution in formula_10. So: 2. The buyer's valuation has a normal distribution with mean 0 and standard deviation 1. formula_15 is monotonically increasing, and crosses the "x"-axis in about 0.75, so this is the optimal price. The crossing point moves right when the standard deviation is larger. A probability distribution function is called regular if its virtual-valuation function is weakly-increasing. Regularity is important because it implies that the virtual-surplus can be maximized by a truthful mechanism. A sufficient condition for regularity is monotone hazard rate, which means that the following function is weakly-increasing: Monotone-hazard-rate

neg: Normal distribution deviation between prior and data mean. The prior distributions are Therefore, the joint prior is The likelihood function from the section above with known variance is: Writing it in terms of variance rather than precision, we get: where formula_339 Therefore, the posterior is (dropping the hyperparameters as conditioning factors): In other words, the posterior distribution has the form of a product of a normal distribution over "p"("μ" | "σ") times an inverse gamma distribution over "p"(σ), with parameters that are the same as the update equations above. The occurrence of normal distribution in practical problems can be loosely classified into