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| The distribution is bell-shaped and parameterized to the [[Statistics/Moments|first and second moments]]. This has useful consequences for estimating the probability that a given value is in the distribution. For example: | The distribution is bell-shaped and parameterized to the [[Analysis/Moment|first and second central moments]]. This has useful consequences for estimating the probability that a given value is in the distribution. For example: |
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| A variable distributed this way is notated (especially in [[Statistics/EconometricsNotation|econometrics]]) like X ~ N(μ, σ^2^). | A variable distributed this way is notated like ''X ~ N(μ, σ^2^)''. |
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| When the mean is 0 and the [[Statistics/Variance|variance]] is 1, the p.d.f. is specifically referred to as the '''standard normal distribution'''. This is defined as {{attachment:stdnorm.svg}}. | The probability density function is given (and commonly notated) as: {{attachment:norm.svg}} Correspondingly, the cumulative probability function is commonly notated as ''Φ(.)'' rather than ''F(.)''. Furthermore the quantile function is commonly notated as ''Φ^-1^(.)''. === Standard Normal Distribution === The '''standard normal distribution''' is specified as ''Z ~ N(0, 1)''. Compare to the calculation for '''Z scores''' that will be compared to the standard normal distribution: ''Z = (x - μ)/σ''. The probability density function simplifies to: {{attachment:stdnorm.svg}}. This is graphed as: |
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More generally, the p.d.f. is given by ''f(x) = (1/σ) * φ(x-μ/σ)''. The c.d.f. for the standard normal distribution is notated as ''Φ(.)'', while the c.d.f. for the generic normal distribution is sometimes notated as ''F(.)''. |
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| The [[Statistics/Moments|first and second moments]] are intrinsic to the distribution definition. | The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = μ''. [[Analysis/Variance|Variance]] is given as ''Var[X] = σ^2^''. |
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| === Normality of Sampling Distributions === For a population parameter whose population distribution is normal with mean ''μ'' and variance ''σ^2^'', the sampling distribution of the sample statistic which estimates that parameter is also normal. In particular, ''X̅'' is the sample statistic that estimates ''μ'', and its sampling distribution is normal with mean ''X̅ = μ'' and variance ''σ^2^,,X̅,, = σ^2^,,X,,/n'', where ''n'' is the sample size. By the central limit theorem, even when the population distribution in question is '''not''' normal, with a sufficiently large sample size the sampling distribution of the sample statistic which estimates that parameter '''is'''. A note on variance of a sampling distribution (i.e., ''σ^2^,,X̅,,'' not ''σ^2^,,X,,''). This is a function of both the sample variance ''σ^2^,,X,,'', which is usually itself estimated using sample variance (i.e., ''σ^2^,,X,, = s^2^,,X̅,,''), and the sample size ''n''. Consider a census of the population; there is only one possible sample so the sampling distribution has only a single level with 100% certainty. As the sample size decreases, the number of possible samples grows. The number of levels in the sampling distribution is also expected to grow accordingly. |
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| The standard normal distribution is referenced for '''Z scores''' (alternatively called '''Z statistics'''). As an example, for a two-tailed test and a [[Statistics/TestStatistic|significance level]] of 5%, the critical Z score value is 1.96. | As noted above, '''Z scores''' (alt. ''Z statistics'') are test statistics for which likelihood can be calculated using the standard normal distribution. As an example, for a two-tailed test and a [[Statistics/TestStatistic|significance level]] of 5%, the critical Z score value is 1.96. |
Normal Distribution
The normal distribution is a bell-shaped continuous probability distribution function that is parameterized to a mean and standard deviation.
Contents
Description
The distribution is bell-shaped and parameterized to the first and second central moments. This has useful consequences for estimating the probability that a given value is in the distribution. For example:
- 68.27% of the cumulative distribution is within 1 standard deviation of the mean
- 95.45% within 2
- 99.73% within 3
A variable distributed this way is notated like X ~ N(μ, σ2).
The probability density function is given (and commonly notated) as:
Correspondingly, the cumulative probability function is commonly notated as Φ(.) rather than F(.). Furthermore the quantile function is commonly notated as Φ-1(.).
Standard Normal Distribution
The standard normal distribution is specified as Z ~ N(0, 1).
Compare to the calculation for Z scores that will be compared to the standard normal distribution: Z = (x - μ)/σ.
The probability density function simplifies to:
.
This is graphed as:
Moments
The expected value is given as E[X] = μ.
Variance is given as Var[X] = σ2.
Usage
Normality of Sampling Distributions
For a population parameter whose population distribution is normal with mean μ and variance σ2, the sampling distribution of the sample statistic which estimates that parameter is also normal. In particular, X̅ is the sample statistic that estimates μ, and its sampling distribution is normal with mean X̅ = μ and variance σ2X̅ = σ2X/n, where n is the sample size.
By the central limit theorem, even when the population distribution in question is not normal, with a sufficiently large sample size the sampling distribution of the sample statistic which estimates that parameter is.
A note on variance of a sampling distribution (i.e., σ2X̅ not σ2X). This is a function of both the sample variance σ2X, which is usually itself estimated using sample variance (i.e., σ2X = s2X̅), and the sample size n. Consider a census of the population; there is only one possible sample so the sampling distribution has only a single level with 100% certainty. As the sample size decreases, the number of possible samples grows. The number of levels in the sampling distribution is also expected to grow accordingly.
Probability Tests
As noted above, Z scores (alt. Z statistics) are test statistics for which likelihood can be calculated using the standard normal distribution.
As an example, for a two-tailed test and a significance level of 5%, the critical Z score value is 1.96.