Compares the various grading methods in a normal distribution. Includes: Standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nine, percent in stanine

In statistics, a standard score indicates by how many standard deviations an observation or datum is above or below the mean. It is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing; however, "normalizing" can refer to many types of ratios; see normalization (statistics) for more.

Standard scores are also called z-values, z-scores, normal scores, and standardized variables; the use of "Z" is because the normal distribution is also known as the "Z distribution". They are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with μ = 0 and σ = 1), though they can be defined without assumptions of normality.

The z-score is only defined if one knows the population parameters, as in standardized testing; if one only has a sample set, then the analogous computation with sample mean and sample standard deviation yields the Student's t-statistic.

The standard score is not the same as the z-factor used in the analysis of high-throughput screening data though the two are often conflated.

Calculation from raw score[link]

The standard score of a raw score x is

Failed to parse (Missing texvc executable; please see math/README to configure.): z = {x- \mu \over \sigma}

where:

μ is the mean of the population;

σ is the standard deviation of the population.

The quantity z represents the distance between the raw score and the population mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above.

A key point is that calculating z requires the population mean and the population standard deviation, not the sample mean or sample deviation. It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest. But knowing the true standard deviation of a population is often unrealistic except in cases such as standardized testing, where the entire population is measured. In cases where it is impossible to measure every member of a population, the standard deviation may be estimated using a random sample.

Applications[link]

The z-score is often used in the z-test in standardized testing – the analog of the Student's t-test for a population whose parameters are known, rather than estimated. As it is very unusual to know the entire population, the t-test is much more widely used.

Also, standard score can be used in the calculation of prediction intervals. A prediction interval [L,U], consisting of a lower endpoint designated L and an upper endpoint designated U, is an interval such that a future observation X will lie in the interval with high probability, i.e.

Failed to parse (Missing texvc executable; please see math/README to configure.): P(L<X<U) =\gamma,

for example γ = 0.95 (95%). For the standard score Z of X it gives:

Failed to parse (Missing texvc executable; please see math/README to configure.): P\left( \frac{L-\mu}{\sigma} < Z < \frac{U-\mu}{\sigma} \right) = \gamma.

By determine the quantile z such that

Failed to parse (Missing texvc executable; please see math/README to configure.): P\left( -z < Z < z \right) = \gamma

it follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): L=\mu-z\sigma,\ U=\mu+z\sigma

Standardizing in mathematical statistics[link]

In mathematical statistics, a random variable X is standardized by subtracting its expected value Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}[X]

and dividing the difference by its standard deviation Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma(X) = \sqrt{\operatorname{Var}(X)}:

Failed to parse (Missing texvc executable; please see math/README to configure.): Z = {X - \operatorname{E}[X] \over \sigma(X)}

If the random variable under consideration is the sample mean of a random sample Failed to parse (Missing texvc executable; please see math/README to configure.): \ X_1,\dots, X_n

of X:

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{X}={1 \over n} \sum_{i=1}^n X_i

then the standardized version is

Failed to parse (Missing texvc executable; please see math/README to configure.): Z = \frac{\bar{X}-\operatorname{E}[X]}{\sigma(X)/\sqrt{n}}.

See normalization (statistics) for other forms of normalization.

See also[link]

• Standard normal deviate
• Z-test

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (May 2012)

References[link]

Further reading[link]

• Carroll, Susan Rovezzi; Carroll, David J. (2002). Statistics Made Simple for School Leaders (illustrated ed.). Rowman & Littlefield. ISBN 978-0-8108-4322-6. http://books.google.com/?id=gccHkMDikb0C. Retrieved 7 June 2009 
• Richard J. Larsen and Morris L. Marx (2000) An Introduction to Mathematical Statistics and Its Applications, Third Edition, ISBN 0-13-922303-7. p. 282.

External links[link]

• Interactive Flash on the z-scores and the probabilities of the normal curve by Jim Reed

The Normal is the recording artist name used by English music producer Daniel Miller, a film editor at the time, who is best known as the founder of the record label Mute Records.

Background[link]

In 1977, Miller had split up with his girlfriend. A friend suggested that he read a book he himself had just finished. The book was Crash (1973) by J.G. Ballard.^[1] He felt that Ballard's writing took him five minutes into the future; the novel was to be a major influence in the music he would produce as The Normal.^[1] Miller was disillusioned by the fact you needed to learn three chords to be in a punk band, so he decided to purchase a synthesiser. His thinking was that you only needed to learn to press one key on a synthesiser.^[1] After buying a Korg 700s synthesiser from Macari's music shop in London,^[1] Miller recorded and released a single under the name The Normal. This was "T.V.O.D."/"Warm Leatherette". Both tracks were minimalist electronic songs influenced by the Crash novel.^[1] He wanted the sound of the recordings to be visual, like driving along a highway between large buildings then going through a tunnel.^[1] The single was recorded in Daniel Miller's house using a TEAC four track tape recorder and the Korg synthesiser.^[2]

"Live at West Runton Pavilion", The Normal's second release, done with another Mute Records act, Robert Rental, wasn't well-received. A strange release, it was a one-sided album (side two was left blank) of improvised electronic noises, in a plain purple dust jacket. Marat Records released the record in Germany as Daniel Miller Robert Rental Live, with a black and white picture sleeve, catalogue No. Marat Rough 017.

Influence on other artists[link]

Songs by The Normal have been covered and performed by many notable artists such as Grace Jones, Chicks on Speed/Hell on a split-7", Giddle and Boyd Rice and in 2006 by Trent Reznor with Jeordie White and Peter Murphy.^[3] A recent cover of "Warm Leatherette" was performed live by Duran Duran in November 2007, and as part of an electro medley during their 2008 tour,^[4] in support of their album Red Carpet Massacre.

Discography[link]

• The Normal - "T.V.O.D." / "Warm Leatherette" (Mute Records 1978)
• Robert Rental & The Normal - Live At West Runton Pavilion, 6-3-79 (Rough Trade 1980)

References[link]

Notes[link]

External links[link]

• The Normal at Mute Records Official site
• The Normal at Allmusic
• The Normal Discography at discogs.com