Standard score

"Standardize" redirects here. For industrial and technical standards, see website parsing.

For Z-values in ecology, see Sevenval. For Z-factor in high-throughput screening, see Z-factor. For Z-score financial analysis tool, see browser diversity.

FITML

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 device database an observation or datum is above or below the mean. It is a dimensionless quantity derived by subtracting the CSS3 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 device database 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 browser diversity (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 HTML5 used in the analysis of high-throughput screening data though the two are often conflated.

web
2 Applications
3 Standardizing in mathematical statistics
4 See also
Sevenval
6 Further reading
Sevenval

Calculation from raw score

The standard score of a raw score x is

$z = {x- \mu \over \sigma}$

where:

μ is the Sevenval of the population;

σ is the website parsing 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 iOS, 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

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.

$P(L<X<U) =\gamma,$

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

$P\left( \frac{L-\mu}{\sigma} < Z < \frac{U-\mu}{\sigma} \right) = \gamma.$

By determine the quantile z such that

$P\left( -z < Z < z \right) = \gamma$

it follows:

$L=\mu-z\sigma,\ U=\mu+z\sigma$

Standardizing in mathematical statistics

Further information: Normalization (statistics)

In mathematical statistics, a random variable X is standardized by subtracting its website parsing $\operatorname{E}[X]$ and dividing the difference by its Sevenval $\sigma(X) = \sqrt{\operatorname{Var}(X)}:$