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Information about Standard Score

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


“Standardize” redirects here. For industrial and technical standards, see Standardization.
For Z-values in ecology, see .


In statistics, the standard score, also called the z-score or normal score, 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.

The standard score indicates how many standard deviations an observation is above or below the mean. It allows comparison of observations from different normal distributions, which is done frequently in research.

The standard score is not the same as the z-factor used in the analysis of high-throughput screening data, but is sometimes confused with it.

Formula

The standard score is:    
where: x is a raw score to be standardized
σ is the standard deviation of the population
μ is the mean 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. For example, a population of people who smoke cigarettes is not fully measured.

When a population is normally distributed, the percentile rank may be determined from the standard score and statistical tables.

Standardizing in mathematical statistics

In mathematical statistics, a random variable X is standardized using the theoretical (population) mean and standard deviation:



where μ = E(X) is the mean and σ² = Var(X) the variance of the probability distribution of X.

If the random variable under consideration is the sample mean:



then the standardized version is

References

  • Abdi, H. (2007). Z-scores. In N.J. Salkind (Ed.), Encyclopedia of Measurement and Statistics. Thousand Oaks, CA: Sage.

External links

See also


 •  • [ e]
Statistics
Descriptive statisticsMean (Arithmetic, Geometric) - Median - Mode - Power - Variance - Standard deviation
Inferential statisticsHypothesis testing - Significance - Null hypothesis/Alternate hypothesis - Error - Z-test - Student's t-test - Maximum likelihood - Standard score/Z score - P-value - Analysis of variance
Survival analysisSurvival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models
Probability distributionsNormal (bell curve) - Poisson - Bernoulli
CorrelationPearson product-moment correlation coefficient - Rank correlation (Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient)
Regression analysisLinear regression - Nonlinear regression - Logistic regression

standardization or standardisation can have several meanings depending on its context. Common use of the word standard implies that it is a universally agreed-upon set of guidelines for interoperability.
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
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In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number.
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In statistics, mean has two related meanings:
  • the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
  • the expected value of a random variable, which is also called the population mean.

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In statistics, a statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population.
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In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma).
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In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma).
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In statistics, the Z-factor is a measure of the quality or power of a high-throughput screening (HTS) assay. It is not the same as the z-score. [1]
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High-throughput screening (HTS), is a method for scientific experimentation especially used in drug discovery and relevant to the fields of biology and chemistry.

Purpose and method


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In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma).
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In statistics, mean has two related meanings:
  • the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
  • the expected value of a random variable, which is also called the population mean.

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A standardized test is a test administered and scored in a standard manner. The tests are designed in such a way that the "questions, conditions for administering, scoring procedures, and interpretations are consistent" [1]
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normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale: the mean ("average",
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percentile rank of a score is the percentage of scores in its frequency distribution which are lower. For example, a test score which is greater than 90% of the scores of people taking the test is said to be at the 90th percentile.
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Mathematical statistics uses probability theory and other branches of mathematics to study statistics from a purely mathematical standpoint.

Mathematical statistics is the subject of mathematics that deals with gaining information from data.
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A random variable is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.
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In statistics, mean has two related meanings:
  • the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
  • the expected value of a random variable, which is also called the population mean.

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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. The arithmetic mean is what students are taught very early to call the "average".
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The Z-score formula for predicting bankruptcy was developed in 1968 by Edward I. Altman, a financial economist and professor at the Leonard N. Stern School of Business at New York University.
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The Z-test is a statistical test used in inference which determines if the difference between a sample mean and the population mean is large enough to be statistically significant.
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In statistics, the Z-factor is a measure of the quality or power of a high-throughput screening (HTS) assay. It is not the same as the z-score. [1]
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See also moment (physics).


The concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f(x) of a real variable about a value c is
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In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k
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In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample).
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A t test is any statistical hypothesis test in which the test statistic has a Student's t distribution if the null hypothesis is true.

History

The t statistic was introduced by William Sealy Gosset for cheaply monitoring the quality of beer brews.
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
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Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample and the measures. Together with simple graphics analysis, they form the basis of virtually every quantitative analysis of data.
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