In statistics, the **standard score** is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores.

- Calculation
- Applications
- Z-test
- Prediction intervals
- Process control
- Comparison of scores measured on different scales: ACT and SAT
- Percentage of observations below a z-score
- Cluster analysis and multidimensional scaling
- Principal components analysis
- Relative importance of variables in multiple regression: Standardized regression coefficients
- Standardizing in mathematical statistics
- T-score
- See also
- References
- Further reading
- External links

It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This process of converting a raw score into a standard score is called **standardizing** or **normalizing** (however, "normalizing" can refer to many types of ratios; see normalization for more).

**Standard scores** are most commonly called ** z-scores**; the two terms may be used interchangeably, as they are in this article. Other terms include

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, a raw score *x* is converted into a standard score by^{ [2] }

where:

*μ*is the mean of the population.*σ*is the standard deviation of the population.

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

Calculating **z** using this formula requires the population mean and the population standard deviation, not the sample mean or sample deviation. But knowing the true mean and standard deviation of a population is often unrealistic except in cases such as standardized testing, where the entire population is measured.

When the population mean and the population standard deviation are unknown, the standard score may be calculated using the sample mean and sample standard deviation as estimates of the population values.^{ [3] }^{ [4] }^{ [5] }^{ [6] }

In these cases, the **z**-score is

where:

- is the mean of the sample.
*S*is the standard deviation of the sample.

In either case, since the numerator and denominator of the equation must both be expressed in the same units of measure, and since the units cancel out through division, **z** is left as a dimensionless quantity.

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.

The 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.

For the standard score *Z* of *X* it gives:^{ [7] }

By determining the quantile z such that

it follows:

In process control applications, the Z value provides an assessment of how off-target a process is operating.

When scores are measured on different scales, they may be converted to z-scores to aid comparison. Dietz et al.^{ [8] } give the following example comparing student scores on the (old)SAT and ACT high school tests. The table shows the mean and standard deviation for total score on the SAT and ACT. Suppose that student A scored 1800 on the SAT, and student B scored 24 on the ACT. Which student performed better relative to other test-takers?

SAT | ACT | |
---|---|---|

Mean | 1500 | 21 |

Standard deviation | 300 | 5 |

The z-score for student A is

The z-score for student B is

Because student A has a higher z-score than student B, student A performed better compared to other test-takers than did student B.

Continuing the example of ACT and SAT scores, if it can be further assumed that both ACT and SAT scores are normally distributed (which is approximately correct), then the z-scores may be used to calculate the percentage of test-takers who received lower scores than students A and B.

"For some multivariate techniques such as multidimensional scaling and cluster analysis, the concept of distance between the units in the data is often of considerable interest and importance … When the variables in a multivariate data set are on different scales, it makes more sense to calculate the distances after some form of standardization."^{ [9] }

In principal components analysis, "Variables measured on different scales or on a common scale with widely differing ranges are often standardized."^{ [10] }

Standardization of variables prior to multiple regression analysis is sometimes used as an aid to interpretation.^{ [11] } (page 95) state the following.

"The standardized regression slope is the slope in the regression equation if X and Y are standardized… Standardization of X and Y is done by subtracting the respective means from each set of observations and dividing by the respective standard deviations… In multiple regression, where several X variables are used, the standardized regression coefficients quantify the relative contribution of each X variable."

However, Kutner et al.^{ [12] } (p 278) give the following caveat: "… one must be cautious about interpreting any regression coefficients, whether standardized or not. The reason is that when the predictor variables are correlated among themselves, … the regression coefficients are affected by the other predictor variables in the model … The magnitudes of the standardized regression coefficients are affected not only by the presence of correlations among the predictor variables but also by the spacings of the observations on each of these variables. Sometimes these spacings may be quite arbitrary. Hence, it is ordinarily not wise to interpret the magnitudes of standardized regression coefficients as reflecting the comparative importance of the predictor variables."

In mathematical statistics, a random variable *X* is **standardized** by subtracting its expected value and dividing the difference by its standard deviation

If the random variable under consideration is the sample mean of a random sample of *X*:

then the standardized version is

In educational assessment, **T-score** is a standard score Z shifted and scaled to have a mean of 50 and a standard deviation of 10.^{ [13] }^{ [14] }^{ [15] }

In bone density measurements, the T-score is the standard score of the measurement compared to the population of healthy 30-year-old adults.^{ [16] }

In probability theory and statistics, **kurtosis** is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations.

In probability theory, a **normal****distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

In statistics, the **standard deviation** is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In probability theory and statistics, **skewness** is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.

In probability theory and statistics, **variance** is the expectation of the squared deviation of a random variable from its mean. In other words, it measures how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , or .

In probability theory and statistics, the **multivariate normal distribution**, **multivariate Gaussian distribution**, or **joint normal distribution** is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be *k*-variate normally distributed if every linear combination of its *k* components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

In probability theory, a **log-normal distribution** is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then *Y* = ln(*X*) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, *X* = exp(*Y*), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics.

In probability and statistics, **Student's t-distribution** is any member of a family of continuous probability distributions that arise when estimating the mean of a normally-distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student".

In probability theory, **Chebyshev's inequality** guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/*k*^{2} of the distribution's values can be *k* or more standard deviations away from the mean. The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.

In statistics, **correlation ** or **dependence ** is any statistical relationship, whether causal or not, between two random variables or bivariate data. In the broadest sense **correlation** is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

In probability theory and statistics, a **standardized moment** of a probability distribution is a moment that is normalized. The normalization is typically a division by an expression of the standard deviation which renders the moment scale invariant. This has the advantage that such normalized moments differ only in other properties than variability, facilitating e.g. comparison of shape of different probability distributions.

In statistics, the **Pearson correlation coefficient**, also referred to as **Pearson's r**, the

A ** Z-test** is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-test tests the mean of a distribution. For each significance level in the confidence interval, the

In statistics, an **effect size** is a number measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event happening. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.

In statistics and optimization, **errors** and **residuals** are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "theoretical value". The **error** of an observed value is the deviation of the observed value from the (unobservable) *true* value of a quantity of interest, and the **residual** of an observed value is the difference between the observed value and the *estimated* value of the quantity of interest. The distinction is most important in regression analysis, where the concepts are sometimes called the **regression errors** and **regression residuals** and where they lead to the concept of studentized residuals.

In statistical inference, specifically predictive inference, a **prediction interval** is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis.

In probability theory and statistics, the **coefficient of variation** (**CV**), also known as **relative standard deviation** (**RSD**), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage, and is defined as the ratio of the standard deviation to the mean . The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R. In addition, CV is utilized by economists and investors in economic models.

The following is a glossary of terms used in the mathematical sciences statistics and probability.

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 an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.

In statistics, the ** t-statistic** is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's

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