Standard score - wikipedia Standard score `` Standardize '' redirects here. For industrial and technical standards, see Standardization. For Fisher z - transformation in statistics, see Fisher transformation. For Z - values in ecology, see Z - value. For z - transformation to complex number domain, see Z - transform. For Z - factor in high - throughput screening, see Z - factor. For Z - score financial analysis tool, see Altman Z - score. 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, the standard score is the signed number of standard deviations by which the value of an observation or data point is above the mean value of what is being observed or measured. Observed values above the mean have positive standard scores, while values below the mean have negative standard scores. The standard score is a dimensionless quantity obtained 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 for more ). Standard scores are also called z - values, z - scores, normal scores, and standardized variables. They are most frequently used to compare an observation to a standard normal deviate, though they can be defined without assumptions of normality. 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 Student 's t - statistic. Contents 1 Calculation from raw score 2 Applications 3 Standardizing in mathematical statistics 4 T - score 5 See also 6 References 7 Further reading 8 External links Calculation from raw score The standard score of a raw score x is z = x − μ σ ( displaystyle z = ( x - mu over sigma ) ) 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 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. It measures the sigma distance of actual data from the average. The Z value provides an assessment of how off - target a process is operating. Applications Main article : Z - test 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 γ ( displaystyle gamma ), i.e. P ( L < X < U ) = γ, ( displaystyle P ( L < X < U ) = gamma, ) For the standard score Z of X it gives : P ( L − μ σ < Z < U − μ σ ) = γ. ( displaystyle P left ( ( \frac ( L - mu ) ( sigma ) ) < Z < ( \frac ( U - mu ) ( sigma ) ) \right ) = gamma. ) By determining the quantile z such that P ( − z < Z < z ) = γ ( displaystyle P left ( - z < Z < z \right ) = gamma ) it follows : L = μ − z σ, U = μ + z σ ( displaystyle 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 expected value E ⁡ ( X ) ( displaystyle operatorname ( E ) ( X ) ) and dividing the difference by its standard deviation σ ( X ) = Var ⁡ ( X ) : ( displaystyle sigma ( X ) = ( sqrt ( operatorname ( Var ) ( X ) ) ) : ) Z = X − E ⁡ ( X ) σ ( X ) ( displaystyle Z = ( X - operatorname ( E ) ( X ) over sigma ( X ) ) ) If the random variable under consideration is the sample mean of a random sample X 1,..., X n ( displaystyle X_ ( 1 ), dots, X_ ( n ) ) of X : X  ̄ = 1 n ∑ i = 1 n X i ( displaystyle ( \bar ( X ) ) = ( 1 over n ) sum _ ( i = 1 ) ^ ( n ) X_ ( i ) ) then the standardized version is Z = X  ̄ − E ⁡ ( X ) σ ( X ) / n ( displaystyle Z = ( \frac ( ( \bar ( X ) ) - operatorname ( E ) ( X ) ) ( sigma ( X ) / ( sqrt ( n ) ) ) ) ). T - score `` T - score '' redirects here. It is not to be confused with t - statistic. 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. In bone density measurements, the T - score is the standard score of the measurement compared to the population of healthy 30 - year - old adults. See also Omega ratio Standard normal deviate References Jump up ^ E. Kreyszig ( 1979 ). Advanced Engineering Mathematics ( Fourth ed. ). Wiley. p. 880, eq. 5. ISBN 0 - 471 - 02140 - 7. Jump up ^ E. Kreyszig ( 1979 ). Advanced Engineering Mathematics ( Fourth ed. ). Wiley. p. 880, eq. 6. ISBN 0 - 471 - 02140 - 7. Jump up ^ John Salvia ; James Ysseldyke ; Sara Witmer ( 29 January 2009 ). Assessment : In Special and Inclusive Education. Cengage Learning. pp. 43 --. ISBN 0 - 547 - 13437 - 1. Jump up ^ Edward S. Neukrug ; R. Charles Fawcett ( 1 January 2014 ). Essentials of Testing and Assessment : A Practical Guide for Counselors, Social Workers, and Psychologists. Cengage Learning. pp. 133 --. ISBN 978 - 1 - 305 - 16183 - 2. Jump up ^ Randy W. Kamphaus ( 16 August 2005 ). Clinical Assessment of Child and Adolescent Intelligence. Springer. pp. 123 --. ISBN 978 - 0 - 387 - 26299 - 4. Jump up ^ `` Bone Mass Measurement : What the Numbers Mean ''. NIH Osteoporosis and Related Bone Diseases National Resource Center. National Institute of Health. Retrieved 5 August 2017. Further reading Carroll, Susan Rovezzi ; Carroll, David J. ( 2002 ). Statistics Made Simple for School Leaders ( illustrated ed. ). Rowman & Littlefield. ISBN 978 - 0 - 8108 - 4322 - 6. Retrieved 7 June 2009. Larsen, Richard J. ; Marx, Morris L. ( 2000 ). An Introduction to Mathematical Statistics and Its Applications ( Third ed. ). p. 282. ISBN 0 - 13 - 922303 - 7. External links Interactive Flash on the z - scores and the probabilities of the normal curve by Jim Reed Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Central limit theorem Moments Skewness Kurtosis L - moments Count data Index of dispersion Summary tables Grouped data Frequency distribution Contingency table Dependence Pearson product - moment correlation Rank correlation Spearman 's rho Kendall 's tau Partial correlation Scatter plot Graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q -- Q plot Run chart Scatter plot Stem - and - leaf display Radar chart Data collection Study design Population Statistic Effect size Statistical power Sample size determination Missing data Survey methodology Sampling stratified cluster Standard error Opinion poll Questionnaire Controlled experiments Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment Uncontrolled studies Observational study Natural experiment Quasi-experiment Statistical inference Statistical theory Population Statistic Probability distribution Sampling distribution Order statistic Empirical distribution Density estimation Statistical model L space Parameter location scale shape Parametric family Likelihood ( monotone ) Location -- scale family Exponential family Completeness Sufficiency Statistical functional Bootstrap U V Optimal decision loss function Efficiency Statistical distance divergence Asymptotics Robustness Frequentist inference Point estimation Estimating equations Maximum likelihood Method of moments M - estimator Minimum distance Unbiased estimators Mean - unbiased minimum - variance Rao -- Blackwellization Lehmann -- Scheffé theorem Median unbiased Plug - in Interval estimation Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife Testing hypotheses 1 - & 2 - tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons Parametric tests Likelihood - ratio Wald Score Specific tests Z ( normal ) Student 's t - test Goodness of fit Chi - squared Kolmogorov -- Smirnov Anderson -- Darling Lilliefors Jarque -- Bera Normality ( Shapiro -- Wilk ) Likelihood - ratio test Model selection Cross validation AIC BIC Rank statistics Sign Sample median Signed rank ( Wilcoxon ) Hodges -- Lehmann estimator Rank sum ( Mann -- Whitney ) Nonparametric anova 1 - way ( Kruskal -- Wallis ) 2 - way ( Friedman ) Ordered alternative ( Jonckheere -- Terpstra ) Bayesian inference Bayesian probability prior posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator Correlation Regression analysis Correlation Pearson product - moment Partial correlation Confounding variable Coefficient of determination Regression analysis Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines ( MARS ) Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity Generalized linear model Exponential families Logistic ( Bernoulli ) / Binomial / Poisson regressions Partition of variance Analysis of variance ( ANOVA, anova ) Analysis of covariance Multivariate ANOVA Degrees of freedom Categorical / Multivariate / Time - series / Survival analysis Categorical Cohen 's kappa Contingency table Graphical model Log - linear model McNemar 's test Multivariate Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model Factor analysis Multivariate distributions Elliptical distributions Normal Time - series General Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality Specific tests Dickey -- Fuller Johansen Q - statistic ( Ljung -- Box ) Durbin -- Watson Breusch -- Godfrey Time domain Autocorrelation ( ACF ) partial ( PACF ) Cross-correlation ( XCF ) ARMA model ARIMA model ( Box -- Jenkins ) Autoregressive conditional heteroskedasticity ( ARCH ) Vector autoregression ( VAR ) Frequency domain Spectral density estimation Fourier analysis Wavelet Survival Survival function Kaplan -- Meier estimator ( product limit ) Proportional hazards models Accelerated failure time ( AFT ) model First hitting time Hazard function Nelson -- Aalen estimator Test Log - rank test Applications Biostatistics Bioinformatics Clinical trials / studies Epidemiology Medical statistics Engineering statistics Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification Social statistics Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics Spatial statistics Cartography Environmental statistics Geographic information system Geostatistics Kriging Portal Commons WikiProject Retrieved from `` https://en.wikipedia.org/w/index.php?title=Standard_score&oldid=803723193 '' Categories : Statistical ratios Talk Contents Interaction About Wikipedia Català Čeština Deutsch Español Euskara Français 한국어 Italiano עברית Lietuvių Nederlands 日本 語 Norsk Polski Русский සිංහල Slovenščina Basa Sunda தமிழ் Türkçe اردو 中文 Edit links This page was last edited on 4 October 2017, at 06 : 12. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: the descriptive statistic that indicates the typical distance of a score from the mean is the
Answer: standard score