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Z-VALUE

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Project Owner : Shyam.C
Created Date : Sat, 17/03/2012 - 11:06
Project Description :

 

In statistics, a standard score indicates 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 ornormalizing; 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.

 

Applications

The z-score is most 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.


Percentile ranks and prediction intervals

With a population that is normally distributed with known mean and known variance, the percentile rank and prediction interval may be determined from the standard score.

With known mean and known variance, prediction intervals can be calculated by subtracting from or adding to the mean (µ) with the standard deviation (σ) multiplied by a standard score (z) that is specific for what prediction intervals are desired:

Prediction
interval
Standard
score (z)
50% 0.67[1]
68% 1.00[1]
90% 1.64[1]
95% 1.96[1]
99% 2.58[1]
Diagram showing the cumulative distribution function for the normal distribution with mean (µ) of 0 and variance (σ2) of 1. The prediction interval for any standard score corresponds numerically to (1 − (1 − Φµ,σ2(standard score))*2). For example, a standard score of x = 1.96 gives Φµ,σ2(1.96) = 0.9750 corresponding to a prediction interval of (1 − (1 − 0.9750)·2) = 0.9500 = 95%.
  • Lower limit of prediction interval = µ − σz
  • Upper limit of prediction interval = µ + σz

About 68.27% of the values lie within 1 standard deviation of the mean. Similarly, about 95.45% of the values lie within 2 standard deviations of the mean. Nearly all (99.73%) of the values lie within 3 standard deviations of the mean. This is known as the 68-95-99.7 rule.

For example, to calculate the 95% prediction interval for a normal distribution with a mean (µ) of 5 and a standard deviation (σ) of 1, then the lower limit of the prediction interval is approximately 5 ‒ (1*2) = 3, and the upper limit is approximately 7, thus giving a prediction interval of approximately 3 to 7.


Standardizing in mathematical statistics

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

Z = {X - \mu \over \sigma}

where \mu = \operatorname{E}[X] is the mean and \sigma = \sqrt{\operatorname{Var}(X)} the standard deviation of the probability distribution of X.

If the random variable under consideration is the sample mean:

\bar{X}={1 \over n} \sum_{i=1}^n X_i

then the standardized version is

Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}.

See normalization (statistics) for other forms of normalization.

A common name for standard score is the z-score. It is often used in statistics.

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