Z-Score

A standardised statistic representing how many standard deviations an observation lies above or below the mean of its distribution.

Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.

A z-score tells us how far an observation lies from the mean of its distribution, measured in standard deviations. A z-score of two means the value sits two standard deviations above the mean, and one of negative one and a half means it lies one and a half below, so it locates any value on a common scale regardless of the original units. This lets values from different distributions be compared, and it maps directly onto the standard normal distribution for finding probabilities. An observation's distance from the mean in standard deviations is what it captures. Kirkwood and Sterne (2003) describe this.

Source: Kirkwood & Sterne 2003

A z-score is calculated by subtracting the mean of the distribution from the observation and dividing the result by the standard deviation, giving the number of standard deviations the observation lies from the mean. So a z-score is calculated as the observation minus the mean, divided by the standard deviation, which standardises the value onto a scale with mean zero and standard deviation one, and this computation converts values from any distribution with a known mean and standard deviation into comparable z-scores, allowing their positions and, for normal data, their probabilities to be assessed on the common standard normal scale.

Source: Casella & Berger 2002

A z-score indicates how far, in standard deviation units, an observation lies from the mean, and in which direction: a large positive z-score indicates a value well above the mean, a large negative one a value well below, and a z-score near zero a value close to the mean. So a z-score indicates the position and relative unusualness of an observation within its distribution, with larger absolute values indicating more extreme values, and for normally distributed data the z-score also corresponds to a probability, showing how likely a value that extreme is, which is why z-scores are used to identify outliers and compare values across distributions.

Source: Casella & Berger 2002

A z-score is used to standardise observations so that values from different distributions can be compared on a common scale, to identify unusual or extreme values, and, for normally distributed data, to find probabilities and percentiles from the standard normal distribution. So a z-score is used for standardisation, comparison, and probability calculation, since converting values to z-scores places them on the standard normal scale, which allows, for example, a person's measurement to be compared with a reference or the probability of a value as extreme to be determined, making the z-score widely used in statistics and in interpreting individual values.

Source: Casella & Berger 2002

A z-score relates to the standard normal distribution as the standardised value that, for normally distributed data, follows the standard normal distribution with mean zero and standard deviation one. Converting an observation to a z-score places it on this reference scale, from which probabilities and percentiles can be read. So a z-score and the standard normal distribution are directly connected, since standardising a normal variable produces z-scores distributed as standard normal, which is why the standard normal distribution provides the probabilities associated with z-scores, and why z-scores are the basis for finding how likely a value is and for inference about normal or approximately normal quantities.

Source: Casella & Berger 2002