For example, a Z-score of 1 means that the value is 1 standard deviation from the mean. The resulting distribution of Z-scores is referred to as a Z distribution (or standard normal distribution), where the Z-score represents the number of standard deviations (not the value of the standard deviation) that a given value is from the mean. Where μ is the mean, σ is the standard deviation, and x is the value being converted.
Z-scores and Z distributionsĪ normally distributed random variable can be standardized by converting its values to Z-scores using the following formula, However, since all normal distributions can be converted to a standard normal distribution, and since the associated probabilities have already been computed and compiled into Z tables for a standard normal distribution, it is possible to reference Z tables rather than having to integrate the pdf to determine the probability of outcomes within a given interval occurring. This is useful because, typically, it is necessary to integrate the probability density function (pdf) of a random variable to determine the probabilities of outcomes within an interval for the case of a normal distribution, this is particularly difficult. It tells us the probability that values in a normal distribution lie below, above, or between values on the standard normal distribution. between negative infinity and Z).Home / probability and statistics / hypothesis test / z table Z tableĪ Z table, also referred to as a standard normal table, is a table of the values of the cumulative distribution function of a normal distribution. This table gives a probability that a statistic is less than Z (i.e. Note that for z = 1, 2, 3, one obtains (after multiplying by 2 to account for the interval) the results f ( z) = 0.6827, 0.9545, 0.9974, If X is a random variable from a normal distribution with mean μ and standard deviation σ, its Z-score may be calculated from X by subtracting μ and dividing by the standard deviation: The standard normal distribution, represented by Z, is the normal distribution having a mean of 0 and a standard deviation of 1. Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal (known as a z-score) and then use the standard normal table to find probabilities. It is used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution. In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ, the cumulative distribution function of the normal distribution. Table of probabilities related to the normal distribution
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