In statistics, a standard normal table, also called the unit normal table or Z table,^[1] is a mathematical table for the values of Φ, the cumulative distribution function of the normal distribution. 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. 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.^[2]

Normal and standard normal distribution

Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. The standard normal distribution, represented by Z, is the normal distribution having a mean of 0 and a standard deviation of 1.

Reading a Z table

Table examples

Cumulative from minus infinity to Z

The values correspond to the shaded area for given Z

This table gives a probability that a statistic is between minus infinity and Z.

${\displaystyle f(z)=\Phi (z)}$

The values are calculated using the cumulative distribution function of a standard normal distribution with mean of zero and standard deviation of one, usually denoted with the capital Greek letter ${\displaystyle \Phi }$ (phi), is the integral

${\displaystyle \Phi (z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-t^{2}/2}\,dt}$

${\displaystyle \Phi }$(z) is related to the error function, or erf(z).

${\displaystyle \Phi (z)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {z}{\sqrt {2}}}\right)\right]}$

Note that for z = 1, 2, 3, one obtains (after multiplying by 2 to account for the [−z,z] interval) the results f (z) = 0.6827, 0.9545, 0.9974, characteristic of the 68–95–99.7 rule.

Examples of use

A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5. Only a cumulative from mean table is available.

• What is the probability that a student scores an 82 or less?
  $${\displaystyle {\begin{aligned}P(X\leq 82)&=P\!\!\left(Z\leq {\frac {82-80}{5}}\right)\\&=P(Z\leq 0.40)\\[2pt]&=0.15542+0.5\\[2pt]&=0.65542\end{aligned}}}$$

  
• What is the probability that a student scores a 90 or more?
  $${\displaystyle {\begin{aligned}P(X\geq 90)&=P\!\!\left(Z\geq {\frac {90-80}{5}}\right)\\&=P(Z\geq 2.00)\\[2pt]&=1-P(Z\leq 2.00)\\[2pt]&=1-(0.47725+0.5)\\[2pt]&=0.02275\end{aligned}}}$$

  
• What is the probability that a student scores a 74 or less?
  $${\displaystyle {\begin{aligned}P(X\leq 74)&=P\!\!\left(Z\leq {\frac {74-80}{5}}\right)\\&=P(Z\leq -1.20)\end{aligned}}}$$

  
  Since this table does not include negatives, the process involves the following additional step:
  $${\displaystyle {\begin{aligned}\qquad \qquad \quad ={}&P(Z\geq 1.20)\\[2pt]={}&1-(0.38493+0.5)\\[2pt]={}&0.11507\end{aligned}}}$$

  
• What is the probability that a student scores between 74 and 82?
  $${\displaystyle {\begin{aligned}P(74\leq X\leq 82)&=P(X\leq 82)-P(X\leq 74)\\[2pt]&=0.65542-0.11507\\[2pt]&=0.54035\end{aligned}}}$$

  
• What is the probability that an average of three scores is 82 or less?
  $${\displaystyle {\begin{aligned}P(X\leq 82)&=P\left(Z\leq {\frac {82-80}{5/{\sqrt {3}}}}\right)\\&=P(Z\leq 0.69)\\[2pt]&=0.2549+0.5\\[2pt]&=0.7549\end{aligned}}}$$

  

See also

References