Standard Normal Z Table

From the normal to the standard normal distribution

A generic normal distribution $\mathcal{N} ( x ; \mu , \sigma )$ can always be transformed into its standardized form $N ( x ; 0 , 1 )$ by introducing the standard variable $Z$, defined as

This transformation, known as standardization, expresses each value of the continuous random variable $X$ in terms of the number of standard deviations it lies away from the mean $\mu$. As a result, the new variable $Z$ follows a standard normal distribution with mean $0$ and standard deviation $1$.

The process of standardization is particularly useful because it places different normal distributions on a common scale. Once the variable $X$ has been transformed into the standard variable $Z$, probabilities and critical values can be derived directly from the standard normal distribution $N ( x ; 0 , 1 )$.

This allows statistical problems to be solved using a single universal reference table (Z table), simplifying calculations and comparisons across different contexts.

Standard Z Table

The standard Z table lists, for each cell, the cumulative probability to the left of a given $Z$ value. In other words, it shows the probability that a standard normal variable takes on a value less than or equal to a specified Z-score. These values correspond to the area under the standard normal curve from negative infinity up to the chosen point on the horizontal axis, represented by the value of $z$.

The complete Z table is extensive, as it contains values for a wide range of Z-scores. For brevity, only a portion of the table is shown below as an illustrative extract.

Several online resources, both static and interactive, allow users to compute cumulative probabilities for different values of $z$.

Using the Z table is straightforward.

• The row corresponding to the $Z$ value contains the integer part and the first decimal place
• The column represents the remaining decimal places starting from the second one.
• The cell at the intersection of the selected row and column gives the cumulative probability up to the specified value of $z$.

Example 1

Let us consider a simple example to illustrate how the Z table is used to determine the cumulative probability corresponding to a given value of the standardized variable $Z$. Let us consider the case of a standardized variable $Z$ such that:

To find the cumulative probability to the left of the standardized variable $z$, we locate the value 0.1 in the row and 0.06 in the column of the Z table. The intersection of these two entries gives the corresponding probability value, which in this case is $0.5636$.

This means that approximately 56.36% of the observations in a standard normal distribution fall below this value of $z$. We can therefore state that the probability of the standardized variable $Z$ being less than $z = 0.16$ is equal to $0.5636$, that is:

Since the total area under the standard normal curve equals $1$, and the distribution is symmetric with respect to its mean, we can immediately deduce that

Moreover, the probability that $Z$ lies between $0$ and $0.16$ is obtained by subtracting the cumulative probability up to $Z = 0$ from that up to $Z = 0.16$:

Substituting the corresponding values gives

Finally, due to the symmetry of the normal distribution about its mean, the probability that $Z$ lies within the interval $- 0.16 < Z < 0.16$ is twice the probability of being between $0$ and $0.16$:

Example 2

Suppose that the average lifetime of a rechargeable battery is $\mu = 8.0$ hours, with a standard deviation of $\sigma = 1.2$ hours, and that battery life follows a normal distribution. We want to calculate the probability that a randomly selected battery lasts less than $6.5$ hours.

To find the probability $P ( X < 6.5 )$, we need to determine the area under the normal curve to the left of $6.5$. For this purpose, we apply the standardization formula:

where $x$ represents the observed value of the random variable, which in this case corresponds to 6.5 hours. We have:

We can then use the Z table to find $P ( Z < - 1.25 )$, which gives the cumulative probability associated with this value of $z$.

From the intersection of the selected row and column, we obtain a probability value equal to $0.1056$. The figure below illustrates where this probability is located under the standard normal curve, corresponding to the cumulative area to the left of $z = - 1.25$.

From the standard normal table, we obtain

This means that approximately 10.56% of all batteries are expected to last less than 6.5 hours.