Z Score Table

Z score table is also known as the z table or the standard normal distribution table. It is used to indicate the percentage of values that fall below a particular z score value. In other words, it is used to show the values that lie to the left of a z score on a standard normal distribution curve.

There can be two types of z score tables, namely, the positive z table and the negative z table. In this article, we will learn more about the z score table, its formulas, how to use this table, and see some associated examples.

A z score table can be defined as a mathematical table that is used to give the percentage of values (usually expressed as decimals) that lie below a certain z score value. Hence, a z table is used to give the values of the probability distribution function of a normal distribution. There are an infinite number of normal distributions and it is very difficult to print the probability tables for each distribution. Thus, the common practice is to convert a normal distribution to a standard normal distribution and then use the z score table to find its probability.

The z score needs to be calculated before using the z table. The z score is used to denote the number of standard deviations by which a raw score lies above or below the mean. The z score can be determined for both the sample data and population data. The z score formulas that are used in the z score table are given as follows:

Z score for population data:

z = \(\frac{x-\mu}{\sigma}\)

\(\mu\) = population mean

\(\sigma\) = population standard deviation

x = raw score

Z score for sample data:

z = \(\frac{x-\overline{x}}{S}\)

\(\overline{x}\) = sample mean

S = sample standard deviation

x = raw score

A standard normal distribution bell curve is centered at the mean. The value of the mean is zero for a standard normal distribution. The z score at the mean will also always be 0. The z table represents the area to the left of the z score under the bell curve. The curve extends infinitely in both directions (left and right). Due to this, there can be two types of z score tables as given below:

Negative Z Score Table

A negative z score table is used to find the values that lie to the left of the z score below the mean. The z score will be negative for such values. The negative z table is given below:

Positive Z Score Table

A positive z score table is used to depict the values that lie to the left of the z score above the mean on the standard normal distribution curve. These values have a positive z score and the z table is given below.

The z score table follows a 68 - 95 - 99 rule. This implies that if the sample space of the data set is large then 68% of the data will lie between a z score of -1 and 1. Similarly, around 95% will lie between -2 and 2 while 99% lies between -3 and 3. The graph of the z score table can be given as follows:

The value of the z score can either be negative or positive. This implies that the raw score is either to the left of the mean or to the right respectively. Thus, it is necessary to choose the correct z table in order to accurately map the z score. Suppose a z score is given as P(z ≤ 0.57) and the area to the left of this value needs to be determined. The steps to use the z score table are as follows:

• Choose the correct z score table. As the z score in the example is positive, the positive z table is used.
• Take the first two digits of the z score and find them in the first column of the z table. Here, 0.5 is taken and it is the 6^th entry in the first column.
• Next, take the second digit after the decimal and map it in the first row. This value is 0.07 and it is the 8^th entry in the first row.
• Now find the intersection of the row of the first two digits and the column of the second decimal value in the z score table. This value is 0.71566 which is the required answer.

If the probability P(z ≥ 0.57) needs to be determined, this is given as follows:

P(z ≥ 0.57) = 1 - P(z < 0.57)

= 1 - 0.7157

P(z ≥ 0.57) = 0.2843

Related Articles:

• Probability and Statistics
• Hypothesis Testing
• F Test
• ANOVA Test
• Z Test

Important Notes on Z Score Table

• A z score table is a mathematical table that is used to display the percentage of values that fall below a particular z score.
• There can be two types of z score tables - positive and negative.
• The z table gives the area under a standard normal distribution curve to the left of the z score.

What is the Meaning of a Z Score Table?

A z score table can be defined as a mathematical table that is used to display the percentage of values (in decimals) that fall below a particular z score.

What are the Types of a Z Score Table?

A z table has two types - negative and positive. The negative z table is used for negative z score values and the positive table is used for positive values.

What is the Z Score Table Formula?

The formula to find the z score which can then be mapped on the z score table is given as z = \(\frac{x-\mu}{\sigma}\).

Is the Standard Normal Table the Same as a Z Score Table?

Yes, the z score table and the standard normal table are different names used to represent the same mathematical table. It checks the percentage of values that fall below the z score.

What Does the Z Score Table Tell You?

The z score table is used to denote the values that fall below a given z score. In other words, it is used to represent the probability distribution of a standard normal distribution.

How to Interpret a Z Score Table?

A positive z score table is used to depict values that lie above the mean of the distribution while a negative z table represents all values that fall below the mean.

How to Find a Value on the Z Score Table?

The steps to find a value on the z score table are as follows:

• Find the correct table using the sign.
• Using the first two digits of the score find the required row.
• Using the second digit after the decimal, find the required column.
• The intersection of the row and the column will give the required value.