Z Score Table: Definition, Use and Z Score Calculator

A Z score table (or Z Table), also known as a standard normal table or unit normal table, is a mathematical table for the different values of ɸ which indicate the values of the cumulative distribution function of the normal distribution. The Z-score (also called the standard score) serves as an indication of the number of standard deviations a raw score lays above or below the mean. When the mean of the Z-score is calculated, it is always equal to zero, whereas the standard deviation or variance is always in increments of 1.

Since the variety of normal distribution is far beyond the realm of measurement, it is simply not possible to print probability tables for each and every normal distribution. Therefore, a commonly used technique is to first convert a normal to a standard normal and then find probabilities with the help of the z-score table.

Table of Contents

What is Z Score?

Z-score (also called a standard score) gives you an idea of how far from the mean a data point is.

Explanation

There is a different set of values associated with every set of data. For example, the heights of human beings can range from eighteen inches to eight feet, and their weights can range from one pound to more than five hundred pounds. As a result of these wide ranges, it can be extremely tricky and tough to analyze data.

Therefore, the normal curve is standardized and set to have a mean of zero and a standard deviation of one. After this, a Z-table can be used to find percentages under the curve. It can be used to calculate the area under the standard normal curve for any value between the mean (zero) and any Z-score.

Z-score formula

The Z-score formula can be used to compare the results from a test to a so-called normal population. Let’s say X is a random variable from a normal distribution, with mean μ and standard deviation σ. In this case, its Z-score can be calculated by subtracting the mean from X and dividing the result by the standard deviation, as:

Z =(X-μ)/σ

Where

X is the test value
μ is the mean, and
σ is the standard deviation (SD) for the average of a sample from a population “n”.

How to interpret the Z-score?

Here is how you can interpret Z-scores:

A Z-score lesser than zero indicates that the element is less than the mean.
A Z-score greater than zero indicates that the element is greater than the mean.
A Z-score equal to zero indicates that the element is equal to the mean.
A Z-score equal to 1 signifies that the element in question is 1 standard deviation greater than the mean. Similarly, a Z-score equal to 2 indicates that the element in question is 2 standard deviations greater than the mean.
A Z-score equal to -1 signifies that the element in question is 1 standard deviation less than the mean. Similarly, a Z-score equal to -2 indicates that the element in question is 2 standard deviations less than the mean.
In case the number of elements in the set happens to be quite large, around 68% of the elements have a Z-score between -1 and 1, around 95% have a Z-score between -2 and 2, and around 99% have a Z-score between -3 and 3.

Explanation using an example

Here is a solved example to help you understand the concept of Z-score better:

Problem: Let’s say the test score of students in an examination has a mean of 70 with a standard deviation of 12. Calculate the probable percentage of students who scored higher than 85.

Solution:

To solve this problem, we can calculate the Z-score for the given data with the help of the formula mentioned above.

Z =(X-μ)/σ

Here

X is the test value = 85
μ is the mean = 70, and
σ is the standard deviation (SD) for the average of a sample from a population “n” = 12.

In this problem, the population (number of students) doesn't matter at all.

By putting the values,

Z = (85-70)/12

Z = 15/12

Z = 1.25

How to use this Z Score to Calculate Result?

In the Z score table (download the table PDF here), you will have to find the value corresponding to the row of 1.2 to the columns of 0.05, as shown in this image:

When you through the Z-score table, you will find out that the fraction of the data within this score is 0.89435. This implies that 89.435% of the students who took the examination have gotten marks less than 85. Thus, the percentage of students who scored above 85 marks = (100-89.435) % = 10.565%.

This is the answer.

Thus, we can use the Z-score table to find out the percentage of values below (to the left) a Z-score in a standard normal distribution.

Types of Z-score tables

There are two kinds of Z-score tables:

Positive Z-score table

In this Z Score Table, the observed value is above the mean of total values.

Negative Z-score table

In this the observed value is below the mean of total values.

These tables have been particularly designed for a standard normal distribution.

Use and Application of Z-scores

Speaking of the normal distribution of a variable, the mean is the center of distribution and the standard deviation indicates the existing variability.

The Z-table can be used to find out the areas for a calculated Z-score if you intend to determine the probability of a particular value in order to calculate the area under any normal distribution. By doing this, you can find out the chances of a value to occur. Also, you must remember that not all Z-score tables are identical.

Z-scores are frequently used in financial accounting and by businesses to determine financial positioning, market volatility, and much more. Usually, a Z-score of less than 1.8 represents severe losses in a form or is an indicator of the company going bankrupt sooner or later. On the other hand, a high Z-score of around 3 is a testament to the company’s solid financial positioning. With its help, analysts can also adapt data scores from different data sets for comparing them with greater ease.

Free Tool: Z Score Calculator

I hope that this guide helps you understand the significance of Z-score tables and the concept of Z-score itself, along with their practical applications. Keep in mind, though, that one of the weaknesses of Z-scores is that they invariably assume that all the distributions are normal distributions. If this assumption proves to be false, then the scores can’t be interpreted as a standard proportion of the given distribution from which they had been calculated. Apart from this, Z-scores are undoubtedly a valuable tool for businesses and companies to improve their performance.