UPDATED BY
Brennan Whitfield | Aug 04, 2023

Imagine a scenario where we compare the standardized test results from two students. Let’s call them Zoe and Mike. Zoe took the ACT and scored a 25, while Mike took the SAT and scored 1150. Which of the test takers scored better? And what proportion of people scored worse than Zoe and Mike?

How to Use a Z-Table

A z-table tells you the area underneath a normal distribution curve, to the left of the z-score. In other words, it tells you the probability for a particular z-score. To use one, first turn your data into a normal distribution. Then find the matching z-score to the left of the table and align it with the z-score at the top of the table. The result gives you the probability.

To be able to utilize a z-table and answer these questions, you have to turn the scores on the different tests into a standard normal distribution N(mean = 0, std = 1)

SAT and ACT scores normal distribution graphs
ACT and SAT score means and normal distributions. Note, these aren’t the actual score means and distribution in real life. | Image: Michael Galarnyk  ​​​​​​

Since these scores on these tests have a normal distribution, we can convert both of them into standard normal distributions by using the following formula.

 

Z-Score Formula

The z-score formula is as follows:

Z-score formula
Equation to find the z-score in a standard deviation. | Image: Michael Galarnyk

With this formula, you can calculate z-scores for Zoe and Mike.

Zoe and Mike's z-score results
Zoe (left) and Mike (right) z-score totals. | Image: Michael Galarnyk

Since Zoe has a higher z-score than Mike, Zoe performed better on her test.

 

How to Interpret Z-Score

A z-score is used to determine how many standard deviations a data point is from the mean value in a distribution. The standard deviation is a measure of how data points are dispersed in relation to the mean. Z-scores can be used with any distribution, but may be the most informative when applied to a symmetric, normal distribution (known as a bell curve or Gaussian distribution). 

If a z-score is positive, the observed data point is above the mean. If a z-score is negative, the data point is below the mean. If a z-score is 0, the data point is equal to the mean. For example, a z-score of +1.0 shows that the data point is one standard deviation above the mean, while a z-score of -1.0 shows the data point is one standard deviation below the mean.

 

How to Use a Z-table 

Reading a Z-Table

  1. Take the whole number before the decimal point and the first digit after the decimal point of the z-score, and find this value on the left-most column of the z-table.
  2. Take the second digit after the decimal point of the z-score, and find this value on the top row of the z-table.
  3. Go to the intersection of the values found in steps 1 and 2 — the number shown at this intersection is the z-score probability.

While we know that Zoe performed better than Mike because of her higher z-score, a z-table can tell you in what percentile each of the test takers are in. The following partial z-table — cut off to save space — can tell you the area underneath the curve to the left of our z-score. This is the probability.

table of standard normal probabilities for positive z scores
The table entries for z is the area under the standard normal curve to the left of z. (Red: Mike; Blue: Zoe). | Image: Michael Galarnyk

More on Data ScienceUnderstanding Boxplots

 

How to Find Zoe’s Z-Score Probability

To use the z-score table, start on the left side of the table and go down to 1.2. At the top of the table, go to 0.05. This corresponds to the value of 1.2 + .05 = 1.25. The value in the table is .8944 which is the probability. Roughly 89.44 percent of people scored worse than Zoe on the ACT.

 

How to Find Mike’s Z-Score Probability

Mike’s z-score was 1.0. To use the z-score table, start on the left side of the table and go down to 1.0. Now at the top of the table, go to 0.00. This corresponds to the value of 1.0 + .00 = 1.00. The value in the table is .8413, which is the probability. Roughly 84.13 percent of people scored worse than Mike on the SAT.

It is important to keep in mind that if you have a negative z-score, you can simply use a table that contains negative z-scores.

A tutorial on normal distribution and z-tables. | Video: Michael Galarnyk

 

How to Create a Z-Table 

This section will answer where the values in the z-table come from by going through the process of creating a z-score table. Please don’t worry if you don’t understand this section. It’s not important if you just want to know how to use a z-score table.

 

Finding the Probability Density Function

This is very similar to the 68–95–99.7 rule, but adapted for creating a z-table. Probability density functions (PDFs) are important to understand if you want to know where the values in a z-table come from. A PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable’s PDF over that range. That is, it’s given by the area under the density function but above the horizontal axis, and between the lowest and greatest values of the range.

This definition might not make much sense, so let’s clear it up by graphing the probability density function for a normal distribution. The equation below is the probability density function for a normal distribution

probability density function equation
Probability density function for a normal distribution. | Image: Michael Galarnyk

Let’s simplify it by assuming we have a mean (μ) of zero and a standard deviation (σ) of one (standard normal distribution).

probability density function equation for a standard normal distribution
Probability density function for a standard normal distribution. | Image: Michael Galarnyk

This can be graphed using any language, but I choose to graph it using Python.

# Import all libraries for this portion of the blog post
from scipy.integrate import quad
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline

x = np.linspace(-4, 4, num = 100)
constant = 1.0 / np.sqrt(2*np.pi)
pdf_normal_distribution = constant * np.exp((-x**2) / 2.0)
fig, ax = plt.subplots(figsize=(10, 5));
ax.plot(x, pdf_normal_distribution);
ax.set_ylim(0);
ax.set_title('Normal Distribution', size = 20);
ax.set_ylabel('Probability Density', size = 20);
normal distribution with probability density function graph
Normal distribution with probability density function graph. Notice how similar it looks to the image in the earlier z-score table. | Image: Michael Galarnyk

The graph above does not show you the probability of events but their probability density. To get the probability of an event within a given range, you need to integrate.

More on Data ScienceUnderstanding Train Test Split

 

Finding the Cumulative Distribution Function

Recall that the standard normal table entries are the area under the standard normal curve to the left of z (between negative infinity and z).

cumulative distribution function in a normal distribution graph
Remember that the table entries are the area under the standard normal curve to the left of z. | Image: Michael Galarnyk

To find the area, you need to integrate. Integrating the PDF gives you the cumulative distribution function (CDF), which is a function that maps values to their percentile rank in a distribution. The values in the table are calculated using the cumulative distribution function of a standard normal distribution with a mean of zero and a standard deviation of one. This can be denoted with the equation below.

z-table value equation with cumulative distribution function
Equation integrating the PDF with the cumulative density function. | Image: Michael Galarnyk

This is not an easy integral to calculate by hand, so I am going to use Python to calculate it. The code below calculates the probability for Zoe, who had a z-score of 1.25, and Mike, who had a z-score of 1.00.

def normalProbabilityDensity(x):
    constant = 1.0 / np.sqrt(2*np.pi)
    return(constant * np.exp((-x**2) / 2.0) )

zoe_percentile, _ = quad(normalProbabilityDensity, np.NINF, 1.25)
mike_percentile, _ = quad(normalProbabilityDensity, np.NINF, 1.00)
print('Zoe: ', zoe_percentile)
print('Mike: ', mike_percentile)
z-table code
Code to find Zoe and Mike’s z-score. | Screenshot: Michael Galarnyk 

As the code below shows, these calculations can be done to create a z-table.

code to create the z-table and the z-table
The standard normal table created in this tutorial. | Image: Michael Galarnyk

One important point to emphasize is that calculating this table from scratch when needed is inefficient, so we usually resort to using a standard normal table from a textbook or online source.

 

Why Are Z-Tables Important?

A z-table is able to present what percentage of data points in a distribution fall below a measured z-score. It can also be used to compare two different z-scores from different distributions, or to shed light on other possible data probabilities based on hypothetical z-scores. Z-tables can be helpful for comparing data points and averages in cases like test scores, health vitals or financial investments.

 

Frequently Asked Questions

How do you calculate the Z-score?

You can calculate a z-score using the following formula:

z = (x-μ) / σ

What are the two different Z-tables?

The two different types of z-tables include positive z-tables and negative z-tables. A positive z-table is used to find the probability of values falling below a positive z-score. A negative z-table is used to find the probability of values falling below a negative z-score.

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