Adapting Accessible Z-Tables: Math Problems and Low Vision

In introductory statistics and quantitative reasoning classes, students often learn how to calculate z-scores and use a standard normal table (also known as a z-table) to find the probability of a value falling above or below that point. Instructors typically provide a z-table for students on a formula sheet/equations sheet or use a z-table from a textbook, which offer inconsistent accessibility and can be difficult or impossible for students with visual impairments or other print disabilities to access with assistive technology.

For students and/or instructors searching for an accessible z-table, I designed my own accessible standard normal probability table files that are optimized for low vision and non-visual access. These accessible z-tables can be enlarged with screen magnification or browser zoom, accessed with text-to-speech or a screen reader, read with a braille display, and/or customized to support student access needs and preferences for reading information. Most importantly, the z-tables can be downloaded for offline use and provide equitable access to z-score and probability data, so students aren’t limited to reading from a PDF.

Based on my experiences as a legally blind statistics student/statistics student with low vision, here are several options for adapting accessible z-table and accessible standard normal table templates for use with assistive technology. All files are free to download and use and can be implemented as part of alternate format accommodations (sometimes called alternative format accommodations) or as an embedded resource available to all students.

Why are z-tables hard to see?

Also known as a standard normal table, the z-table is used to find the probability of a value falling above or below a given point. A typical z-table consists of 11 columns and anywhere between 35 and 70 rows, with each cell containing decimals. Values are placed in close proximity and in small print sizes to fit on a single page. This dense table can be challenging to read for students who use large print or that have other visual processing issues, as it requires careful visual tracking, visual search, and scanning skills.

Another barrier to access for z-tables is that they are often shared as PDF documents in a low-resolution format that contains scanned images or blurry/pixelated text, which can be difficult to read even without a visual impairment – highlighting text as an anchor tool is not possible on a scanned image that does not contain actual text. The non-responsive nature of a PDF makes it even more difficult for students to enlarge the font size or adjust the spacing of numbers within the table, not to mention keep track of specific rows or columns.

If a physical copy is created from a low-resolution digital file, attempting to enlarge the text will only intensify the pixelated or blurred appearance, making text even more difficult to access. Using display scaling to enlarge a physical copy or low-resolution file to increase the font size also creates a frustrating experience, as the resulting table will likely take up several dozen pages or have information cut off at the edge due to the large size. While using an original file can make it easier to create high resolution copies of an accessible z-table, students who use assistive technologies may prefer an accessible electronic format.

Related links

• File Formats For Low Vision and Print Disabilities
• How To Create An Accessible Formula Sheet
• Adapting Digital Equations: Math Problems and Low Vision
• How I Take Math Tests With Double Vision
• Five Apps I Use In Statistics Classes As A Low Vision Student

Export z-table as a csv or spreadsheet file

One option for adapting an accessible z-table is to export the values as a CSV file, which is a plain text file that contains a list of data separated by commas. This can be converted into a spreadsheet in Microsoft Excel or Google Sheets, used with statistical programming languages like Python or R to locate values, imported into database software, or used to create a table. Most z-tables display values rounded to five decimal places, though formula sheets for AP Statistics or high school statistics courses display values rounded to four decimal places.

Linked below are two free accessible z-table csv files, which are free to download and modify. The Z-Table file is rounded to five decimal places with a range of -.3.99 to 3.99 and contains headers. The Standard Normal Probabilities table (z-table) is rounded to four decimal places with a range of -3.49 to 3.49 and contains headers; this matches the Standard Normal Probabilities table listed on the AP Statistics formula sheet

Related links

• Standard Normal Probabilities: AP Statistics (csv)
• Z Table: Statistics (csv)

Use a line guide or focus window to block out text

A focus window or line guide blocks out surrounding text on a page or displays a certain number of lines at a time so the reader can focus on one section at a time. There are a few different ways to use this strategy for reading a z-table, including:

• Using an index card as a line guide to focus on a specific row or column. This can be helpful for positioning a magnifying glass or video magnifier too
• Purchasing or making a typoscope, which is typically made of black plastic and comes in multiple sizes. This is designed for reading a small section of text, such as a single line, or for use as a signature guide, so it would primarily be used for reading a single value instead of as a tool for locating rows/columns
• Positioning two pages of construction paper, one in a vertical orientation and one in a  horizontal orientation. The student can move the horizontal paper down the page to view rows, and use the vertical paper to block out surrounding columns.

Another strategy I have used when reading an accessible z-table is a line focus window, which displays between 1 and 5 lines of text on the screen at a time. When reading a z-table in Microsoft Word, use the line focus tool in Immersive Reader (enabled in the View tab) to view one line of the table at a time, which can help block out distracting text. To display a z-table in Microsoft Word, users can either create a table using the CSV files linked in this post, or copy and paste the HTML tables linked in another section.

Related links

• Adapting Page Layouts: Math Problems and Low Vision
• Digital Rulers For Dysgraphia and Low Vision
• Highlighting Information With Low Vision
• Designing Accessible Documents With Microsoft Word

Transpose rows and columns

In a traditional z-table with the example value of 1.44, the tenths value of the z-score is displayed in the leftmost column (1.4), while the top row of the table corresponds to the hundredths value of the z-score (0.04). One option for making a z-table easier to read for students with low vision is to transpose the rows and columns sot that the first part of the z-score is shown in the top row (1.4), while the second part of the z-score is displayed in the leftmost column (0.04), creating a table with fewer rows but more columns.

This layout could be helpful for students who want to use a line guide or typoscope to block out surrounding text, or for filtering data in a spreadsheet program like Excel or Sheets. To create a large print copy for printing, open the CSV in Excel and auto fit the width of the cell (using the keyboard shortcut Alt+H, followed by pressing O and I), then adjust the font size as needed. Users can also convert the CSV file to a table, which would range from A1:BS11 for the AP Statistics z-table, and A1:CC11 for the statistics z-table.

Linked below are two free accessible z-table csv files, which are free to download and modify; the layout has been transposed so the first two digits of the z-score are columns and the last digits are rows. The Z-Table file is rounded to five decimal places with a range of -.3.99 to 3.99 and contains headers. The Standard Normal Probabilities table (z-table) is rounded to four decimal places with a range of -3.49 to 3.49 and contains headers; this matches the Standard Normal Probabilities table listed on the AP Statistics formula sheet

Related links

• Standard Normal Probabilities: AP Statistics (csv, transposed)
• Z Table: Statistics (csv, transposed)

Two-column accessible Z-table for assistive technology

Another strategy to reduce the visual complexity of a z-table is to display the table values as a sorted list. Instead of having columns labeled 0.01, 0.02, and so on, this design transforms the multi-column layout into two columns; the first column has the corresponding z-score (1.51, 1.52, etc.), and the second column has the probability. This can be helpful for students who are still learning to access tables with a screen reader or braille display and need a more streamlined table, as well as students who are impacted by dense visual layouts due to cortical/cerebral visual impairment (CVI), neurological visual impairment, or that have other access needs related to visual processing issues.

Linked below are two free accessible z-table csv files, which are free to download and modify; these are different from the typical z-table layout with multiple columns, but have the same values. The Z-Table file is rounded to five decimal places with a range of -.3.99 to 3.99 and contains headers. The Standard Normal Probabilities table (z-table) is rounded to four decimal places with a range of -3.49 to 3.49 and contains headers; this matches the Standard Normal Probabilities table listed on the AP Statistics formula sheet

Related links

• Standard Normal Probabilities: AP Statistics two-column design (csv)
• Two-column Z-Table: Statistics (csv)

Access Z-table with HTML

HTML formats are designed to be read in the web browser and offer a responsive layout with options for using browser zoom, enlarging the text size, adjusting line spacing, and other digital text preferences. This can be easier than reading a PDF document, which has fewer options for magnification or adjusting the size or appearance of text. Another bonus is that the user can search for specific rows using the Ctrl + F keyboard shortcut, which can help with locating values.

I created a basic HTML table for each of the accessible z-tables in this post, which can be customized using the Developer Tools menu in a web browser (keyboard shortcut Ctrl + Shift + I). Once the Developer Tools menu is open, select the text box for the body section of the page to customize characteristics such as:

• Font (font-family: arial provides an Arial font)
• Font size (font-size: 24pt is 24 pt font)
• Line height (line-height: 2 is double spaced)

To use this HTML table efficiently with a screen reader, use the Ctrl + F keyboard shortcut to search for the first part of the z-score, including the negative sign. This will bring the cursor to the row of interest, which makes it easier to locate the probability in the table. The two-column z-table layouts are also included as HTML output below.

To save the HTML table for offline use, save the webpage (or use Ctrl + S) as a complete HTML file. The file can be opened in the web browser, but does not require an internet connection. If desired, users can also copy and paste the HTML source code (Ctrl + U in Microsoft Edge) for further modification.

Related links

• CSS Accessibility Tips for Low Vision
• Standard Normal Probabilities: AP Statistics (HTML)
• Z-Table: Statistics (HTML)
• Z-Table: Statistics (HTML, transposed)
• Standard Normal Table: AP Statistics (HTML, transposed)
• Standard Normal Probabilities: AP Statistics (HTML, two-column)
• Z-Table: Statistics (HTML, two-column)

More resources on accessible statistics for low vision

• There is no shortage of online z-tables and z-score calculators, but these tools are usually prohibited during tests and exams because they are considered outside resources. One resource I like is ZTable.io – Z-Score Table (Standard Normal Table), which can be useful for homework or lessons.
• Learning R programming? Check out Tips For Teaching R Programming To A Low Vision Student and Pre-Teaching Programming Languages To Visually Impaired Students
• Interested in more introductory statistics resources? Read Implementing UDL For Data Science Lessons: Free Resource List and Free Math Websites For Low Vision Students: Instructional Technology
• Want to read more about low vision math accessibility? Check out my Math Problems for Low Vision Series by browsing the tag Math Problems for Accessibility Archives | Veronica With Four Eyes (veroniiiica.com)

Published September 13, 2024. Updated September 2026