How to Use a Z Table: Step-by-Step for Any Problem

Key takeaways

• A z table gives you an area under the normal curve (a probability).
• Most tables give area to the left of z. Some give area from 0 to z—you must check the title first.
• “Between two values” is usually bigger-left minus smaller-left.

How to use a z table is confusing at first because the table looks like a grid of random decimals. You’re not behind. You just need a clear method.

After this guide, you’ll know exactly what to read (row/column), what area you’re finding (left/right/between), and how to avoid the usual traps.

Quick answer

1. Round z to 2 decimals (like 1.23).
2. Find the row for the first part (1.2) and the column for the last digit (0.03).
3. Read the table value (that’s an area/probability).
4. Convert if needed:
   • Right tail = 1 − left tail.
   • Between = left(z₂) − left(z₁).

What a z table is (and what it is NOT)

Z-score (standard score): how many standard deviations (typical spread) a value is from the mean (average).

Standard normal distribution: a normal (bell-shaped) curve with mean 0 and standard deviation 1.

Z table (standard normal table): a table that matches each z-score to an area under the curve (probability).

Step 0 (non-negotiable): Check what your table shows

Some z tables show area to the LEFT of z. Others show area from 0 to z. The title usually tells you.

Here’s the quick guide:

If you’re not sure which one you have, look at the value for z = 0.00:

• If it’s about 0.5000, it’s likely left-of-z.
• If it’s about 0.0000, it’s likely 0-to-z.

How to read rows and columns

Most z tables are set up like this:

• Row = z to the ones and tenths place (like 1.2)
• Column = z to the hundredths place (like 0.03)
• Cell = the area/probability

Example: z = 1.23

• Row: 1.2
• Column: 0.03
• Cell: that’s your probability (depending on your table type).

Step-by-step method

Step 1: Figure out what area the problem wants

Common question types:

• Left-tail: “less than,” “below,” “at most” → $P(Z < z)$
• Right-tail: “greater than,” “above,” “at least” → $P(Z > z)$
• Between: “between a and b” → $P(z_1 < Z < z_2)$

Step 2: If needed, convert to a z-score

If your problem gives raw x-values, use:$$z=\frac{x-\mu}{\sigma}$$

• $x$ = your value
• $\mu$ = mean (average)
• $\sigma$ = standard deviation (how spread out the data is)

(If your problem already gives z, skip this.)

👉 Use our z-score calculator to check your answer.

Step 3: Look up z in the table

• Round z to two decimals.
• Find the row + column.
• Read the cell value.

Step 4: Convert the table value to what you need

This depends on the table type.

If your table is “area to the LEFT of z”:

• Left-tail: use it as-is.
• Right-tail: $P(Z>z)=1-P(Z\le z)$
• Between: $P(z_1<Z<z_2)=P(Z\le z_2)-P(Z\le z_1)$

If your table is “area from 0 to z”:

• Use symmetry and “half the curve is 0.5.”
• For $z>0$: left-tail $=0.5 + \text{table}(z)$
• For $z<0$: left-tail $=0.5 – \text{table}(|z|)$

Step 5: Quick self-check (do this every time)

• Probabilities must be between 0 and 1.
• If z is positive, the left-tail should be greater than 0.5 (for left-of-z tables).
• If z is negative, the left-tail should be less than 0.5.

👉If your calculator asks for x, mean (μ), and standard deviation (σ) instead of a z-score, use the Normal Distribution Calculator to compute the probability directly from the original values.

What to write on your paper (mini-checklist)

Copy this format:

1. Given: (write the z or compute it)
2. Wanted: left / right / between
3. Table lookup: row __ , column __ → value __
4. Convert: (if right-tail or between)
5. Answer: __ (final probability)

Worked examples (step-by-step)

Example 1 (very easy): Left of z

Find $P(Z \le 1.23)$ using a left-of-z table.

1. Wanted: left-tail.
2. Look up z = 1.23 → row 1.2, column 0.03.
3. Table value = your answer.

Why: Left-of-z tables already give $P(Z\le z)$

Example 2 (typical): Right of z

Find $P(Z > 1.23)$ using a left-of-z table.

1. Look up $P(Z \le 1.23)$ from the table.
2. Convert to right-tail:

$$P(Z>1.23)=1-P(Z\le 1.23)$$

Why: Left + right = 1 (the whole area).

Example 3 (exam-style): Between two z-values

Find $P(-0.50 < Z < 1.20)$ using a left-of-z table.

1. Get left area at the top value: $P(Z \le 1.20)$.
2. Get left area at the bottom value: $P(Z \le -0.50)$.
3. Subtract:

$$P(-0.50<Z<1.20)=P(Z\le1.20)-P(Z\le-0.50)$$

Why: “Between” is the area up to the bigger point minus the area up to the smaller point.

Tip for negative z: Use symmetry if your table only lists positive z values. The normal curve is symmetric.

Example 4 (common “trap”): “Area from 0 to z” table

Your table shows area from 0 to z (not left-of-z). You need $P(Z \le 1.23)$.

1. Look up the table value for 1.23 (this is area from 0 to 1.23).
2. Add 0.5:

$$P(Z\le1.23)=0.5+\text{table}(1.23)$$

Why: Half the curve is left of 0, so you add that half.

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Practice set (with answer key)

Directions

Assume you have a left-of-z table unless stated.

1. Find $P(Z \le 0.45)$.
2. Find $P(Z > 0.45)$.
3. Find $P(Z \le -1.10)$.
4. Find $P(Z > -1.10)$.
5. Find $P(-0.80 < Z < 0.80)$.
6. Find $P(1.25 < Z < 2.00)$.
7. Trap: A student says $P(Z>1.50)$ equals the table value at 1.50. What should they do instead?
8. If your table is 0-to-z, find $P(Z \le -0.30)$.

Answer key (short)

1. Table (0.45)
2. $1-$Table(0.45)
3. Use symmetry or negative row (depends on your table).
4. $1-$Answer to #3
5. Table(0.80) − Table(-0.80)
6. Table(2.00) − Table(1.25)
7. Use $1-$Table(1.50)
8. $0.5-$Table(0.30) (because it’s a 0-to-z table)

Two short solutions (so you can check your thinking):

• #2: Right tail means “everything else,” so subtract the left area from 1.
• #5: “Between” means bigger-left minus smaller-left.

Common mistakes

If you get stuck…

• “I don’t know what the question wants.”
  Circle the words: below/less than (left), above/greater than (right), between (subtract).
• “My answer is negative.”
  A probability cannot be negative. Re-check: did you subtract in the wrong order?
• “I keep mixing up left vs right.”
  Write this on your paper: Right = 1 − Left (for left-of-z tables).
• “My calculator gives a different value.”
  Small differences can happen because tables round. If you need a precise value, use a calculator tool.
• “I think I need a tool, not a table.”
  Use the Normal Distribution Calculator to confirm your result, especially for tricky “between” problems.

Next steps

• z table — Use this when your teacher gives you a printed table.
• standard normal distribution table — Another name teachers use for the same table.
• standard normal distribution — Helps you understand why z = 0 is the center and why symmetry works.
• Normal Distribution Calculator — Fast checks for left, right, and between probabilities.
• Best High-Quality Notebooks for Math Notes — If you lose steps easily, a good layout for “Given / Wanted / Work / Answer” can help.

How we know

• This uses the standard classroom idea that probability under a normal curve is an area, often found using a z-table.
• Definitions of z-scores match reputable learning references.
• The “right tail = 1 − left tail” method follows how cumulative probability tables work.
• The “between = bigger-left minus smaller-left” method is a standard approach taught in normal-distribution lessons.
• The “0-to-z” table conversion uses symmetry, as described in a statistics standards handbook.

Use this the right way

Use this guide to learn the method and show clear work. If this is for a project or paper, cite any tools or references you use. Don’t copy answers without understanding the steps.

Study tools that can help

• If you get distracted easily: a simple timer (25 minutes on, 5 minutes off) can help you finish one set of problems.
• If your steps get messy: a notebook with clear lines or grid spacing can keep your row/column lookups neat.
• If you want a fast accuracy check: use the online calculator after you do it by hand once.
• If you want better math notes: Best High-Quality Notebooks for Math Notes

References

• National Institute of Standards and Technology. (n.d.). Cumulative distribution function of the standard normal distribution. NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm
• OpenStax. (2023). Introductory Statistics 2e: 6.1 The standard normal distribution. https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution
• OpenStax. (2023). Introductory Statistics 2e: 6.2 Using the normal distribution. https://openstax.org/books/introductory-statistics-2e/pages/6-2-using-the-normal-distribution