Z Score vs Percentile: Differences and How to Convert

Key takeaways

• A z-score tells how many standard deviations a value is from the mean.
• A percentile tells what percent of values are at or below a score.
• You can convert z-score to percentile by finding the area to the left on the standard normal curve. You can convert percentile to z-score by working backward from that left-tail area.

z score vs percentile can feel confusing at first because both describe where a score stands. But they do not say the same thing.

This guide shows the difference, how to convert each one, and how to solve problems step by step. You’re not behind. You just need a clear method.

Quick answer

A z-score tells distance from the mean in standard deviation units. A percentile tells rank, or the percent of values below a score.

To convert z-score to percentile:

1. Find the z-score.
2. Use a z-table or calculator.
3. Read the area to the left.
4. Change it to a percent.

To convert percentile to z-score:

1. Change the percentile to a decimal.
2. Find that left-tail area in the z-table, or use inverse normal.
3. Read the matching z-score.

What is a z-score?

A z-score (standardized score: how far a value is from the mean, measured in standard deviations) tells position in a distribution. PSU defines a z-score as the distance between an individual score and the mean in standard deviation units.

Formula

$$z = \frac{x-\mu}{\sigma}$$

or, for sample data,$$z = \frac{x-\bar{x}}{s}$$

Symbols:

A standard deviation is a measure of spread. It tells how far values usually sit from the mean.

How to interpret a z-score

• z = 0 means the score is exactly at the mean.
• z = 1 means the score is 1 standard deviation above the mean.
• z = -2 means the score is 2 standard deviations below the mean.

👉Check out our Z-table generator if you want a faster way to find z-scores and associated probabilities.

What is a percentile?

A percentile (a cut point that divides ordered data into hundredths) tells rank. If a student is in the 80th percentile, that means the student scored the same as or higher than about 80% of the group. It does not mean the student got 80% correct.

Percentile interpretation

• 50th percentile = median level
• 25th percentile = lower quarter
• 90th percentile = very high relative standing

Z-score vs percentile

A z-score is more about how far. A percentile is more about how many below. Under a normal distribution, the two are linked by the standard normal curve.

When are z-score and percentile connected?

They connect most cleanly when you are working with the standard normal distribution (bell-shaped curve with mean 0 and standard deviation 1). In that setting, each z-score matches one left-tail area, and that area is the percentile in decimal form.

For example:

• $z=0$ matches about 0.5000, so that is the 50th percentile
• $z=1.00$ matches about 0.8413, so that is about the 84th percentile
• $z=-1.00$ matches about 0.1587, so that is about the 16th percentile

How to convert z score to percentile

Method

1. Start with the z-score.
2. Go to the z-table.
3. Find the area to the left of that z-score.
4. Turn the decimal into a percent.

What to write on your paper

• Given: $z=\_\_\_$
• Look up $P(Z \le z)$
• Convert decimal to percent
• Write the interpretation in words

Quick self-check

• If your z-score is positive, your percentile should be above 50.
• If your z-score is negative, your percentile should be below 50.

How to convert percentile to z score

Method

1. Change the percentile to a decimal.
   • Example: 90th percentile = 0.90
2. Find that area inside the z-table.
3. Read the matching z-score from the row and column.
4. If needed, use that z-score in another formula to find the raw score.

What to write on your paper

• Given percentile = $\_\_\_\%$
• Decimal area = $\_\_\_$
• Table value closest to that area = $\_\_\_$
• So $z \approx \_\_\_$

Quick self-check

• A percentile above 50 should give a positive z-score.
• A percentile below 50 should give a negative z-score.

Worked examples

Example 1

Find the percentile for $z=0.50$.

Step 1: Look up 0.50 in the z-table.
The area to the left is about 0.6915.

Step 2: Change to a percent.
$0.6915 = 69.15\%$

Answer: A z-score of 0.50 is about the 69th percentile.

Why: About 69% of values are below a score that is 0.5 standard deviations above the mean.

Example 2: Typical homework level

Find the percentile for $z=-1.20$.

Step 1: Look up $-1.20$ in the z-table.
The area to the left is about 0.1151. This is close to the table values shown around $-1.2$.

Step 2: Change to a percent.
$0.1151 = 11.51\%$

Answer: $z=-1.20$ is about the 12th percentile.

Why: The score is below the mean, so only a small percent of scores fall below it.

Example 3: Convert percentile to z-score

Find the z-score for the 90th percentile.

Step 1: Change 90% to a decimal.
$0.90$

Step 2: Find the closest left-tail area in the z-table.
PSU’s example shows the 10th percentile is about $z=-1.28$, so by symmetry the 90th percentile is about $z=1.28$. OpenStax also shows the 90th percentile uses a z-score of about 1.28.

Answer: The 90th percentile is about z=1.28.

Example 4: Exam-style raw score problem

A test has mean $70$ and standard deviation $10$. What score is at the 84th percentile?

Step 1: Convert percentile to z-score.
84th percentile is close to area 0.8413, so $z \approx 1.00$z.

Step 2: Use the z-score formula backward.$$x = \mu + z\sigma$$$$x = 70 + (1)(10) = 80$$

Answer: The score is 80.

Why: A score one standard deviation above the mean is around the 84th percentile in a normal distribution.

Beginner-safe method for homework and quiz problems

If the problem gives a raw score

1. Write the mean and standard deviation.
2. Compute the z-score.
3. Use the z-table or calculator.
4. Read the area to the left.
5. Write the percentile.

If the problem gives a percentile

1. Change the percentile to a decimal.
2. Find the matching z-score.
3. If needed, use $x=\mu+z\sigma$.
4. Check if the final answer makes sense.

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Practice set

Directions

Answer the following. Assume the z-table gives area to the left.

1. What percentile is $z=0$?
2. What percentile is $z=1.00$?
3. What percentile is $z=-0.50$?
4. Find the z-score for the 50th percentile.
5. Find the z-score for the 10th percentile.
6. A class has mean 75 and standard deviation 8. What percentile is a score of 83?
7. A class has mean 60 and standard deviation 5. What score is at the 90th percentile?
8. A class has mean 100 and standard deviation 15. What score is at the 16th percentile?
9. Common trap: A student is at the 80th percentile. Does that mean the student got 80% on the test?
10. If $z=-2$, is the percentile above or below 50?

Answer key

1. 50th percentile
2. About 84th percentile
3. About 31st percentile
4. z=0
5. z≈−1.28
6. About 84th percentile
   Short solution: $z=(83-75)/8=1$, so percentile $\approx 84\%$.
7. About 66.4
   Short solution: 90th percentile gives $z \approx 1.28$. Then $x=60+1.28(5)=66.4$.
8. About 85
   Since the 16th percentile is near $z=-1$, $x=100+(-1)(15)=85$.
9. No
10. Below 50

Common mistakes

If you get stuck…

“I don’t know what formula to use.”
Use $z=\frac{x-\mu}{\sigma}$​ when you have a score and want position. Use the table backward when you have a percentile and want z.

“My answer is negative but it shouldn’t be.”
A negative z-score is often fine. It just means the score is below the mean.

“I keep mixing up z-score and percentile.”
Ask: “Do I need distance from the mean, or rank?” Distance means z-score. Rank means percentile.

“My calculator gives a different value.”
Check whether your tool gives:

• area to the left
• area to the right
• area between two values
  That changes the answer format.

“I do not understand the z-table.”
Review how rows, columns, and left-tail areas work before doing more problems.

When to review basics:
If the words mean, standard deviation, or normal distribution still feel fuzzy, review those first.

When to ask for help:**
If you keep getting the sign wrong, or your class uses a different type of z-table, ask your teacher or tutor to show one example with your class notes.

Next steps

• Review How to Use a Z-Table to learn how to read rows, columns, and left-tail areas clearly.
• Study Standard Normal Distribution if you want the bigger picture behind z-scores and bell curves.
• Try the Z-Score Calculator to check homework answers and see each value faster.
• Use the Normal Distribution Calculator when your problem gives a mean and standard deviation and asks for area or cutoff values.
• Practice with the Z-Table Tool if you want a faster way to match z-scores and probabilities.

How we know

• This guide uses standard classroom definitions for z-scores and percentiles from university and open textbook resources.
• The conversion method matches the standard normal table approach used in statistics teaching.
• Worked examples were checked against published normal-table examples, including the 10th and 90th percentile pattern.
• Explanations avoid claims that one method is always “better.” Each one answers a different question.
• Common mistakes listed here are typical learning errors in coursework, especially mixing up left-tail area, rank, and raw score.

Use this the right way

Use this guide to learn the idea, not just copy answers. Practice the steps on your own paper. If you use this for classwork or a write-up, make sure your final work matches your teacher’s method and instructions. If you include outside help in research or reports, cite your sources and avoid plagiarism.

Sources

Penn State Eberly College of Science. (n.d.). 2.2.8 – z-scores | STAT 200. https://online.stat.psu.edu/stat200/lesson/2/2.2/2.2.8

OpenStax. (2020). Statistics 6.2: Using the normal distribution. https://openstax.org/books/statistics/pages/6-2-using-the-normal-distribution

OpenStax. (2023). Contemporary mathematics 8.6: The normal distribution. https://openstax.org/books/contemporary-mathematics/pages/8-6-the-normal-distribution