Z score: How to Calculate and Interpret (Examples)

Key Takeaways
A z score tells how many standard deviations (typical spread) a value is from the mean.
Use $z=\frac{x-\text{mean}}{\text{SD}}$ . Keep the sign (+/−).
z-scores help you compare values from different groups.

Your z score is the first thing teachers want when they say “standardize this value.” You’ll learn one simple method and see worked examples. You’re not behind. You just need a clear method.

Quick answer

A z score tells how far a value is from the mean, measured in standard deviations.
Use this formula: $\;z=\frac{x-\mu}{\sigma}\;$ (population) or $\;z=\frac{x-\bar{x}}{s}\;$ (sample).
Step 1: subtract the mean. Step 2: divide by the standard deviation.
Positive z means “above the mean.” Negative z means “below the mean.”

What is a z score?

A z score (standard score) is the number of standard deviations a value is above or below the mean.

Key words

Mean (average value).
Standard deviation (how spread out the data is).
Standardize (convert to a common scale).
Normal distribution (bell-shaped pattern).
Standard normal (a normal curve with mean 0 and SD 1).

The z score formula (and what symbols mean)

Symbol	Means	Plain-English meaning
$x$	data value	the score you’re converting
$\mu$	population mean	true average (whole group)
$\sigma$	population SD	true spread (whole group)
$\bar{x}$	sample mean	average of your sample
$s$	sample SD	spread of your sample
$z$	z score	“how many SDs away”

Most homework uses the sample version: $\;z=\frac{x-\bar{x}}{s}$ .

How to read z score

Positive z score (+) → the value is above the average
Example: z = +1 means the score is 1 standard deviation higher than the average.
Zero (0) → the value is exactly at the average
Negative z score (−) → the value is below the average
Example: z = −2 means the score is 2 standard deviations lower than the average.

Why we use z scores

To compare fairly. You can compare scores from different tests or groups.
(Example: 85 in Math vs. 90 in English — z scores show which is better relative to each class.)
To see how unusual a value is.
A z near 0 = common.
A z near ±2 or ±3 = rare or extreme.
To understand position in the data.
It shows if you are below average, average, or above average — and by how much.

In short: a z score answers, “Where does this value stand compared to everyone else?”

Bell curve with z-scores labels with meaning

How to calculate a z score (step-by-step)

Step-by-step method

Write the formula you will use.
- Population: $z=\frac{x-\mu}{\sigma}$
- Sample: $z=\frac{x-\bar{x}}{s}$
Plug in numbers for $x$ x, mean, and SD.
Subtract: $x-\text{mean}$
Divide by SD.
Round the way your teacher wants (often 2 decimals).

What to write on your paper

✅ Formula used (population or sample)
✅ Subtraction line: $x-\text{mean}=\_\_\_$
✅ Divide by the standard deviation (s or σ).
✅ Final z score with sign (+/−)

Quick self-check (to verify your answer)

If $x$ is above the mean, z should be positive.
If $x$ is below the mean, z should be negative.
If $x=$ mean, then $z=0$ .

Worked examples

Example 1 (very easy)

Mean $=50$ , SD $=10$ , value $x=60$ .

$z=\frac{x-\text{mean}}{\text{SD}}$
$z=\frac{60-50}{10}$
$z=\frac{10}{10}=1$
Answer: $z=1$
Why: 60 is 1 SD above 50.

Example 2 (typical homework)

A quiz score is $x=72$ . Class mean $\bar{x}=65$ . Class SD $s=7$

$z=\frac{x-\bar{x}}{s}$
$z=\frac{72-65}{7}$
$z=\frac{7}{7}=1$
Answer: $z=1$
Meaning: 72 is 1 SD above the class average.

Example 3 (negative z score)

Heights: mean $=160$ , SD $=8$ . You are $x=148$

$z=\frac{148-160}{8}$
$z=\frac{-12}{8}=-1.5$
Answer: $z=-1.5$
Why: 148 is below the mean, so z must be negative.

Example 4 (common “trick”: choose the right SD)

A data set has mean $=20$ =20.

If SD $=4$ =4 and $x=28$ x=28, then $z=\frac{28-20}{4}=2$ z=428−20=2.
If SD $=8$ =8 and $x=28$ x=28, then $z=\frac{28-20}{8}=1$ z=828−20=1.
Point: Same $x$ x, different SD → different z. SD controls the scale.

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Practice set (with answer key)

Directions: Find the z score. Round to 2 decimals.

Mean=30, SD=5, $x=40$
Mean=30, SD=5, $x=25$
$\bar{x}=12$ , $s=3$ , $x=18$
$\bar{x}=80$ , $s=10$ , $x=80$ (trap: don’t overthink)
Mean=200, SD=20, $x=170$
$\bar{x}=50$ , $s=4$ , $x=57$
Mean=10, SD=2, $x=6$
Mean=100, SD=25, $x=112.5$

Answer key

$z=2.00$
$z=-1.00$
$z=2.00$
$z=0.00$
$z=-1.50$
$z=1.75$
$z=-2.00$
$z=0.50$

Short solutions (2 items only):

#5: $z=\frac{170-200}{20}=\frac{-30}{20}=-1.50$
#6: $z=\frac{57-50}{4}=\frac{7}{4}=1.75$

Common mistakes

Mistake	Why it happens	Do this instead
Forgetting the negative sign	You subtract in the wrong order	Always do $x-\text{mean}$ , not mean − x.
Using the wrong SD (sample vs population)	Both formulas look similar	If the problem gives $\bar{x}$ and $s$ , use the sample formula.
Dividing before subtracting	Rushing	Do it in order: subtract first, then divide.
Thinking z must be between −1 and 1	Confusing z with probability	z can be any real number (like 2.4, −3.1).
Rounding too early	Small rounding errors grow	Keep extra decimals, round at the end.
Mixing units	Mean/SD in one unit, x in another	Match units first (cm with cm, minutes with minutes).

If you get stuck…

“I don’t know what formula to use.”
If you see $\mu,\sigma$ → population. If you see $\bar{x},s$ → sample.
“My answer is negative but it shouldn’t be.”
Check: is $x$ below the mean? If yes, negative is correct.
“I keep mixing up steps.”
Say it out loud: “Subtract the mean. Divide by SD.”
“My calculator gives a different value.”
Check parentheses: type $(x-\text{mean})/\text{SD}$ .
“My z score is huge.”
Re-check SD. A tiny SD can make z look big.
When to review prerequisites:
If “standard deviation” feels unclear, review that first (see Next steps).
When to ask for help:
If your class uses a special rounding rule or a special SD formula, ask your teacher or tutor. That’s normal.

Next steps

Z-score calculator — check homework fast and reduce arithmetic errors.
z score to percentile — learn how to turn z into a percentile rank (great for normal curve questions).
percentile to z score — go the other direction (common in exam problems).
standard deviation — strengthen the skill that controls the whole z-score scale.

How we know

z-score meaning and sign rules match standard intro statistics explanations (how many SDs from the mean).
Both sample and population forms are shown the way university stats lessons present them.
Steps were checked against worked examples and standard interpretations (positive above mean, negative below).
Definitions align with widely used open textbooks and education platforms.

Use this the right way

Use z scores to understand your work, not to copy answers. Try the steps first, then check with a calculator. If you use z scores in a research paper, cite your sources and explain what your numbers mean in words.

Study tools that can help

If you make sign mistakes often: use the Z-score calculator after you do it by hand once.
If you get overwhelmed: use a simple homework checklist (one page) and tick each step.
If you’re time-crunched: do 3 practice items daily (not 30 in one night). Consistency helps more than cramming (no promises—just a better study habit).

References

OpenStax. (2020). 6.1 The standard normal distribution. In Statistics. Rice University. https://openstax.org/books/statistics/pages/6-1-the-standard-normal-distribution

National Institute of Standards and Technology. (2012). ISO 13528 ZSCORE (Dataplot reference manual). https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/zscore.htm

Z score: How to Calculate and Interpret (Examples)

Quick answer