How to Determine a Z-Score- Step-by-Step Guide
What Is a Z-Score and Why You Need to Know It
A Z-score tells you how many standard deviations a data point sits from the mean of a dataset. That's it. Nothing fancy.
You use it to compare values from different populations, spot outliers, or standardize scores for analysis. If you're working with data, you'll eventually need this.
The Z-Score Formula
Here's the only equation you need:
Z = (X - μ) / σ
Where:
- X is the value you're analyzing
- μ is the population mean
- σ is the population standard deviation
That's the whole thing. Three variables, one division operation.
Step-by-Step: How to Calculate a Z-Score
Step 1: Find Your Mean
Add up all your values and divide by how many values you have.
Example: Your dataset is 4, 8, 12, 16, 20. Sum = 60. Mean = 60 / 5 = 12
Step 2: Calculate the Standard Deviation
For the population standard deviation:
- Subtract the mean from each value to get deviations
- Square each deviation
- Find the average of those squared deviations
- Take the square root
Using our example (mean = 12):
- Deviations: -8, -4, 0, 4, 8
- Squared: 64, 16, 0, 16, 64
- Average: 160 / 5 = 32
- Square root: σ ≈ 5.66
Step 3: Plug Into the Formula
Say you want the Z-score for X = 20.
Z = (20 - 12) / 5.66 = 8 / 5.66 = 1.41
The value 20 is 1.41 standard deviations above the mean.
What Your Z-Score Means
A Z-score of 0 means the value sits exactly at the mean.
A positive Z-score means the value is above the mean. A negative one means it's below.
The further from zero, the more unusual the value. Most statistical software flags values with |Z| greater than 2 or 3 as potential outliers.
Using Sample Data Instead of Population Data
Sometimes you don't have the entire population. You have a sample.
Use s (sample standard deviation) instead of σ. The calculation is almost identical—except you divide by n-1 instead of n when finding the average of squared deviations.
The formula becomes:
Z = (X - x̄) / s
Where x̄ is your sample mean.
Z-Score Reference Table
Here's what common Z-scores represent in terms of percentage of data below that point:
| Z-Score | % Below This Value | Interpretation |
|---|---|---|
| -3.0 | 0.13% | Extreme low outlier |
| -2.0 | 2.28% | Low outlier threshold |
| -1.0 | 15.87% | Below average |
| 0 | 50% | Exactly at the mean |
| 1.0 | 84.13% | Above average |
| 2.0 | 97.72% | High outlier threshold |
| 3.0 | 99.87% | Extreme high outlier |
About 68% of data falls between Z = -1 and Z = 1. About 95% falls between -2 and 2. About 99.7% falls between -3 and 3. This is the empirical rule.
Quick Practical Example: Test Scores
You scored 85 on a test. The class average was 72 with a standard deviation of 8.
Z = (85 - 72) / 8 = 13 / 8 = 1.625
You scored higher than about 95% of the class. That's a solid performance.
Now compare: Your friend scored 90 on a different test. Class average was 85, standard deviation was 3.
Z = (90 - 85) / 3 = 5 / 3 = 1.67
Your friend's relative performance was actually slightly better, even though the absolute score was lower. This is why Z-scores are useful for comparison.
How to Get Started
- Collect your data — Know whether you're working with a full population or a sample
- Calculate the mean — Sum divided by count
- Find the standard deviation — Use population formula (σ) or sample formula (s) depending on your data
- Apply the formula — Subtract mean, divide by standard deviation
- Interpret — Check the table to see how unusual your value is
For large datasets, use a calculator or spreadsheet. Excel has =STDEV.P() for population and =STDEV.S() for samples. Google Sheets has the same functions.
When Z-Scores Don't Apply
Z-scores assume your data is roughly normally distributed. If your data is heavily skewed or has multiple peaks, Z-scores will give you misleading results.
For non-normal data, consider rank-based methods or transformations before standardizing.