What is a Z-Score?
A z score is also called a standard score. It helps you compare different sets of data fairly. For example, you can use it to compare a student's score on a math test to their score on a history test, even if the tests have totally different averages and grading scales. It converts raw data into a universal scale based on the normal distribution.
Z-Score Formula
The formula to calculate a z score is simple.
z = (x - μ) / σ
Here is what each letter means in that formula:
- x is your raw score. This is the specific data point you are looking at.
- μ (mu) is the mean. This is the average of your entire data set.
- σ (sigma) is the standard deviation. This tells you how spread out your data is.
You subtract the mean from your raw score. Then, you divide that answer by the standard deviation. If you are working with a smaller sample instead of a whole population, you simply use the average of your sample. You can easily find this using our sample mean calculator.
How to Calculate a Z-Score
If you want to do the math by hand, just follow these easy steps.
- Find the mean of your data set.
- Find the standard deviation of your data set.
- Take your specific raw score and subtract the mean from it.
- Take that result and divide it by the standard deviation.
- The final number is your z score. You can then look up this final number on a standard normal distribution table to find the exact probability for that score.
Worked example
A commuter logs a 41-minute trip. City records show a mean commute of 28 minutes with a standard deviation of 5 minutes.
z = (41 − 28) / 5 = 13 / 5 = 2.60
A z score of 2.60 means this trip is 2.60 standard deviations above the city average. About 99.5% of commutes in that distribution are shorter. Enter 41, 28, and 5 in the calculator above to match this result.
Positive vs. Negative Z-Scores
The plus or minus sign in front of your z score tells you the direction of your data point.
- Positive z score: Your raw score is above the average.
- Negative z score: Your raw score is below the average.
- Zero z score: Your raw score is exactly equal to the average.
Understanding the direction helps you quickly see where a person or item falls within a group.
Z-Score and the Standard Normal Distribution
When you calculate z scores for a whole data set, they form a standard normal distribution. This looks like a classic bell curve. The very middle of the curve is zero. Most of the data falls between -1 and +1.
How to Read a Z-Table
A z table shows you the area under the bell curve to the left of your z score. This area represents a percentage or probability. You use the left column of the table to find the first two digits of your z score. You use the top row to find the second decimal number. Where those two points meet is your probability. Our calculator does this lookup for you automatically. If you are running a hypothesis test, you can simply compare this probability to your significance level to make your final decision.
Frequently Asked Questions
What is a good z score?
A z score between -1 and 1 is considered typical and normal. A score lower than -2 or higher than 2 is considered unusual. A score lower than -3 or higher than 3 is considered extremely rare.
Can a z score be greater than 3?
Yes, a z score can be any number from negative infinity to positive infinity. However, scores beyond 3 or negative 3 are very rare in a normal distribution.
What does a z score of 2 mean?
A z score of 2 means your data point is exactly 2 standard deviations above the mean. It also means the score is higher than about 97.5% of all the other data points in the set.
How do you convert a z score to a percentage?
You convert it by using a z table to find the area under the curve. You then multiply that decimal number by 100. For example, an area of 0.8413 equals 84.13%.
Is a higher z score always better?
Not always. It depends entirely on what you are measuring. If you are measuring test scores or sales, a higher z score is better. If you are measuring error rates or race times, a lower z score is better.