Z-score Calculator
PrintIn statistics, a z-score (also called a standard score, normal score, or standard normal value) is a dimensionless quantity that describes a data point’s relationship to the mean of a group of values. A z-score measures the signed, fractional number of standard deviations a raw score lies above or below the mean. If a z-score is positive, the data point lies above the mean; if negative, it lies below the mean; and if it is exactly zero, it is identical to the mean.
Our free Z-score Calculator is a multi-mode statistical tool. You can calculate the standard z-score from raw distribution parameters, convert z-scores to five distinct probability ranges (percentiles), or find the exact probability between two bounded z-scores.
Z-score Formulas and Calculations
To calculate a z-score, subtract the mean from the raw score and divide the result by the standard deviation. The formulas vary slightly in notation depending on whether you are studying an entire population or a representative sample:
1. Population Z-score Formula
Use this equation when you know the parameters of the entire population:
z = (x - μ) / σ
Where:
- x: The raw score or data point (such as a test score, height, or blood pressure reading).
- μ: The population mean.
- σ: The population standard deviation.
2. Sample Z-score Formula
Use this equation when you are estimating parameters from a sample:
z = (x - x¯) / s
Where x¯ is the sample average and s is the sample standard deviation.
Z-score Calculation Example
Calculate the z-score of a student who scored 85 on a test, where the class mean (μ) is 75 and the standard deviation (σ) is 5:
z = (85 - 75) / 5 = 10 / 5 = 2.0
A z-score of 2.0 indicates the student’s test score lies exactly 2.0 standard deviations above the class average.
What is a Z-Table?
A z-table (also called a standard normal table or unit normal table) lists standardized probabilities associated with different z-scores. Because the normal distribution curve is symmetrical, Z-tables only list positive z-values to save space. The displacement left or right of the mean shares identical area bounds under the curve.
The table below provides the probability that a value lies between the mean (0) and a positive z-score (0 to Z):
| Z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |
|---|---|---|---|---|---|---|
| 0.0 | 0.00000 | 0.00399 | 0.00798 | 0.01197 | 0.01595 | 0.01994 |
| 0.5 | 0.19146 | 0.19497 | 0.19847 | 0.20194 | 0.20540 | 0.20884 |
| 1.0 | 0.34134 | 0.34375 | 0.34614 | 0.34849 | 0.35083 | 0.35314 |
| 1.1 | 0.36433 | 0.36650 | 0.36864 | 0.37076 | 0.37286 | 0.37493 |
| 1.5 | 0.43319 | 0.43448 | 0.43574 | 0.43699 | 0.43822 | 0.43943 |
| 2.0 | 0.47725 | 0.47778 | 0.47831 | 0.47882 | 0.47932 | 0.47982 |
How to Read the Z-Table
To find the probability of a value falling between the mean (0) and a z-score of 1.12:
- Locate the row matching the tenth’s place: 1.1.
- Locate the column matching the hundredth’s place: 0.02.
- Find the intersection of row 1.1 and column 0.02. The value is 0.36864.
This means there is a 36.864% chance that a random data point in a normally distributed population will have a z-score between 0 and 1.12.
Calculate dataset standard deviations on our Standard Deviation Calculator or compute general descriptive parameters with the Statistics Calculator.
Frequently Asked Questions (FAQ)
What is the difference between a z-score and a standard deviation?
Standard deviation measures the overall spread or variation of values within a dataset. A z-score measures how many of those standard deviations a single, specific data point lies away from the mean.
What does a negative z-score represent?
A negative z-score indicates that the raw score is less than the mean. For example, a z-score of -1.5 means the data point is 1.5 standard deviations below the average value of the dataset.
Why do we convert raw scores to z-scores?
Standardizing raw scores allows researchers to compare data points from completely different scales. For example, you can use z-scores to compare a student’s SAT score (out of 1600) with their ACT score (out of 36) to see on which test they performed better relative to other test-takers.
What percentage of data lies within 1, 2, and 3 standard deviations of the mean?
According to the Empirical Rule (or the 68-95-99.7 rule) for normal distributions:
- Approximately 68.27% of data falls within 1 standard deviation (z-scores between -1 and +1).
- Approximately 95.45% of data falls within 2 standard deviations (z-scores between -2 and +2).
- Approximately 99.73% of data falls within 3 standard deviations (z-scores between -3 and +3).