What is a z-score?
Z-score, otherwise known as the standard score, is the number of standard deviations by which a data point is above (or below) the mean. It is defined as the distance between the mean score and the experimental data point, expressed in terms of standard deviation (SD). In statistical data analysis it is also called z value, standardized score, and normal score.
How to calculate z-score: the formula
To find the z-score, you first need to calculate a data set's mean and standard deviation.
- Mean (μ) is the sum of all values divided by the number of data points:
μ = Σx / n - Standard deviation (σ) measures the spread of the data:
σ = √[Σ(x − μ)² / n]
To find the z-score, apply the following formula:
z = (x − μ) / σ
where x is the raw data value, μ is the mean, and σ is the standard deviation.
Calculating z-score: an example
Suppose four students scored 50, 53, 62, and 70 points on a test. What is the z-score of the result 62?
- Find the mean: μ = (50 + 53 + 62 + 70) / 4 = 58.75
- Calculate (x − μ)² for each value:
- (50 − 58.75)² = 76.5625
- (53 − 58.75)² = 33.0625
- (62 − 58.75)² = 10.5625
- (70 − 58.75)² = 126.5625
- Calculate the standard deviation: σ = √[(76.5625 + 33.0625 + 10.5625 + 126.5625) / 4] = √(246.75 / 4) ≈ 7.854
- Apply the z-score formula: z = (62 − 58.75) / 7.854 ≈ 0.4138
How to use the z-score calculator?
- Select the calculation mode — choose what you want to find: the z-score, the data value x, the mean μ, or the standard deviation σ.
- Enter the known values in the corresponding fields.
- Click Calculate. The result will appear on the right along with probability values from the standard normal table.
The calculator also supports the reverse operation — if you know the z-score and need to find the original value, the mean, or the standard deviation, simply select the appropriate mode.
What is a z-score table?
A z-score table (also called a standard normal table) shows the area to the left of the given z-score under the standard normal distribution curve. The first column lists z-values to one decimal place, and the top row provides the second decimal place.
For example, for z = 0.41:
- Find z = 0.4 in the first column.
- Find 0.01 in the top row.
- The area to the left is 0.6591 — meaning there is a 65.91% probability that a randomly selected value falls at or below this score.
- The area to the right (p-value) is 1 − 0.6591 = 0.3409, or 34.09%.
Z-score calculator and six sigma methodology
In quality control and process improvement, the six sigma methodology uses z-scores to measure how many standard deviations a process is from its specification limits. A six sigma process has a z-score of 6, meaning nearly all (99.99966%) of outputs fall within specification — corresponding to fewer than 3.4 defects per million opportunities.
Z-scores help quality engineers assess process capability and pinpoint areas for improvement.
FAQs
What does a z-score of 0 mean?
A z-score of 0 means the data point is exactly at the mean of the distribution.
Can z-scores be negative?
Yes. A negative z-score means the value is below the mean; a positive z-score means it is above the mean.
What is a good z-score?
There is no universally "good" z-score — it depends on context. In academic testing, a z-score above 1 typically indicates above-average performance. In quality control, a z-score above 3 or even 6 is desirable.
What is the difference between z-score and t-score?
The z-score is used when the population standard deviation is known; the t-score (from the t-distribution) is used when working with sample standard deviations and smaller sample sizes. As the sample size grows, the t-distribution approaches the normal distribution.
How do I interpret z-scores in terms of percentiles?
- z = −2 → approximately 2.28th percentile
- z = −1 → approximately 15.87th percentile
- z = 0 → 50th percentile (median)
- z = +1 → approximately 84.13th percentile
- z = +2 → approximately 97.72nd percentile