Z-Score Calculator
Calculate the z-score for any value in a data set and find the percentile.
Value and mean can be any real numbers.
The average or center value.
Standard deviation must be greater than zero.
Z-score
1.5
Above mean
Distance from mean
15
z = (x − μ) ÷ σ
Raw value x
85
Mean
70
Standard deviation
10
Absolute z-score
1.5
Direction
Above mean
Normal percentile estimate
93.32%
Assumes approximately normal data.
Left-tail probability
93.32%
Normal-distribution estimate.
Right-tail probability
6.68%
Normal-distribution estimate.
Two-tail probability
13.36%
Normal-distribution estimate.
Formula used
z = (x − μ) ÷ σ
z = (85 − 70) ÷ 10
Normal Curve Visual
This visual is a simple normal-curve sketch. Percentile estimates assume the data is approximately normally distributed.
Z-Score Formulas
Z-Score
z = (x − μ) ÷ σ
Sample Z-Score
z = (x − x̄) ÷ s
Distance from Mean
Distance = x − μ
Reverse Z-Score
x = μ + zσ
Absolute Z-Score
|z| = absolute distance in standard deviation units
Variable Explanations
z
Z-score or standard score.
x
Raw value or data point.
μ
Population mean.
x̄
Sample mean.
σ
Population standard deviation.
s
Sample standard deviation.
x − μ
Distance from the mean.
|z|
Distance from the mean regardless of direction.
What a Z-Score Means
Standardized value
A z-score standardizes a value relative to a mean and standard deviation.
Typical or unusual
It shows how unusual or typical a value is within a dataset.
Positive
Positive z-scores are above average.
Negative
Negative z-scores are below average.
Zero
z = 0 means the value equals the mean.
Comparable scales
Z-scores make values from different scales easier to compare.
Worked Examples
x = 85, mean = 70, SD = 10
Formula: z = (x − μ) ÷ σ
Substitution: (85 − 70) ÷ 10
Answer: z = 1.5
85 is 1.5 standard deviations above the mean.
x = 60, mean = 70, SD = 10
Formula: z = (x − μ) ÷ σ
Substitution: (60 − 70) ÷ 10
Answer: z = -1
60 is 1 standard deviation below the mean.
Value equals mean
Formula: z = (x − μ) ÷ σ
Substitution: (70 − 70) ÷ 10
Answer: z = 0
The value is exactly average.
Reverse z-score
Formula: x = μ + zσ
Substitution: 50 + 2 × 5
Answer: x = 60
A z-score of 2 is 10 units above the mean.
Compare test scores
Formula: standardize each score
Substitution: score gaps ÷ class SD
Answer: compare z-scores
Z-scores compare values from different scales.
Negative value example
Formula: z = (x − μ) ÷ σ
Substitution: (-5 − -10) ÷ 2
Answer: z = 2.5
Negative raw values are valid.
Decimal SD example
Formula: z = (x − μ) ÷ σ
Substitution: (12.4 − 10.1) ÷ 1.5
Answer: z ≈ 1.533
Decimals are supported.
Normal percentile estimate
Formula: normal CDF
Substitution: z = 1
Answer: ≈ 84.13%
This assumes approximately normal data.
Positive, Negative, and Zero Z-Scores
Positive z-score
The value is above the mean.
Negative z-score
The value is below the mean.
Zero z-score
The value equals the mean.
Absolute z-score
Shows distance regardless of direction.
Near zero
Close to average.
Large z-score
May indicate an unusual value or outlier depending on context.
Z-Scores, Percentiles, and Normal Distribution
Standard normal
Z-scores can be used with the standard normal distribution.
Percentile estimate
Percentiles estimate how much of a normal distribution falls below a value.
Normality assumption
Percentile estimates assume the data is approximately normal.
Skewed data warning
If data is skewed or non-normal, percentile interpretation may be misleading.
Still useful
Z-scores still standardize values even without a normal distribution.
Probability claims
Probability claims need distribution assumptions.
Common Use Cases
Common Z-Score Mistakes
Understanding Your Result
Z-score
Number of standard deviations from the mean.
Positive z-score
Value is above the mean.
Negative z-score
Value is below the mean.
Distance from mean
Raw difference before standardizing.
Absolute z-score
Distance regardless of direction.
Percentile
Estimated share below the value under normality assumptions.
Reverse z-score
Raw value corresponding to a z-score.
Frequently Asked Questions
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