Outlier Calculator — Z-Score Method
Identify outliers in your data using the Z-score method: paste your numbers, choose a threshold, and see which values lie beyond k standard deviations from the mean, plotted on a normal curve.
Z-score threshold
Values with |Z-score| above the threshold
- 1
Mean (μ)
(sum of 9 values) ÷ 9 = 20,2222Average of all data points. - 2
Population std dev (σ)
√(Σ(xᵢ − 20,2222)² ÷ 9) = 12,7956 - 3
Lower fence
20,2222 − 2 × 12,7956 = -5,3691 - 4
Upper fence
20,2222 + 2 × 12,7956 = 45,8135 - 5
Outliers detected
count of values with |Z| > 2 = 1
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The Z-score outlier method computes Z = (x − mean) / std_dev for each data point and flags any value with |Z| > threshold. A threshold of 2 flags the outermost ~5% of normally distributed data; 3 flags ~0.3%. The method assumes roughly normal data — for skewed or small samples, use the IQR method instead.
Τύπος
How this is calculated
The Z-score method flags outliers by measuring how many standard deviations each value is from the mean. For each data point x, compute its Z-score: Z = (x − μ) / σ, where μ is the mean of the data set and σ is the population standard deviation. Any point with |Z| greater than your chosen threshold is considered an outlier. With a threshold of 2, roughly 4.6% of values in a perfectly normal distribution would be flagged; at 3, only about 0.3% would be.
The threshold you choose reflects the trade-off between sensitivity and specificity. A threshold of 2 is commonly used in exploratory data analysis. A threshold of 3 is a conservative choice often used when outliers are rare and harmful to flag incorrectly — this is sometimes called the "three-sigma rule". For heavily skewed distributions or small samples, the Z-score method can be unreliable because extreme values distort the mean and standard deviation themselves; in those cases the IQR (interquartile range) method is more robust.
This calculator uses the population standard deviation (dividing by n, not n−1), which is appropriate when the data set is the entire population of interest. For a sample, the sample standard deviation (dividing by n−1) would be used; for large samples the difference is negligible.
Συχνές ερωτήσεις
The Z-score method measures distance from the mean in standard deviation units and assumes data is roughly normally distributed. The IQR (interquartile range) method uses quartile-based fences and is more robust to skewness and small samples — it is less sensitive to the extreme values being tested. For clean, large, approximately normal data the two methods largely agree; for skewed or small data sets the IQR method is preferred.
A threshold of 2.0 is a common default for exploratory analysis — it flags the outermost 4.6% of a normal distribution. For stricter detection, use 2.5 or 3.0. If you are doing quality control or anomaly detection where false positives are costly, 3.0 (the three-sigma rule) is standard.
The Z-score method works better on larger samples (n > 30). For small samples, extreme values have an outsized effect on the mean and standard deviation, which can mask or inflate the Z-score of outliers. For small samples, Grubbs' test or the IQR method with Tukey fences is preferable.
Γνωστό και ως
TG we-Calculate Editorial Team. (2026). Outlier Calculator — Z-Score Method [Online calculator]. TG we-Calculate. https://we-calculate.com/el/calculator/outlier-calculator
TG we-Calculate Editorial Team. "Outlier Calculator — Z-Score Method." TG we-Calculate. 2026. https://we-calculate.com/el/calculator/outlier-calculator.
TG we-Calculate Editorial Team, "Outlier Calculator — Z-Score Method," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/el/calculator/outlier-calculator
@misc{wecalculate_outlier_calculator, title = {Outlier Calculator — Z-Score Method}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/el/calculator/outlier-calculator}}, year = {2026}, note = {TG we-Calculate} }
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