σ = 0 means perfection
If every value is identical, σ collapses to zero. In that world, z-scores are undefined—there’s no “spread” to measure.
No variance
Standard Deviation & Z-Score — Paste Numbers → mean/σ/z
Data & Options
Tips: Supports commas, spaces, tabs, new lines, and scientific notation (e.g., 1e-3). Extra whitespace is ignored.
Affects display only.
Uses the dataset’s current mean and chosen σ.
Tip: After calculating, use x→z and z→x for one-off conversions.
Results
Results will appear here.
Formulas & Notes
Mean: \( \bar{x} = \dfrac{1}{N}\sum_{i=1}^{N} x_i \)
Population variance: \( \sigma^2 = \dfrac{1}{N}\sum (x_i - \bar{x})^2 \), Standard deviation: \( \sigma = \sqrt{\sigma^2} \)
Sample variance: \( s^2 = \dfrac{1}{N-1}\sum (x_i - \bar{x})^2 \), Standard deviation: \( s = \sqrt{s^2} \)
Z-score (using chosen σ): \( z = \dfrac{x - \bar{x}}{\sigma} \)
If σ = 0 (all values equal), z-scores are undefined. We’ll flag that gently instead of shouting 😌.
What Are Standard Deviation and Z-Scores?
Standard deviation measures how spread out your data are around the mean. A small standard deviation (σ) means most values sit close to the average; a large σ means the data are more dispersed.
- Population variance: \( \sigma^2 = \dfrac{1}{N}\sum_{i=1}^{N}(x_i-\bar{x})^2 \), standard deviation: \( \sigma=\sqrt{\sigma^2} \)
- Sample variance: \( s^2 = \dfrac{1}{N-1}\sum_{i=1}^{N}(x_i-\bar{x})^2 \), standard deviation: \( s=\sqrt{s^2} \)
What Does a Z-Score Mean?
A z-score tells you how many standard deviations a value sits above or below the mean: \( z = \dfrac{x-\bar{x}}{\sigma} \). In normally distributed data, the 68–95–99.7 rule applies.
Quick Example
If \( \bar{x}=70 \) and \( \sigma=4 \), then for \( x=76 \), \( z=(76-70)/4=1.5 \) — about the 93rd percentile under a normal model.
When to Use Population vs Sample
- Population σ (divide by N): whole group.
- Sample s (divide by N−1): subset used to estimate a population.
Common Pitfalls (and How This Tool Helps)
- Mixed separators: commas, spaces, tabs, new lines — supported. Scientific notation too.
- All values equal: if σ = 0, z-scores are undefined (we’ll explain why).
- Rounding: change decimal places for cleaner reporting.
- Outliers: large |z| suggests anomalies.
From Z to Percentiles
Map z-scores to percentiles with the normal CDF (e.g., z=0 → 50th, ±1 → 84th/16th, ±2 → 97.5th/2.5th).
5 Fun Facts about Standard Deviation & z-scores
√N noise shrink
Sampling error often shrinks with 1/√N. Double your sample, your standard error drops ~30%—why big datasets feel steadier.
Precision boost
±2σ is not a wall
In a normal curve, about 5% of points live beyond ±2σ. Outliers happen naturally; context decides if they’re anomalies or signals.
Tail reality
Z-scores stack
Add independent z-scores (as squared sums) for combined uncertainty: \(z_\text{total} = \sqrt{z_1^2 + z_2^2}\) in many detection problems.
Signal math
σ sets your z ruler
Using sample σ (N−1) vs population σ (N) changes every z. Pick the divisor first—then interpret the scores.
Method matters