Starlight Tools

Z-Score and Standard Deviation Calculator

Calculate a z-score from x, mean, and standard deviation, or paste a dataset to calculate mean, sample/population standard deviation, variance, and z-scores. You can also reverse a z-score to x, compute a sample mean z-score, and see normal-model percentiles and probabilities.

Calculator Mode

Affects display only.
Used for single z-score modes.

Normal percentiles and probabilities assume the variable is approximately normally distributed. Outlier thresholds depend on context and the distribution.

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: \( s = \sqrt{s^2} \)

Raw-score z-score: \( z = \dfrac{x - \mu}{\sigma} \)

Reverse solve: \( x = \mu + z\sigma \)

Sample mean z-score: \( z = \dfrac{\bar{x} - \mu}{\sigma/\sqrt{n}} \)

True z-scores conventionally use a known population mean and population standard deviation. Dataset z-scores computed from a pasted sample are standardized sample scores based on estimates from the data. If standard deviation is 0, z-scores are undefined.

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: \( z = \dfrac{x-\mu}{\sigma} \). A negative z-score is below the mean; a positive z-score is above the mean. In approximately normal data, the normal CDF turns z into percentile and probability values.

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: \(|z| > 2\) is often unusual, and \(|z| > 3\) is often treated as a possible outlier, but context matters.

From Z to Percentiles

Map z-scores to percentiles with the normal CDF (e.g., z=0 → 50th, +1 → about 84th, -1 → about 16th, +2 → about 97.7th). These probabilities assume an approximately normal distribution.

Step-by-Step Examples

Raw score to z-score

For \(x=76\), \(\mu=70\), and \(\sigma=4\): \(z=(76-70)/4=1.5\). Under a normal model, that is about the 93rd percentile, so 76 is moderately above average.

x to z

Dataset standard deviation

For 68, 70, 72: mean \(=70\). Squared deviations are 4, 0, and 4, so the sum of squared deviations is 8. Population SD \(=\sqrt{8/3}=1.633\); sample SD \(=\sqrt{8/2}=2\).

dataset

Z-score to raw score

For \(z=1.5\), \(\mu=70\), and \(\sigma=4\): \(x=70+(1.5)(4)=76\). A z-score of 1.5 puts the value 1.5 standard deviations above the mean.

z to x

Sample mean z-score

For sample mean \(=74\), population mean \(=70\), population SD \(=8\), and \(n=16\), the standard error is \(8/\sqrt{16}=2\). The z-score is \((74-70)/2=2\), an unusual result under the normal model.

sample mean

FAQ

How do I calculate a z-score from mean and standard deviation?

Use \(z=(x-\mu)/\sigma\). Enter x, the mean, and a standard deviation greater than zero in the first mode.

What percentile is a z-score?

The percentile below a z-score is the standard normal CDF value. For example, \(z=1\) is about the 84th percentile when the normal model applies.

What does a negative z-score mean?

A negative z-score means the value is below the mean. A z-score of -2 is two standard deviations below the mean.

Is a z-score the same as a standard score?

Yes. A z-score is a standard score expressed in standard deviation units from the mean.

When should I use a z-score vs a t-score?

Use a z-score when the population standard deviation is known or the normal approximation is justified. Use a t-score when you estimate the standard deviation from a small sample.

Can I calculate z-scores from a sample dataset?

Yes, but they are standardized sample scores because the mean and standard deviation are estimated from the dataset. They are useful for comparing values inside that sample, but they are not the same as z-scores from known population parameters.

What z-score counts as an outlier?

A common rule of thumb is that \(|z| > 2\) is unusual and \(|z| > 3\) is a possible outlier. Use domain context and inspect the distribution before making decisions.

Which separators can I use in a dataset?

Use commas, semicolons, spaces, tabs, or new lines. Scientific notation such as 1e-3 is supported.

Is my data private?

Yes. Calculations run locally in your browser, with no upload or storage by this calculator.

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