X N 2.9 0.78 Z Score Calculator

This z-score calculator helps you determine how many standard deviations a data point (x) is from the mean (μ) of a normal distribution. The z-score is a dimensionless quantity that describes a position on a normal distribution curve.

What is a z-score?

A z-score (also called a standard score) measures how many standard deviations an element is from the mean. Z-scores allow you to compare values from different normal distributions. A z-score of 0 indicates the value is exactly at the mean, while positive and negative values indicate positions above and below the mean, respectively.

Z-scores are widely used in statistics, quality control, and data analysis to identify outliers, compare performance, and make inferences about populations.

How to calculate z-score

The z-score formula is straightforward:

Z-score formula

Z = (x - μ) / σ

Where:

Z = z-score
x = individual data point
μ = population mean
σ = population standard deviation

The formula calculates how many standard deviations (σ) a data point (x) is from the mean (μ). A positive z-score indicates the data point is above the mean, while a negative z-score indicates it's below the mean.

Interpreting z-scores

Z-scores provide several key insights:

Position relative to mean: A z-score of 0 means the value is exactly at the mean.
Direction: Positive z-scores are above average, negative below.
Magnitude: The absolute value shows how far from the mean the value is.
Standardization: Z-scores allow comparison across different distributions.

In practical terms:

Z-scores between -2 and +2 cover about 95% of normal distributions.
Values beyond ±3 are considered outliers.
Z-scores are used in hypothesis testing, quality control, and performance evaluation.

Worked example

Let's calculate the z-score for x = 2.9 with μ = 2.9 and σ = 0.78:

Example calculation

Z = (2.9 - 2.9) / 0.78 = 0 / 0.78 = 0

Result: The z-score is 0, indicating the value is exactly at the mean.

This example shows how a z-score of 0 means the data point is exactly at the average of the distribution.

FAQ

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. It indicates the value is average or typical for that distribution.

How do I interpret negative z-scores?

Negative z-scores indicate values below the mean. The more negative the score, the further below average the value is. For example, a z-score of -1.5 means the value is 1.5 standard deviations below the mean.

Can z-scores be used for non-normal distributions?

Z-scores are specifically designed for normal distributions. For non-normal data, other standardization methods like min-max scaling or robust z-scores may be more appropriate.

What's the difference between z-score and standard deviation?

Standard deviation measures the dispersion of data points around the mean, while z-scores measure how far individual data points are from the mean in terms of standard deviations. Z-scores standardize individual values.