Z-Score Calculator

What is a Z-Score?

A z-score (also called a standard score) indicates how many standard deviations an element is from the mean. It allows us to calculate the probability of a score occurring within a normal distribution and compare two scores that are from different normal distributions.

Formula and Its Meaning

The formula for calculating a z-score is:

\[z = \frac{x - \mu}{\sigma}\]

Where:

• \(z\) is the z-score
• \(x\) is the value of the element
• \(\mu\) is the population mean
• \(\sigma\) is the population standard deviation

Calculation Steps

1. Determine the value (x) you want to standardize.
2. Calculate or obtain the mean (µ) of the population.
3. Calculate or obtain the standard deviation (σ) of the population.
4. Subtract the mean from the value: (x - µ)
5. Divide the result by the standard deviation: (x - µ) / σ

Example Calculation

Let's calculate the z-score for a value of 75 in a dataset with a mean of 70 and a standard deviation of 5.

\[z = \frac{75 - 70}{5} = \frac{5}{5} = 1\]

This z-score of 1 indicates that the value is one standard deviation above the mean.

Visual Representation

This graph represents a standard normal distribution. The red dashed line indicates the mean (µ), and the green point shows a z-score of 1, which is one standard deviation above the mean.