Z Score Calculator

This z-score calculator converts a raw score or a sample mean into a standardized z-score. It also provides a clear, step-by-step explanation to help you understand how the calculation was done.

Want to convert the z-score into probability? Use the Z Score Probability Calculator for instant results.

How to Use the Z Score Calculator

This calculator allows you to compute a z-score for a single raw score, a sample mean, or multiple raw scores. Follow the steps below depending on the type of calculation you want to perform.

• Step 1. Select the type of calculation you want to perform from the dropdown.
  • Select “Single Raw Score” if you want to convert one value into a z-score
  • Select “Sample Mean” if you want to calculate a z-score of a sample mean
  • Select “Set of Raw Scores” if you want to standardize several data points at once
• Step 2. Fill in the input fields depending on the option you chose from the dropdown
  • For a single raw score, you’ll enter the raw score (X), the population mean (μ), and the population standard deviation (σ).
  • To convert the sample mean to a z score, you’ll enter the sample mean (x̄), the population mean (μ), the population standard deviation (σ), and the sample size (n).
  • To convert multiple raw scores to z, you paste the values and indicate whether they are from a sample or a population
• Step 3. Click the “Calculate” button

The calculator will instantly compute the appropriate z-score and provide a clear, step-by-step explanation of how each of the solutions was obtained. This will consequently help you verify results and understand how to calculate z-scores manually.

How to Find the Z Score from a Raw Score

To calculate a z-score from a raw score, you need three parameters: the raw score (X), the mean (μ), and the standard deviation (σ). In this case, you can easily convert a raw score to a z-score using the formula: Z =(X-μ)/σ

Example 1. A student scored 78 on a math test. The class average was 81 with a standard deviation of 9. Find how this student’s score compares to the rest of the class in terms of standard deviations.

Solution

In this case, we need to find a z-score from raw scores.

From the question, we know that:

• Raw score, X = 78
• Population mean, μ = 81
• Population standard deviation, σ = 9

Using the z-score formula, we have: z = (78-81)/9

= -0.3333

Alternatively, to find this value quickly with the z-score calculator, follow these steps:

1. Select “Single raw score.”
2. Enter the parameters: X = 78, μ = 81, and σ = 9
3. Click Calculate

The calculator instantly returns a z-score value of -0.3333 as shown below. You can also click the “step-by-step explanation” to learn how the value was computed.

How to Find the Z-Score from a Sample Mean

Sometimes you need to determine how unusual a sample mean is compared to a population. In this case, you calculate a z-score using the formula: z = (x̄-μ​)/(σ/√n)

Where:

• x̄ is the sample mean
• μ is the population mean
• σ is the population standard deviation
• n is the sample size

Example 2. A researcher studies the average time students spend studying per week. The population average study time is 12 hours with a standard deviation of 4 hours. A random sample of 25 students has an average study time of 14 hours. Determine how unusual this sample mean is.

Solution

In this case, we need to find a z-score from a sample mean.

From the question, we know that:

• Sample mean, x̄ = 14
• Population mean, μ = 12
• Population standard deviation, σ = 4
• Sample size, n = 25

Using the z-score formula for sample means, we have:

z = (14-12)/(4/√25)

=2/0.8

=2.5

This means the sample mean is 2.5 standard errors above the population mean.

Alternatively, to find this value quickly with the standard z-score calculator, follow these steps:

1. Select “Sample mean.”
2. Enter the parameters: sample mean = 14, μ = 12, σ = 4, and n = 25
3. Click Calculate

The calculator instantly returns a z-score of 2.5 (as shown below). You can also click the “step-by-step explanation” to see how the value was computed.

How to Interpret Z Scores

After calculating the z-score, the next step is to interpret it. Here’s how the z-scores should be interpreted:

• A positive z-score means the value is above the mean.
• A negative z-score means the value is below the mean.
• A z-score of 0 means the value is exactly at the mean.

For instance, the student’s z-score of 1.5 indicates the score is 1.5 standard deviations above the class average.

Want to Learn More About Z Scores? Check out our Z Score Tutorial for step-by-step examples and easy-to-understand explanations.

Related. How to find z-scores using Excel

Frequently Asked Questions

Reference

Z-score in statistics | Engineering mathematics. (2025, July 12). GeeksforGeeks.