Z Score Calculator

Want to convert a z-score to a probability instead? Use the z-score probability calculator.

How to Use the Z Score Calculator

Finding a z-score from raw data or a sample mean is simple with this calculator. Just follow these steps:

1. Choose the correct calculation type. You should choose the Raw Score option if you want to convert a single raw score to a z-score or the Sample mean option if you want to find the standard score of the sample mean.
2. Enter the required values in the input fields.
3. Click Calculate.

The calculator will instantly return the correct z-score, interpretation, and show you how the value was computed manually.

Example 1: Raw Score

Suppose a student scores 85 on a test. The class mean is 70, and the standard deviation is 10. Find the z-score.

Solution

To calculate the z score using the calculator, follow these steps:

1. Choose the Raw Score option
2. Enter Raw score, x = 85, Mean, μ = 70, and standard deviation, σ = 10
3. Click Calculate

The calculator gives a z-score of 1.50 and provides its interpretation. It will also show you how to apply the z-score formula and obtain the same value manually.

Example 2: Sample Mean

Suppose a sample of 25 students has a mean score of 74. The population mean is 70, and the population standard deviation is 10. Calculate the z-score of the sample mean.

Solution

To calculate the standard score (z-score) of the sample mean using the calculator:

• Choose the Sample mean option
• Enter Sample mean, x̄ = 74, population mean, μ = 70, population standard deviation, σ = 10, and sample size, n = 25
• Click Calculate

The calculator will instantly return the correct z score as 2.00 and provide its interpretation. It will also show you how to get the same solution for your data by applying the correct formula.

What Is a Z Score?

A z-score is a statistical measure that tells you how many standard deviations a value is from the mean. It can take both positive and negative values, with a positive z score implying that the value is above the mean and a negative z score suggesting that the value is below the mean. However, if you ever find a z-score of 0, it means that the value is exactly the same as the mean.

For example, a z score of 2 means the value is 2 standard deviations above the mean. A z score of -1.5 means the value is 1.5 standard deviations below the mean.

Z scores are useful because they place different values on the same standard scale. This makes it easier to compare scores from different distributions. For example, you can compare exam scores from two different classes using z scores, even if the classes had different means and standard deviations.

Z Score Formula

For an individual raw score, the z-score formula is z = (x-μ)/σ

Where:

• x is the raw score
• μ is the population mean
• σ is the population standard deviation
• z is the z score, also known as the standard score

In other words, to find a z score from a single raw score, simply subtract the mean from the raw score and divide the result by the standard deviation.

Z Score Formula for a Sample Mean

Sometimes, you may need to calculate a z-score for sample data. In this case, the z-score formula changes to account for the sample size by reflecting the standard error of the mean. Thus, the formula becomes:

When you are calculating a z score for a sample mean, use:

Where:

• x̄ is the sample mean
• μ is the population mean
• σ is the population standard deviation
• n is the sample size
• $\sigma /\sqrt{n}$ is the standard error of the mean

This formula is used when the value you want to standardize is a sample mean instead of one individual observation. The denominator is called the standard error. It measures how much the sample means are expected to vary from the population mean.

Note: Use the z approach when the population standard deviation is known or when the normal approximation is appropriate. If the population standard deviation is unknown and the sample size is small, a t statistic may be more appropriate.

What Does the Z Score Mean?

A z-score tells you how far a value is from the mean in standard deviation units. Here’s a quick guide on how to interpret z-score values:

• 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 equal to the mean.

For example, a z score of 1 means the value is one standard deviation above the mean, and a z score of -2 means the value is two standard deviations below the mean.

Note. When working with a sample mean, the z score tells you how far the sample mean is from the population mean in standard error units.

Raw Score vs Sample Mean Z Score

Use the raw score option when you are working with one individual value. For example, you may want to know how far one test score, height, weight, or measurement is from the average. However, if you’re working with the average of the sample, use the sample mean option. This is useful when you want to compare a sample mean with a known population mean.

The calculator changes the formula depending on the option you choose. As such, you should always make sure you select the correct calculation type before entering your values.

Frequently Asked Questions