Enter the raw data point, mean, and standard deviation into the calculator below to determine the z-score, also known as the standard score.
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Z-Score Formula
The following formula is used to calculate the z-score:
- Where μ is the mean of the distribution (population mean; often estimated by the sample mean)
- and σ is the standard deviation of the distribution (population standard deviation; often estimated by the sample standard deviation)
- x is the raw data point
Probabilities (areas) and common critical z-values can be found using a z-table such as the one shown below.
Z Score Definition
A z-score, also known as a standard score, is a statistic that describes the signed number of standard deviations a data point is above (positive) or below (negative) the mean.
A z-score standardizes a value by expressing it relative to the mean and standard deviation of its distribution. You can compute z-scores using population parameters (μ and σ) or, commonly in practice, using sample estimates (x̄ and s).
Z-scores are not a “measure of confidence.” However, z-scores are closely connected to the standard normal distribution: z-tables and z critical values (z*) are used to compute probabilities and to build confidence intervals under appropriate assumptions (for example, normality or large-sample conditions).
How to calculate a z-score?
The following is a step by step guide on how to calculate the z-score:
- Determine the mean (μ) of the distribution (or use the sample mean, x̄, if you are working with a sample).
- For this example, we will assume the mean is 20.
- Determine the standard deviation (σ) of the distribution (or use the sample standard deviation, s, if you are working with a sample). For this example let’s assume the deviation is 1.5.
- Measure (or record) your raw data value x. We find that x is 25.
- Plug all of this information into the formula: z = (x − μ) / σ. Here, z = (25 − 20) / 1.5 = 3.33 (rounded).
Analyze your results and apply to future problems.
FAQ
What does a Z-Score indicate?
A Z-Score indicates how many standard deviations an element is from the mean. A positive Z-Score means the data point is above the mean, while a negative Z-Score means it’s below the mean.
Why is the Z-Score important in statistics?
The Z-Score is important because it allows for comparisons between different data sets or within the same data set over time, and it helps identify outliers.
Can Z-Scores be used for all types of data?
You can compute a z-score for any numeric data using a mean and standard deviation. However, using z-scores to look up probabilities (via the standard normal distribution) works best when the data are approximately normally distributed (or when large-sample approximations apply).
How does the Z-Score relate to the standard normal distribution?
The Z-Score is a key part of the standard normal distribution. It is used to determine where a data point lies within the distribution, helping to calculate probabilities and to obtain critical values used in confidence intervals and hypothesis tests.