Z-Score

Introduction

The Z-score, also known as a standard score, is a statistical measurement that describes a value's relationship to the mean of a group of values. Z-score is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

Calculation

The Z-score is calculated using the following formula:

Z = (X - μ) / σ

Where: - Z is the Z-score, - X is the value of the element, - μ is the population mean, - σ is the standard deviation.

The Z-score formula for a sample is slightly different, as the sample mean (x̄) and the sample standard deviation (s) are used instead.

Interpretation

Z-scores are primarily used to compare results from a test to a "normal" population. Results from tests or surveys have thousands of possible results and units; those results can often seem meaningless. For example, knowing that a test taker answered 200 questions correctly out of 500 doesn't tell you much about how he or she did compared to other test takers. By converting raw scores to Z-scores, you can describe exactly where that score lies on a distribution curve.

Applications

Z-scores are commonly used in the fields of psychology, healthcare, education, and any other field that uses statistics. They are especially useful when seeking to compare the relative standings of items from different scales. For example, a Z-score can be used to compare the relative success of a student, measured by GPA, to the relative success of a business, measured by profit.

Advantages and Disadvantages

Z-scores have the advantage of being dimensionless, allowing for direct comparisons across different types of data. They also allow analysts to calculate the probability of a score occurring within a standard normal distribution and compare two scores that are from different normal distributions, which is often not possible when the distributions have different means and standard deviations.

However, Z-scores also have some limitations. They can only be used to compare scores from a normal distribution, and they can be misleading when used with small sample sizes or with data that are not normally distributed.

See Also

- Standard Deviation - Mean - Normal Distribution - Statistical Analysis

A photo of a normal distribution curve with various Z-scores marked on it.