A z-score (also known as a standard score or z-value) measures how many standard deviations a data point is from the mean of a distribution. It’s a way of standardizing individual data points so they can be compared, regardless of the scale of the data.
How to calculate the z-score:
The formula for the z-score of a data point is:
\(z = \frac{X - \mu}{\sigma}\)
Where:
- \(X\) = the value of the data point.
- \(\mu\) = the mean (average) of the dataset.
- \(\sigma\) = the standard deviation of the dataset.
Steps to calculate the z-score:
- Find the mean of the dataset.
- Find the standard deviation of the dataset.
- Subtract the mean from the data point you’re interested in.
- Divide the result by the standard deviation.
Example:
Imagine you have the following dataset of test scores: [80, 85, 90, 95, 100]
Mean (μ): ((80 + 85 + 90 + 95 + 100) / 5 = 90)
Standard Deviation (σ): Calculate the standard deviation of the dataset (you could use a standard formula or a calculator for this). Let’s say it’s 7.91.
To find the z-score for a score of 85:
\(z = \frac{85 - 90}{7.91} \approx -0.63\)
This means the score of 85 is 0.63 standard deviations below the mean.
Common uses of the z-score:
Standardizing data: Z-scores are used to standardize data, especially when you want to compare values from different datasets with different units or scales.
Identifying outliers: A data point with a z-score greater than +3 or less than -3 is often considered an outlier because it lies more than 3 standard deviations away from the mean.
Probability and normal distribution: In statistics, z-scores are crucial when working with the normal distribution. They can be used to find the probability of a value occurring within a normal distribution by looking up the z-score in a standard normal distribution table (or using statistical software).
Comparing scores across different distributions: For example, in education, z-scores allow you to compare scores from different tests, even if those tests have different scales (e.g., SAT vs. ACT scores). You can convert raw scores into z-scores to compare relative performance.
Risk assessment and finance: In finance, z-scores can be used to assess the risk or financial stability of a company. The Altman Z-score, for instance, is used to predict the likelihood of bankruptcy.
Quality control and manufacturing: In quality control, z-scores help assess whether a product meets specifications, identifying whether the production process is consistent or if corrective action is needed.