Z-Score Calculator

The z-score is a statistical metric that quantifies the number of standard deviations a particular data point is from the mean of a data set. This measure is vital for comparing individual data points to a distribution, determining how typical or atypical a data point is within that distribution.

Z-Score Formula

The formula to calculate the z-score is as follows:

\[ z = \frac{(x - \mu)}{\sigma} \]

where:

• \(z\) is the z-score,
• \(x\) is the raw data point,
• \(\mu\) is the mean of the population,
• \(\sigma\) is the standard deviation of the population.

Example Calculation

For an example, let's use the following values:

• Population mean (\(\mu\)): 20,
• Standard deviation (\(\sigma\)): 1.5,
• Raw data point (\(x\)): 25.

Plugging these values into the formula gives:

\[ z = \frac{(25 - 20)}{1.5} = \frac{5}{1.5} \approx 3.33 \]

Understanding Z-Scores

A z-score provides a way to compare data points across different normal distributions, offering insights into how far and in what direction those points deviate from their mean. It's extensively used in fields such as statistics, research, and finance to identify outliers, normalize data, and assess probabilities.

Common FAQs

1. What does a high z-score indicate?

   • A high z-score (positive or negative) indicates that the data point is significantly different from the mean, lying far from it in terms of standard deviations.

2. Can z-scores be negative?

   • Yes, a negative z-score indicates that the data point is below the mean.

3. Why are z-scores useful?

   • Z-scores are crucial for statistical analysis, especially in hypothesis testing, standardization of scores for comparison, and in determining probabilities within a normal distribution.

Understanding and calculating z-scores are fundamental skills in statistics, providing insights into data's relative position and significance within a given set. This calculator simplifies the process, making statistical analysis more accessible.