Question

What is a clear explanation of how to calculate z-scores and what do they represent?

Answer 1

A **z-score** (or standard score) is a statistical measure that describes a data point's relation to the mean of a group of data. It tells you how many standard deviations an element is from the mean.

### How to Calculate Z-Scores

The formula for calculating a z-score for a particular data point (X) is:

\[
z = \frac{X - \mu}{\sigma}
\]

Where:
- \( z \) is the z-score.
- \( X \) is the value of the data point.
- \( \mu \) is the mean of the dataset.
- \( \sigma \) is the standard deviation of the dataset.

### Steps to Calculate a Z-Score

1. **Calculate the Mean (\( \mu \))**:
- Sum all the data points and divide by the number of data points (N).

\[
\mu = \frac{\sum X}{N}
\]

2. **Calculate the Standard Deviation (\( \sigma \))**:
- First, find the variance by calculating the average of the squared differences from the mean.
- Then, take the square root of the variance to get the standard deviation.

\[
\text{Variance} = \frac{\sum (X - \mu)^2}{N}
\]
\[
\sigma = \sqrt{\text{Variance}}
\]

3. **Plug Values into the Z-Score Formula**:
- Substitute the values of \( X \), \( \mu \), and \( \sigma \) into the z-score formula.

### Interpretation of Z-Scores

- A **z-score of 0** means the data point is exactly at the mean.
- A **positive z-score** indicates the data point is above the mean, while a **negative z-score** indicates it is below the mean.
- The **magnitude of the z-score** indicates how far away the data point is from the mean in terms of standard deviations. For example:
- A z-score of +2 means the data point is 2 standard deviations above the mean.
- A z-score of -1.5 means the data point is 1.5 standard deviations below the mean.

### Uses of Z-Scores

Z-scores are widely used in statistics for:
- Identifying outliers: A z-score greater than +3 or less than -3 is often considered an outlier.
- Standardizing scores: Converting different datasets to a common scale.
- Statistical inference: Facilitating comparisons between different datasets or distributions.

Understanding z-scores allows for better insights into the relative standing of data points within a dataset.