A z-score (also called a standard score) is a measure in statistics that indicates how many standard deviations a data point is from the mean of a dataset. It’s a way to standardize data, making it possible to compare values from different distributions or datasets.

Here's a breakdown of the concept:

1. Formula: The z-score for a given value $x$ in a dataset is calculated using the formula: $z = \frac{x - \mu}{\sigma}$ where:

   • $x$ is the value for which you're calculating the z-score,
   • $\mu$ (mu) is the mean of the dataset, and
   • $\sigma$ (sigma) is the standard deviation of the dataset.

2. Interpretation:

   • A z-score of 0 means the value is exactly at the mean of the dataset.
   • A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.
   • The magnitude of the z-score tells us how far, in standard deviations, the value is from the mean. For instance, a z-score of 1.5 means the value is 1.5 standard deviations above the mean.

3. Standardization: By converting raw scores to z-scores, we create a standard form where we can compare scores from different datasets or distributions, as all z-scores are expressed in terms of standard deviations from the mean.

4. Applications: Z-scores are often used in fields like psychology, finance, and quality control. For example, they can help in determining probabilities for specific scores in a normal distribution, or in identifying outliers (values with high z-scores, typically above 2 or below -2).

5. Normal Distribution Context: In a normal distribution, about 68% of values fall within 1 standard deviation from the mean (z-scores between -1 and 1), 95% fall within 2 standard deviations (z-scores between -2 and 2), and 99.7% fall within 3 standard deviations (z-scores between -3 and 3).

Would you like a practical example of calculating a z-score, or have any other questions?

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Here are 5 follow-up questions you may find helpful:

1. How do you calculate the z-score if you have a specific value and its mean and standard deviation?
2. Why is the z-score useful in comparing data from different distributions?
3. What does it mean if a value has a z-score of -1.5?
4. How does the z-score relate to probability in a normal distribution?
5. When might a high z-score indicate an outlier in a dataset?

Tip: Always check if your dataset follows a normal distribution before heavily relying on z-scores for interpretation.