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In statistics, the **z-score** helps us understand how a particular data point relates to the average of a data set. It's essentially a measure of how far and in what direction, a data point is from the mean, measured in terms of standard deviations. The formula for calculating a z-score is:\[ z = \frac{(x - \mu)}{\sigma} \]Here:

• \(x\) is the data point.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation.

The z-score tells us the number of standard deviations a data point is from the mean. For example, a z-score of 1.5 means the data point is 1.5 standard deviations above the mean. Calculating the z-score for a data point can help determine where it fits in the overall distribution, helping identify if the data point is typical or unusual.

The **mean** is essentially the average of all data points in a set. It's the central value around which all observations in a data set revolve. When we talk about a normal distribution in statistics, the mean provides a pivotal reference point. On the other hand, the **standard deviation** measures the amount of variation or spread in a set of values. A low standard deviation implies that the data points are close to the mean, whereas a high standard deviation indicates that the data points are more spread out over a wider range. In the context of a normal distribution:

• The data tends to group around the mean, forming a bell-shaped curve.
• About 68% of the data is within 1 standard deviation of the mean.
• Approximately 95% falls within 2 standard deviations.
• And roughly 99.7% is within 3 standard deviations.

Understanding these concepts is crucial as they lay the foundation for interpreting data, allowing us to assess whether certain observations may be considered unusual.

When evaluating data, it's important to determine whether an observation is **unusual**. In a normal distribution, an observation is deemed unusual if it falls outside the range that contains most of the data points. Conventionally:

• Data points within ±2 standard deviations from the mean are considered typical.
• Data points beyond ±3 standard deviations are often flagged as unusual.

For instance, if we calculate a z-score and find it is greater than or less than 3, this indicates the observation is unusual. To compare in our example:

• The mean is 50, and the standard deviation is 15.
• The calculated z-score for an observation of 0 is approximately -3.33.

This z-score indicates that 0 is more than 3 standard deviations below the mean. Over 99.7% of normally distributed data lies within 3 standard deviations from the mean. Since 0 falls outside this range, it is classified as unusual.