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The z-score is a statistical measurement that tells you how many standard deviations an element, \( x \), is from the mean \( \mu \) of a normal distribution. When you compute a z-score, you are standardizing the data allowing for comparison across different normal distributions. To find a z-score, you use the formula:

• \( z = \frac{x - \mu}{\sigma} \)

where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Let's explore how to use this formula in practice. If you have a set of data points and you want to know how they relate to the average (mean), calculate each point's z-score. For example, if \( x = 98 \), \( \mu = 110 \), and \( \sigma = 12 \), substitute these values into the formula:

• \( z = \frac{98 - 110}{12} = \frac{-12}{12} = -1 \)

This tells you that 98 is one standard deviation below the mean of the sample, providing insight into how typical or atypical the value is within that distribution.

Standard deviation, denoted as \( \sigma \), measures the amount of variability or dispersion in a set of values. A low standard deviation means data points are generally close to the mean, while a high standard deviation indicates they are spread out over a wider range of values.

For instance, in the given problem, a standard deviation of 12 points implies the scores tend to deviate from the mean by about 12 points. Understanding this concept can help contextualize z-scores and gauge how likely a particular data point is in the context of the whole dataset.

• A small \( \sigma \) suggests a tightly clustered dataset.
• A large \( \sigma \) indicates more scattered data.

This measure is crucial as it adjusts the scale of z-scores and determines how "rare" or "common" a data point might be within the distribution.

The mean, represented as \( \mu \), is the average of all data points in a dataset. In a normal distribution, the mean is not only the measure of central tendency but also the symmetrical center of the distribution curve.

Calculating the mean involves summing all the values in the dataset and then dividing by the number of values. In our exercise scenario, the mean is given as 110 points. This value represents the central point around which the data is distributed.

• It serves as the reference value for calculating z-scores.
• It acts as a benchmark for identifying how typical a particular score is.

Recognizing the mean's role helps you understand how individual scores relate to the overall sample context, whether they fall below, above, or right at the average.