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The z-score is an essential concept in statistics that measures how many standard deviations an element is from the mean. It is useful for determining the relative position of a data point within a distribution. The formula to find the z-score is: \( z = \frac{x - \mu}{\sigma} \).
Here:

• \( x \) is the value in the dataset you are examining.
• \( \mu \) is the mean of the dataset.
• \( \sigma \) is the standard deviation.

The z-score helps in converting a normal distribution into a standard normal distribution, which simplifies finding probabilities using z-tables. For example, a z-score of 1 means the data point is one standard deviation away from the mean.

The empirical rule is a statistical guideline that applies to normal distributions. It states that:

• Approximately 68% of data falls within one standard deviation of the mean.
• Approximately 95% of data falls within two standard deviations of the mean.
• Approximately 99.7% of data falls within three standard deviations of the mean.

This rule is particularly useful when analyzing datasets with a bell-shaped curve. It helps to understand the spread and variability of the data. For example, in our original exercise, we calculated that 68.26% of the area under the curve falls between \( z = -1 \) and \( z = 1 \), aligning closely with the first point of the empirical rule.

The area under the curve of a probability distribution represents the total probability, which is always equal to 1. For a standard normal distribution, this area helps us understand the probability of different z-scores. We use z-tables to find the area to the left of a specific z-score.

• To find the probability between two z-scores, we take the difference between their corresponding areas in the z-table.

For example, in this exercise, the area between \( z = -1 \) and \( z = 1 \) was found by subtracting the area at \( z = -1 \) from the area at \( z = 1 \): 0.8413 - 0.1587 = 0.6826. This value represents the probability or the proportion of data between these z-scores, and converting it to a percentage, gives us about 68.26%.

Standard deviation is a crucial measure of the spread or dispersion of a set of data points. It indicates how much the values deviate from the mean on average. The formula to calculate the standard deviation is:
\[ \sigma = \sqrt{ \frac{1}{N-1} \sum_{i=1}^{N} (x_i - \mu)^2 } \]
Here:

• \( N \) is the number of data points.
• \( x_i \) is each individual data point.
• \( \mu \) is the mean of the data points.

In the context of standard normal distribution, the standard deviation is normalized, meaning it is set to 1. This standardization simplifies the process of finding probabilities related to various z-scores. For instance, in our original problem, we observe that an element within \( z = -1 \) to \( z = 1 \) lies within one standard deviation of the mean, representing 68.26% of the area under the curve.