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The Z-score is an incredibly useful statistical tool that tells us how far away any given value is from the mean in terms of standard deviations. In a normal distribution, these standard deviations effectively indicate how many steps a particular value is from the average value. Calculating the Z-score is straightforward using the formula:

• \[Z = \frac{X - \mu}{\sigma}\]

Here, \(X\) is the value for which we want to calculate the Z-score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Your Z-score will tell you how unusual or common your value is compared to the rest of the data set. The further the Z-score is from zero, the more exceptional the value is.
In our exercise, the Z-score was calculated by determining how many standard deviations below or above the mean the value 1.54 V was. By determining this, it becomes easier to use additional tools like the Z-table to find probabilities related to that value.

The mean and standard deviation are fundamental concepts in statistics and crucial for understanding normal distributions. The mean (\(\mu\)) is the average or the center point of your data set. It provides a reference point around which the data is distributed.
The standard deviation (\(\sigma\)) measures the dispersion of the data points around the mean. It essentially quantifies the amount of variation or spread in a given data set. A smaller standard deviation means data points are close to the mean, while a larger standard deviation indicates that data points are spread out over a wider range of values.
In the context of our battery voltage example, the mean voltage \(\mu\) is 1.50 V, indicating where the center of our distribution lies, and the standard deviation \(\sigma\) is 0.05 V, showing us the typical deviation of the voltages from the mean. Together, these two metrics help define the shape and spread of the distribution, making it easier to assess where most voltages fall.

To determine probabilities using the normal distribution, one typically uses the Z-table. This table helps translate Z-scores into probabilities or percentages of the distribution.
The Z-table lists the probabilities that a value falls below a specific Z-score in a standard normal distribution where the mean is zero, and the standard deviation is one. This is also known as the cumulative probability. By finding your Z-score and matching it to a point on the Z-table, you can easily find out what percentage of your data falls below that value.
In the given exercise about battery voltages, after calculating the Z-score of 0.8, we referred to the Z-table to discover that approximately 78.81% of batteries have a voltage below 1.54 V. This is a key aspect of probability calculation in statistics because it helps predict the likelihood of certain outcomes within a set of normally distributed data.