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A z-score is a way of expressing a data point in relation to the mean, measured in terms of standard deviations. It's a very useful statistic in the world of normal distributions because it allows us to easily understand and compare different values from various distributions. The formula to calculate a z-score is:

• \( z = \frac{x - \mu}{\sigma} \)

Where:

• \( x \) is the value you're looking at,
• \( \mu \) is the mean of the distribution,
• \( \sigma \) is the standard deviation.

To standardize or find the z-score for a range, such as from \( \mu + \sigma \) to \( \mu + 2\sigma \), you substitute these expressions into the formula to compute the respective z-scores. By doing this, values from different distributions can be directly compared using their z-scores. The z-score helps in determining how far away a particular data point is from the mean in terms of standard deviations.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. When we speak of a normal distribution, the standard deviation tells us how spread out the values are from the mean. The smaller the standard deviation, the closer the data points are to the mean, resulting in a steeper bell-curve shape. Conversely, a larger standard deviation indicates a wider spread, creating a flatter shape on the graph. The standard deviation is pivotal when working with normal distributions because it tells us the 'spread' or 'width' of the data. In statistics, it allows precise statements about where particular fractions of the population, or probabilities, lie in relation to the mean. For instance, within one standard deviation of the mean, approximately 68% of values lie in a normal distribution. When applied to a range, it helps us define intervals such as \( [\mu + \sigma, \mu + 2\sigma] \), quantifying the fraction of data within this space.

The standard normal distribution table, often referred to as a z-table, helps determine the percentage of values below a certain z-score in standard normal distribution. This table is essential for converting z-scores into probabilities.When you calculate a z-score, you can look it up in the z-table to find the cumulative probability up to that z-score. This value tells you the proportion of data below that z-score in a standard normal distribution. Here's how it works:

• Find the z-score in the table (most tables showcase the first digit and first decimal together, with the second decimal being isolated on a second column).
• Locate the cumulative probability corresponding to the z-score.

For intervals, such as between \( z = 1 \) and \( z = 2 \), you find the cumulative probabilities for both values and subtract them: \( P(z = 2) - P(z = 1) \). This result gives you the probability of a data point falling within this specific range. Such tables are an invaluable tool in statistics, particularly when analyzing data and making probability statements within normally distributed data sets.