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Standard deviation is a key concept in statistics that measures how much the values in a data set deviate from the mean of that set. In simpler terms, it tells us how spread out the data is. In the context of a standard normal distribution, the standard deviation is always 1. This is because the standard normal distribution, also known as the Z-distribution, has been standardized.
This means that the original data has been transformed in such a way that the average, or mean, is 0 and the spread of the data, or standard deviation, is 1. These transformations allow us to use the Z-table effectively.

• A lower standard deviation means the data is closer to the mean.
• A higher standard deviation indicates a wider spread of data points.

Using this information, we can better understand probability scores from a standard normal distribution. Each Z-score tells us how many standard deviations away from the mean a particular value is, which gives context to its placement on the curve.

Probability is a measure of the likelihood that a specific event will occur. In the realm of normal distributions, we often discuss probabilities in terms of areas under the curve.
For a standard normal distribution, this area represents the probability that a value, or random variable, falls within a certain interval. When we are calculating probabilities like in the exercise above, we use these intervals to express likelihoods.

• Probability values range from 0 to 1.
• An area under the curve approaching 1 signifies a higher probability.
• In our example, a probability of 0.4474 means there is a 44.74% chance that the random variable falls between the two Z-values.

Understanding this helps us infer what percentage of data lies between two points in a normal distribution. It is fundamental to making informed predictions and decisions based on statistical data.

The Z-table, also known as the standard normal table, is a crucial tool that lists Z-scores alongside their cumulative probabilities. This table helps us quickly find how much of the data falls below a certain Z-score in a standard normal distribution.
To use a Z-table, simply locate the row that corresponds to the first decimal of the Z-score and the column that represents the second decimal. You then find where they intersect to get the probability.

• The values represent cumulative probabilities, including all probabilities at and below a given Z-score.
• In practice, for a \( z = 1.62 \), the probability \( P(Z \leq 1.62) \) is approximately 0.9474, meaning 94.74% of values fall below this Z-score.
• Similarly, \( P(Z \leq 0) \), which corresponds to the mean, is always 0.5 or 50%.

This step-by-step process helps transform complex probability queries into straightforward numerical interpretations, making the Z-table a powerful asset in statistics.