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Standard deviation is a key measure in statistics that helps us understand the amount of variation or dispersion in a set of values. When we apply it to a normal distribution, it tells us how spread out the data points are around the mean (average) value.
In the context of the exercise, with a mean annual spending of $1,994 on reading and entertainment, and a standard deviation of $450, the standard deviation indicates that most adults spend within $450 more or less than the mean. This gives us a range of spending for the majority of people in the group.
If the standard deviation is small, the data points are close to the mean, indicating less variability. A larger standard deviation means the data is more spread out. This tool is crucial for understanding how typical or atypical a particular data point is, relative to the rest of the data.

A Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It is expressed in terms of standard deviations.
The formula to calculate a Z-score is: \[ z = \frac{x - \mu}{\sigma} \] where :

• \( x \) is the value you're examining,
• \( \mu \) is the mean,
• \( \sigma \) is the standard deviation.

In the exercise, when calculating the Z-score for \(2,500, we used the mean \)1,994 and standard deviation \(450. The Z-score tells us how far, in units of standard deviations, \)2,500 is from the mean. By knowing the Z-score, we can use a standard normal distribution table to find the probability or percentage of data points falling above or below that Z-score. In this way, Z-scores can effectively standardize different data sets on a uniform scale, which makes comparisons and interpretations more intuitive.

To calculate the percentages of the data within certain ranges of a normal distribution, we often use Z-scores as an intermediary step. Once we have standardized our data with Z-scores, we use a standard normal distribution table (or calculator) to find the percentage of data points to the left or right of a given Z-score.

• In cases like the exercise, where we need to find the percentage of adults spending more than a certain amount, we first determine the area to the left of the Z-score and subtract it from total one to find the desired percentage to the right.
• For spending ranges, like between $2,500 and $3,000, we find and subtract the percentages for each boundary using their respective Z-scores.
• For amounts less than a specific figure, such as $1,000, we directly use the area to the left of the relevant Z-score.

Calculating these percentages requires understanding the normal distribution's structure and knowing how to leverage the cumulative probability that Z-scores provide. This approach helps in quantifying and interpreting real-world scenarios with precision and ease.