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In the context of a normal distribution, the mean and standard deviation are crucial concepts to understand. The mean, often denoted as \( \mu \), represents the average value in a set of data. In simpler terms, it's what you might call the 'typical' value. For the given problem about electricity consumption, the mean is 1650 kilowatt-hours, suggesting that on average, households in the Boston area use this amount per month during winter.

The standard deviation, denoted as \( \sigma \), measures how much the values in a data set vary from the mean. In this case, the standard deviation is 320 kilowatt-hours. This implies that a majority of the household electricity consumptions are expected to be within 320 units of the mean (either above or below).

Why is this important? Well, understanding the spread and average of your data helps in making informed predictions about individual data points. For instance, even if most households use around 1650 units, the standard deviation tells us that there is still significant variation around this average, providing insights into the diversity of consumption habits among households.

The Z-score is a way of standardizing individual points within a normal distribution. It indicates how far and in what direction, a single data point deviates from the mean, measured in terms of standard deviations. For example, if a household's consumption is exactly at the mean, its Z-score is 0, indicating no deviation.

To compute a Z-score, you can use the formula:\[ Z = \frac{X - \mu}{\sigma} \]

- \( X \) is the value you want to standardize.
- \( \mu \) is the mean of the distribution.
- \( \sigma \) is the standard deviation.

In the problem, Bill Johnson's household is in the 90th percentile of electricity usage, which means only 10% of households use more electricity than he does. Therefore, his Z-score is approximately 1.282, indicating that his consumption is 1.282 standard deviations above the mean. This allows us to precisely understand his consumption level relative to all other households.

Percentiles are a great way to understand how an individual data point relates to the rest of the data in a normal distribution. A percentile indicates the value below which a percentage of data falls. For instance, being in the 90th percentile means a data point is higher than 90% of all other values in the data set.

In our problem about electricity consumption, Bill Johnson's household usage is in the 90th percentile. This shows that about 90% of the other households have lower electricity usage than him. It provides a context to understand where he stands in comparison to the 'average' household. This concept is very useful in various real-world applications, such as determining scores on tests, understanding spending habits, or, as in this problem, consumption patterns.

Understanding percentiles can help make strategic decisions, as they provide perspective not just on individual scores, but on how these scores fit into the larger data distribution, offering a view of both average and extraordinary values within a dataset.