91Ó°ÊÓ

In statistics, the concepts of mean and standard deviation are fundamental when discussing data sets. The mean, or average, is a measure that represents the central point of a dataset. It is calculated by summing all data points and dividing by the number of points. For example, in our electricity consumption scenario, the mean consumption for households is 1650 kilowatt-hours per month.

On the other hand, the standard deviation provides insight into how spread out the data points are around the mean. A low standard deviation indicates that the data points tend to be close to the mean, whereas a high standard deviation suggests they are more spread out. In this case, a standard deviation of 320 kilowatt-hours means that individual household electricity consumption can vary significantly from the mean of 1650 kilowatt-hours. This variability is crucial for understanding the distribution of electricity usage among different households.

Percentiles are a way of understanding the relative position of a data point within a dataset. If a score is in the 90th percentile, it means that it exceeds 90% of the other scores. In the context of our example, Bill's electricity consumption is higher than 90% of the households, indicating he is at the 90th percentile.

The z-score, on the other hand, is a statistical measure that tells us how many standard deviations a particular score is from the mean. It is crucial in transforming a data point from a standard normal distribution to a specific dataset. For the 90th percentile, the z-score is approximately 1.28. This number helps us translate the percentile rank into a real-life measurement using the formula:

• \(X = \mu + Z\sigma\)
• \(\mu\) is the mean
• \(Z\) is the z-score
• \(\sigma\) is the standard deviation

By applying these values, we calculate Bill's electricity consumption.

Analyzing electricity consumption, especially during specific seasons like winter, is essential for energy companies and households alike. Such analyses help in understanding usage patterns and preparing adequate resources to meet demand.

In our scenario, the electric company has provided that the typical household consumes on average 1650 kilowatt-hours per month. Knowing these details helps the company predict overall energy needs and anticipate any potential issues in supply or overly high consumption periods. Understanding these elements is not only beneficial for the company but also for consumers. It allows households like Bill's to compare their usage against typical consumption rates and adjust their habits if necessary.

Using statistics such as mean, standard deviation, percentiles, and z-scores, households and companies can make informed decisions. This means better energy efficiency and resource management across the board.