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Percentile calculation is an essential concept to understand when dealing with data distributions. It helps determine where a particular score or value lies in relation to the entire dataset. In simple terms, the percentile of a value is the percentage of data points in a dataset that are below that particular value. To calculate percentiles, especially in a symmetrical distribution, it is helpful to understand the empirical rule (also known as the 68-95-99.7 rule). This rule provides a quick way to estimate where certain percentiles lie based on the properties of the normal distribution. In the problem, for instance, the 16th percentile can be approximated using the mean minus one standard deviation:

• The mean is 1400 gallons.
• The standard deviation is 300 gallons.
• Using the empirical rule, the 16th percentile is approximately one standard deviation below the mean.

Calculating this, we find the 16th percentile is approximately at 1100 gallons, making it lower compared to most data points.

A symmetric distribution is one where data is evenly distributed on both sides of the mean. Think of it like a mirror image; both halves of the distribution are identical. This characteristic is crucial because it greatly influences how we view concepts like the median and percentiles. In a symmetric distribution, like the one described in the problem, the mean and the median are equal. This is because exactly half of the data lies on each side of the mean. So if you were to divide the data down the middle, you would find the median right at the mean, which is 1400 gallons in this scenario. This symmetry property simplifies calculations significantly, as evidenced by our ability to deduce the value of the median without further computation, once we know the mean. Understanding symmetric distributions can also help in visualizing the spread of data and how it clusters around the central point.

Standard deviation is a measure of how spread out the data points in a distribution are. It tells us the typical distance of each data point from the mean. A smaller standard deviation indicates that the data points are close to the mean, whereas a larger standard deviation signifies that they are spread out over a wider range.In the given problem, the standard deviation is 300 gallons. This mark gives us a sense of the variability in water usage among different homes. When we utilize the empirical rule for normal distributions:

• About 68% of the data falls within one standard deviation (\(mean \pm \sigma\))
• 95% falls within two standard deviations
• 99.7% lies within three standard deviations

Analyzing the standard deviation alongside the mean enables us to make assessments about percentiles and tendencies within a dataset. For example, the 84th percentile can be quickly approximated by adding one standard deviation to the mean, resulting in 1700 gallons in this problem, indicating a relatively high water usage compared to other homes.