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The z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It is used to identify how far and in what direction a particular data point is from the mean. To calculate the z-score, we use the formula:- \( z = \frac{x - \mu}{\sigma} \)where:- \(x\) is the value in question,- \(\mu\) is the mean of the distribution,- \(\sigma\) is the standard deviation.The z-score tells us how many "standard deviations" away \(x\) is from the \(\mu\). If the score is above 0, it indicates the value is above the mean. If it's below 0, it's below the mean. A z-score of 1.82, as in the given problem, signifies the value is 1.82 standard deviations above the mean.

Standard deviation is a key concept in statistics that measures how spread out the values in a data set are around the mean. It provides insight into the variability and dispersion of a dataset.The formula for standard deviation is:- \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \)where:- \(x_i\) are the individual data points,- \(\mu\) is the mean,- \(N\) is the number of data points.A smaller standard deviation indicates that the values tend to be closer to the mean, whereas a larger standard deviation suggests a wider spread of values. In the garage door opener example, a standard deviation of 11 feet tells us there is some variability in how far users typically are from their garage doors when activating the opener.

The mean, often referred to as the average, is a measure of central tendency that summarizes all the values in a dataset. It is calculated by adding up all the values and dividing by the total number of values.The formula for the mean is:- \( \mu = \frac{\sum x_i}{N} \)where:- \(x_i\) represent each data point,- \(N\) is the total number of data points.In statistical contexts, the mean provides a snapshot of the overall distribution of values. For example, in the scenario with the garage door openers, the mean of 30 feet suggests that on average, users attempt to operate their garage door opener from 30 feet away. This measure acts as a baseline, helping us understand how other values in the distribution compare.