91Ó°ÊÓ

The concept of Z-score is a fundamental element in statistics that helps us understand how a particular value compares to a group of values. When we calculate the Z-score, we essentially determine the number of standard deviations a particular data point is from the mean. It's represented by the formula:

• \[ z = \frac{(x - \mu)}{\sigma} \]

where \( x \) is the individual data point, \( \mu \) is the mean of the data, and \( \sigma \) is the standard deviation.

A positive Z-score indicates that the data point is above the mean, while a negative Z-score signifies it is below the mean. In the exercise, the Z-score of 5.32 tells us that the height of the tallest building is 5.32 standard deviations above the average building height. This large positive Z-score highlights the building's unusual height in comparison to the rest.

Standard deviation is an essential statistical tool that quantifies the amount of variation or dispersion in a set of data. A low standard deviation indicates that the data points are generally close to the mean, while a high standard deviation suggests that the data points are spread out over a wider range. The formula for calculating standard deviation in a sample is:

• \[\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (x_i - \mu)^2}\]

Here, \( x_i \) represents each data point, \( \mu \) is the mean, and \( N \) is the total number of data points.

In our exercise example, the standard deviation was given as \( 106.32 \) ft, which provides context for the variability among the building heights within the sample. Understanding standard deviation helps set a benchmark, which is crucial when it comes to determining outliers.

Outliers are data points that differ significantly from other observations. Statistically, one effective method to detect outliers is the three-standard-deviation rule. According to this criterion, any observation with a Z-score greater than 3 or less than -3 may be considered a potential outlier.

In our city building example, the calculated Z-score for the tallest building was \( 5.32 \). Since this Z-score is well above 3, the building is categorized as a potential outlier. This implies that its height is significantly higher than that of the majority of buildings, potentially indicating an anomaly in the dataset.

Recognizing outliers can help in understanding unusual data patterns, making decisions about data cleaning, and correctly interpreting statistical findings.