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The interquartile range, or IQR, measures the spread of the middle 50% of a dataset. It is the difference between the third quartile (Q_3) and the first quartile (Q_1). This range is crucial because it helps understand how data is distributed around the median, or second quartile (Q_2). In our exercise, Q_1 = 252.4 and Q_3 = 454.5. So, the IQR is calculated as follows:
IQR = Q_3 - Q_1 = 454.5 - 252.4 = 202.1
This value of 202.1 specifies that the middle 50% of violent-crime rates fall within this range. Since it excludes the highest and lowest 25% of values, it offers a more robust measure of variability compared to the overall range, which might be affected by outliers. Understanding the IQR can help efficiently interpret the distribution characteristics and identify the extent of variability in a dataset.

Outliers are data points significantly different from others in a dataset. Their identification is essential because they can influence statistical analyses and the interpretation of data. To detect if Washington, D.C.’s violent-crime rate is an outlier, we use the IQR to calculate upper and lower bounds. If a data point lies outside these bounds, it is considered an outlier.
The bounds are determined as follows:
Lower Bound = Q_1 - 1.5 × IQR = 252.4 - 1.5 × 202.1 = -50.75
Upper Bound = Q_3 + 1.5 × IQR = 454.5 + 1.5 × 202.1 = 757.65
Any value below -50.75 or above 757.65 is an outlier. Since Washington, D.C.’s violent-crime rate is 1243.7, it clearly exceeds the upper bound of 757.65, marking it as an outlier. Recognizing outliers helps in understanding extreme variations and can indicate unusual conditions or data entry errors.

Skewness refers to the asymmetry of the data distribution. If data points are not symmetrically distributed around the mean, they exhibit skewness. Specifically, skewness can be detected by identifying if the tail on one side of the mean is longer or shorter. In our exercise, the presence of an extreme value like 1243.7 suggests the data might not be symmetrically distributed.
Given the significant outlier in Washington, D.C., the distribution of violent-crime rates is likely to be right-skewed (positively skewed). This kind of skewness results in a long right tail, meaning that a large number of smaller observations cluster on the left side with few large observations pulling the tail to the right. This is where the median becomes noticeably lower than the mean, signaling the impact of extreme values on the data distribution. Identifying skewness is crucial as it informs appropriate statistical methods and helps to understand underlying patterns in data.