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The range in descriptive statistics refers to the simplest measure of variability. It tells us the spread within a data set by calculating the difference between the maximum and minimum values. In our example, we found the ages of Canadian tourists which are

• Maximum age: 72
• Minimum age: 17

The formula to calculate the range is: \[\text{Range} = \text{Maximum value} - \text{Minimum value}\]Applying the numbers, we get: \[\text{Range} = 72 - 17 = 55\]The range highlights the broadness between the oldest and youngest tourists in this context. However, it doesn't inform us about individual distributions in between. Therefore, while it's a good starting point, additional measures can provide deeper insights.

Mean deviation, also known as the average absolute deviation, helps us understand the consistency of the data relative to the mean. It shows the average distance of each data point from the mean value. In our case, where we calculated a mean age of 43.2 years, we found the absolute deviations for each age. The formula is represented as: \[\text{Mean Deviation} = \frac{\sum |x - \bar{x}|}{N}\]Where

• \( x \) is each data point,
• \( \bar{x} \) is the mean,
• \( N \) is the number of observations.

After calculating all absolute deviations (e.g., \(|32 - 43.2|=11.2\)), and summing them: \( 144 \), the mean deviation is: \[\frac{144}{10} = 14.4\]This measure is advantageous because it's simple and lends a clear understanding of how the ages vary around the mean age, bridging the gap where the range falls short.

Standard deviation provides a deeper insight into data variability than the mean deviation. It represents the average spread of data points from the mean, considering both the direction and weight of the distance due to squaring each deviation. The formula is a square root of variance: \[\text{Standard Deviation } \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{N}}\]In the example, we derived a variance of 112.16 after calculating square deviations like \((32 - 43.2)^2\), summing them: \(1121.6\), and dividing by 10: Thus, the standard deviation becomes: \[\sigma = \sqrt{112.16} \approx 10.59\]This value tells us that on average, each tourist's age deviates approximately 10.59 years from the average age. It's a crucial tool in understanding data spread and assessing risks or variances in fields like finance and quality control, as it considers even slight data fluctuations.