Problem 69 - Basic Statistics for Business and Economics with Student

Question

The ages of a sample of Canadian tourists flying from Toronto to Hong Kong were: \(32,21,60,47,54,17,72,\) \(55,33,\) and 41 a. Compute the range. b. Compute the mean deviation. c. Compute the standard deviation.

Step-by-Step Solution

Answer

a. Range = 55. b. Mean deviation = 14.4. c. Standard deviation \(\approx 10.59\).

Step 1: Identify the Maximum and Minimum Ages

First, identify the maximum and minimum age among the given data set. The ages of the tourists are: \(32, 21, 60, 47, 54, 17, 72, 55, 33, 41\). The maximum age is 72 and the minimum age is 17.

Step 2: Compute the Range

The range is calculated as the difference between the maximum and minimum values. Thus, the range is \(72 - 17 = 55\).

Step 3: Compute the Mean

Calculate the mean of the ages by adding all the values together and dividing by the number of data points. Sum of ages = \(32 + 21 + 60 + 47 + 54 + 17 + 72 + 55 + 33 + 41 = 432\). Number of tourists = 10. Mean age \(\bar{x}\) = \(\frac{432}{10} = 43.2\).

Step 4: Calculate Absolute Deviations

Find the absolute deviation of each age from the mean. \(|32 - 43.2|=11.2, |21 - 43.2|=22.2, |60 - 43.2|=16.8, |47 - 43.2|=3.8, |54 - 43.2|=10.8, |17 - 43.2|=26.2, |72 - 43.2|=28.8, |55 - 43.2|=11.8, |33 - 43.2|=10.2, |41 - 43.2|=2.2\).

Step 5: Compute the Mean Deviation

Add all absolute deviations and divide by the number of data points to find the mean deviation. Sum of absolute deviations = \(11.2 + 22.2 + 16.8 + 3.8 + 10.8 + 26.2 + 28.8 + 11.8 + 10.2 + 2.2 = 144\). Mean deviation = \(\frac{144}{10} = 14.4\).

Step 6: Compute the Variance

Calculate the variance by finding the squared deviation of each age from the mean, adding these squares, and dividing by the number of data points. Squared deviations: \((32-43.2)^2\), \((21-43.2)^2\), \(etc.\). Sum of squared deviations = \(1121.6\). Variance \(\sigma^2 = \frac{1121.6}{10} = 112.16\).

Step 7: Compute the Standard Deviation

The standard deviation is the square root of the variance. Thus, standard deviation \(\sigma = \sqrt{112.16} \approx 10.59\).

Key Concepts

Understanding Range

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.

Exploring Mean Deviation

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.

Analyzing Standard Deviation

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.