Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The weights, in ounces, of ten packages of potato chips are \(\begin{array}{llllllllll}16.1 & 16 & 15.8 & 16 & 15.9 & 16.1 & 15.9 & 16 & 16 & 16.2\end{array}\) Find the average and the median of these weights.

Step-by-Step Solution

Answer

The average and median weight of the ten packages of potato chips are both \(16\) ounces.

Step 1: Data arrangement

Arrange the given weights in an ascending order: \(15.8, 15.9, 15.9, 16, 16, 16, 16.1, 16.1, 16.2\)

Step 2: Calculate the average (mean) weight

To calculate the average, find the sum of the given weights and divide them by 10, since there are 10 packages. Average = \(\frac{15.8+15.9+15.9+16+16+16+16.1+16.1+16+16.2}{10}\) Average = \(\frac{160}{10} = 16\) Average weight = \(16\) ounces.

Step 3: Calculate the median weight

With the values in ascending order, the median is either the middle value or the average of the two middle values if there is an even number of values. Since there are 10 values, an even number, we need to take the average of the 5th and 6th values. Median = \(\frac{16 + 16}{2} = 16\) Median weight = \(16\) ounces.

Step 4: Conclusion

The average weight of the ten packages of potato chips is \(16\) ounces, and the median weight is also \(16\) ounces.

Key Concepts

Mean

In statistics, the mean is a common measure of the central tendency of a data set. It's often referred to as the "average" and serves as a useful indicator of the overall level of a particular set of data points.

To find the mean, add up all the numbers in the data set and then divide by the number of data points you have. This gives you an idea of the "typical" value that represents your data set.

In our exercise, with weights being 16.1, 16, 15.8, 16, 15.9, 16.1, 15.9, 16, 16, and 16.2 ounces, the mean calculation was done by summing up these values and dividing by 10. After calculating, the mean turned out to be 16 ounces.

• The mean is useful for comparing different sets of data.
• It's important to note that mean can be affected by extreme values (outliers) in the data set.
• Always arrange data logically when calculating the mean for accuracy.

Understanding the mean is crucial for interpreting data results and applying this knowledge practically in data analysis.

Median

The median is another important measure of central tendency in statistics. Unlike the mean, the median is the middle value of a data set when arranged in ascending or descending order. This makes it less sensitive to outliers.

To find the median:

• List the numbers from smallest to largest.
• If there is an odd number of items, the median is the middle number.
• If there is an even number, calculate the mean of the two middle numbers.

In the case of the potato chips weights exercise, the data set consists of 10 numbers, making it an even numbered set. By arranging the weights: 15.8, 15.9, 15.9, 16, 16, 16, 16, 16.1, 16.1, 16.2, the median is calculated to be 16 ounces, as it is the average of the 5th and 6th values.

The median offers a quick glance into the dataset without being skewed by outliers and is often used in skewed distributions where the mean might not accurately reflect the center of the data.

Data Analysis

Data analysis is the process of inspecting, cleaning, transforming, and modeling data to gain useful information, inform conclusions, and support decision-making. It is an essential part of statistics and involves multiple steps, such as understanding data context, collecting information, calculating statistics like mean and median, and finally, interpreting results.

In data analysis:

• Start by arranging your data logically to ensure clarity and accuracy.
• Calculate various statistical metrics like mean, median, mode, range to get insights from your data.
• Analyze different patterns or deviations that might show any significant trends or outliers.

Consider the potato chips example: by analyzing the weights, finding the mean and median helped understand both the average weight and the midpoint of the dataset. Such measures provide the basis for comparison and deeper insights into any anomalies or consistencies within the data.

Effective data analysis is vital, as it supports making informed decisions based on evidenced patterns and trends.