Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The total number of pieces of mail delivered (in billions) each year from 2002 through 2006 is given in the following table: $$\begin{array}{lccccc}\hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Number } & 203 & 202 & 206 & 212 & 213 \\ \hline\end{array}$$ What is the average total number of pieces of mail delivered from 2002 through 2006 ? What is the standard deviation for these data?

Step-by-Step Solution

Answer

The average total number of pieces of mail delivered from 2002 through 2006 is \(207.2\) billion, and the standard deviation for these data is \(4.53\) billion pieces of mail.

Step 1: Calculate the Mean

First, add the total number of pieces of mail delivered for each year together: \[203 + 202 + 206 + 212 + 213 = 1036\] Now, divide this sum by the number of years (5) to obtain the mean: \[\frac{1036}{5} = 207.2\] The average total number of pieces of mail delivered from 2002 through 2006 is 207.2 billion.

Step 2: Calculate the Deviations from the Mean

Now, find the deviation of each data point from the mean. Subtract the mean from each year's total number of pieces of mail delivered: \[203 - 207.2 = -4.2\] \[202 - 207.2 = -5.2\] \[206 - 207.2 = -1.2\] \[212 - 207.2 = 4.8\] \[213 - 207.2 = 5.8\]

Step 3: Square the Deviations

Next, square each of these deviations: \[(-4.2)^2 = 17.64\] \[(-5.2)^2 = 27.04\] \[(-1.2)^2 = 1.44\] \[(4.8)^2 = 23.04\] \[(5.8)^2 = 33.64\]

Step 4: Calculate the Mean of the Squared Deviations

Add up the squared deviations and divide by the number of data points (5): \[\frac{17.64 + 27.04 + 1.44 + 23.04 + 33.64}{5} = \frac{102.8}{5} = 20.56\]

Step 5: Calculate the Square Root of the Mean of the Squared Deviations

Take the square root of the mean of the squared deviations to obtain the standard deviation: \[\sqrt{20.56} = 4.53\] The standard deviation for these data is 4.53 billion pieces of mail. In conclusion, the average total number of pieces of mail delivered from 2002 through 2006 is 207.2 billion, and the standard deviation for these data is 4.53 billion pieces of mail.

Key Concepts

Mean Calculation

Understanding how to calculate the mean is essential in statistics, as it provides us with the average value of a dataset. For this example, we calculate the mean of the total number of pieces of mail delivered from the years 2002 to 2006. To find the mean:

• Add up all the data points: the number of mail pieces for each year.
• For the years 2002 to 2006, this addition is: 203 + 202 + 206 + 212 + 213 = 1036.
• Count the total number of data points, in this case, the number of years, which is 5.
• Divide the sum of the data points by the number of data points: \( \frac{1036}{5} = 207.2 \).

Thus, the average number of pieces of mail delivered over these five years is 207.2 billion. This value offers a central location around which the values are spread, helping to summarize the data effectively.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are. Calculating it involves several steps to find how much each data point differs from the mean, on average.

• Find deviations: Subtract the mean from each data point to find the deviation of each year’s mail count from the mean \(207.2\).
• Square the deviations: Squaring the deviations ensures they are positive and emphasizes larger differences.
• Calculate the average of squared deviations: Sum these squared deviations up and divide by the number of data points, 5.
• Take the square root: The final step is obtaining the square root, which returns the value back to the original unit (billions of mail pieces), giving us the standard deviation \( \sqrt{20.56} = 4.53 \).

This 4.53 billion provides insight into the variability of the mail count year over year; smaller values would suggest the data points are closely clustered around the mean, while larger values indicate a wider spread.

Data Analysis

Data analysis involves systematically examining datasets to uncover trends and patterns. With our mail delivery data, we've calculated both the mean and standard deviation, important tools in data analysis.

Why it matters:

• The mean gives us a quick snapshot of the average mail volume over the examined period, providing a baseline for comparison with future data.
• The standard deviation shows variability, indicating the stability (or lack thereof) in mail delivery over these years.

To further analyze, consider additional factors such as trends over a longer period or correlating with other changes (technological advancments or postal regulations), contributing a richer understanding. This approach helps businesses and researchers make informed decisions from data-driven insights, as it distills complex datasets into understandable metrics. In this example, it would help in future planning for resource allocation and service enhancement.