Question

Table 17.2 shows the number of passengers taking a particular daily flight from Boston to Washington over the course of a week. Find the mean number of passengers for the week. $$ \begin{array}{c|c|c|c|c|c|c|c} \hline \text { Day } & \text { Mon } & \text { Tue } & \text { Wed } & \text { Thur } & \text { Fri } & \text { Sat } & \text { Sun } \\ \hline \text { Passengers } & 228 & 110 & 215 & 178 & 140 & 72 & 44 \\ \hline \end{array} $$

Short answer

Answer: 141 passengers

Step-by-step solution

Sum the number of passengers for each day

We have the number of passengers for each day: \(Mon = 228, Tue = 110, Wed = 215, Thur = 178, Fri = 140, Sat = 72, Sun = 44\). We need to find the total number of passengers for the week, so we add them all together: \(Total = 228 + 110 + 215 + 178 + 140 + 72 + 44\)

Calculate the total number of passengers for the week

To find the total number of passengers for the week, we complete the sum from Step 1: \(Total = 987\)

Divide by the number of days

Since we're trying to find the mean number of passengers for the week, we need to divide the total number of passengers (987) by the total number of days in the week (7): $$Mean\,of\,passengers = \frac{987}{7}$$

Find the mean number of passengers

Dividing the total number of passengers by the number of days gives us the mean number of passengers for the week: $$Mean\,of\,passengers = 141$$ So, the mean number of passengers per day for that week is 141 passengers.

Key concepts

Sum of a Data Set

Before we can calculate the mean of a data set, the first step is to find the sum of all the elements in that set. In the context of our exercise, we're given the daily number of passengers on a flight from Boston to Washington over a week. To find the total number of passengers for the entire week, we add together the number of passengers for each day.
Here's a breakdown of the passengers by day:

• Monday: 228 passengers
• Tuesday: 110 passengers
• Wednesday: 215 passengers
• Thursday: 178 passengers
• Friday: 140 passengers
• Saturday: 72 passengers
• Sunday: 44 passengers

To find the total, you simply add them: \[Total = 228 + 110 + 215 + 178 + 140 + 72 + 44 = 987\]Making sure to carefully add each value helps ensure accuracy in your calculations.

Division for Averaging

After determining the sum of the data set, the next step in finding the mean is division, as averaging essentially involves dividing a total by the number of items in the set. The purpose of this step is to distribute the total sum equally across all elements, giving us an average value.
For the weekly passenger data, we have a total of 7 days in the week. Therefore, we divide the total number of passengers, 987, by 7.
Here's the formula for dividing:\[\text{Average} = \frac{\text{Total Sum of Data Set}}{\text{Number of Items in the Data Set}}\]Applying this to our data gives:\[\text{Mean of passengers} = \frac{987}{7}\]This calculation will give us the average number of passengers per day over the week. Division is key to finding averages because it helps distribute a sum evenly across a given set of elements.

Arithmetic Mean

The arithmetic mean, often just called the mean or average, is a central value of a data set. It serves as a simple measure of central tendency, providing a single value that represents the data as a whole. In this exercise, finding the mean allows the airline to understand the typical number of passengers their flight carries daily.
The formula used to compute the arithmetic mean is:\[\text{Arithmetic Mean} = \frac{\text{Sum of all Data Points}}{\text{Total Number of Data Points}}\]In our example, the arithmetic mean of the passengers is 141, calculated by dividing the total, 987, by the number of days, 7.Understanding the mean can be incredibly useful in various fields, from business to science, because it provides a clear summary that facilitates decision-making and insights into data trends.