Problem 21 - Algebra 2

Question

The table lists the areas of some large shopping malls in the United States. Mall \(\qquad\) Gross Leasable Area \(\left(\mathrm{ft}^{2}\right)\) 1 Del Amo Fashion Center, Torrance, CA 3,000,000 2 South Coast Plaza/Crystal Court, Costa Mesa, CA 2,918,236 3 Mall of America, Bloomington, MN 2,472,500 4 Lakewood Center Mall, Lakewood, CA 2,390,000 5 Roosevelt Field Mall, Garden City, NY 2,300,000 6 Gurnee Mills, Gurnee, IL 2,200,000 7 The Galleria, Houston, TX 2,100,000 8 Randall Park Mall, North Randall, OH 2,097,416 9 Oakbrook Shopping Center, Oak Brook, IL 2,006,688 10 Sawgrass Mills, Sunrise, FL 2,000,000 10 The Woodlands Mall, The Woodlands, TX 2,000,000 10 Woodfi eld, Schaumburg, IL 2,000,000 Find the mean, median, and mode of the gross leasable areas.

Step-by-Step Solution

Answer

Mean = 2,373,736.67 ft²; Median = 2,150,000 ft²; Mode = 2,000,000 ft².

Step 1: Confirm Number of Malls

First, count the number of shopping malls listed to ensure all data is considered. There are a total of 12 shopping malls as per the given data.

Step 2: List All Gross Leasable Areas

List all the areas: \(3,000,000; 2,918,236; 2,472,500; 2,390,000; 2,300,000; 2,200,000; 2,100,000; 2,097,416; 2,006,688; 2,000,000; 2,000,000; 2,000,000\). This step organizes all the data we will need to perform calculations.

Step 3: Calculate the Mean

Add all the gross leasable areas and divide by the number of malls. \[\text{Mean} = \frac{3,000,000 + 2,918,236 + 2,472,500 + 2,390,000 + 2,300,000 + 2,200,000 + 2,100,000 + 2,097,416 + 2,006,688 + 2,000,000 + 2,000,000 + 2,000,000}{12}\]Calculate the total sum: 28,484,840 ft². \[\text{Mean} = \frac{28,484,840}{12} = 2,373,736.67\text{ ft}^2\]

Step 4: Find the Median

Sort the listed areas in ascending order:\[2,000,000, 2,000,000, 2,000,000, 2,006,688, 2,097,416, 2,100,000, 2,200,000, 2,300,000, 2,390,000, 2,472,500, 2,918,236, 3,000,000\]Since there are 12 numbers, find the average of the 6th and 7th numbers for the median.Median = \( \frac{2,100,000 + 2,200,000}{2} = 2,150,000 \text{ ft}^2 \)

Step 5: Determine the Mode

Identify the number that appears most frequently. The value 2,000,000 appears three times, more than any other value. Mode = 2,000,000 ft².

Key Concepts

Mean

The mean is a type of average that represents the central value of a data set. To calculate the mean, you sum up all the numbers and then divide by the total count of numbers. In this exercise, we are looking at the gross leasable areas of 12 shopping malls.
The formula for the mean is:

• Add all the gross leasable areas together.
• Then, divide by the total number of malls, which is 12.

When you calculate the mean, each mall's area is considered. It's a way to get an average size if each mall were to have the same area. For these specific malls:\[\text{Mean} = \frac{3,000,000 + 2,918,236 + \dots + 2,000,000}{12} = 2,373,736.67\text{ ft}^2\]
This calculated value depicts that if the total leasable area was equally divided among 12 malls, each would have around 2,373,736.67 square feet.

Median

The median is the middle value in a data set, which separates the higher half from the lower half. To find the median, arrange the numbers in ascending order and determine the middle number. However, if there is an even number of data points, the median will be the average of the two central numbers.

In this problem, the gross leasable areas need arranging so we can find the median:

• Order the areas in increasing order: 2,000,000, 2,000,000, 2,000,000, 2,006,688, 2,097,416, 2,100,000, 2,200,000, 2,300,000, 2,390,000, 2,472,500, 2,918,236, 3,000,000.
• Locate the central numbers since we have 12 values.
• Calculate the median by averaging the 6th and 7th numbers: 2,100,000 and 2,200,000.

The median calculated will be:\[\text{Median} = \frac{2,100,000 + 2,200,000}{2} = 2,150,000\text{ ft}^2\]
This shows that half of the mall areas are below 2,150,000 square feet and half are above.

Mode

The mode is the number in a data set that appears most frequently. It indicates the value that is most common. In some cases, a data set may have more than one mode, or no mode at all if no number repeats. For this analysis, the gross leasable area values are examined to find the most frequent one.

While checking the mall areas in the problem, the occurrence of each area is reviewed:

• Identify any repetitions in the data.
• The area value that appears most frequently in this data set is 2,000,000.

This means the mode is: 2,000,000 ft². The mode provides insight into what is typical in a data set, and in this context, it shows that several malls have a gross leasable area of 2,000,000 square feet, the most common size amongst the malls listed.