Problem 16 - Algebra 2

Question

An electronics store placed an ad in the newspaper showing five flat-screen TVs for sale. The ad says, "Our flat-screen TVs average \(\$ 695 . "\) The prices of the flat- screen TVs are \(\$ 1200, \$ 999, \$ 1499, \$ 695, \$ 695, \$ 1100, \$ 1300,\) and \(\$ 695 .\) Find the mean, median, and mode of the prices.

Step-by-Step Solution

Answer

Mean is $1022.88, median is $1049.5, mode is $695.

Step 1: Understand the Problem

We need to find the mean, median, and mode of a given set of prices for flat-screen TVs.

Step 2: List the Prices

The prices of the flat-screen TVs listed are: \(\\(1200, \\)999, \\(1499, \\)695, \\(695, \\)1100, \\(1300, \\)695\).

Step 3: Calculate the Mean

Add all the prices together and then divide by the number of items. \[\text{Mean} = \frac{1200 + 999 + 1499 + 695 + 695 + 1100 + 1300 + 695}{8}\]\[= \frac{8183}{8} = 1022.875\]

Step 4: Find the Median

First, arrange the prices in ascending order: \(695, 695, 695, 999, 1100, 1200, 1300, 1499\). Since there are 8 prices, the median will be the average of the 4th and 5th values.\[\text{Median} = \frac{999 + 1100}{2} = 1049.5\]

Step 5: Determine the Mode

The mode is the number which appears most frequently. Here, \\(695 appears 3 times, more frequently than any other price.Thus, the mode is \\)695.

Key Concepts

Arithmetic Mean

The arithmetic mean, often simply called the "mean," is a measure of central tendency.It helps us understand the average value in a set of numbers. In this context, it is used to find the average price of flat-screen TVs in the given list.
To calculate the mean, simply sum up all the values and divide the result by the number of values. Here's the step-by-step calculation:

• First, add up all listed prices: 1200 + 999 + 1499 + 695 + 695 + 1100 + 1300 + 695 = 8183.
• Next, count the number of prices, which is 8 in this example.
• Finally, divide the total sum by the number of prices:\[\text{Mean} = \frac{8183}{8} = 1022.875\]

Thus, the arithmetic mean price of the TVs is approximately $1022.88.
This value provides an overall average for the listed TV prices, helping us quickly estimate a central value without getting overwhelmed by all individual prices.

Median

The median is another central tendency measure that provides the middle value in a data set when it is ordered. It differs from the arithmetic mean as it doesn't get influenced by extreme values or outliers.
To find the median, follow these steps:

• Start by arranging the list of prices in ascending order: 695, 695, 695, 999, 1100, 1200, 1300, 1499.
• Since there are 8 values, which is an even number, you need to find the average of the 4th and 5th values in the ordered series.
• The 4th and 5th values are 999 and 1100, respectively. Thus, median computation becomes: \[\text{Median} = \frac{999 + 1100}{2} = 1049.5\]

This indicates that the point at which half of the TV prices are lower and half are higher is $1049.50. The median provides a reliable middle ground value, unaffected by extremely high or low prices.

Mode

The mode represents the most frequently occurring number in a data set. It's useful in identifying the most common value among the prices.
To determine the mode, inspect all values to see which one appears most often. For the TV prices, it is clear that 695 appears three times, which is more frequently than any other given price:

• 695, 695, 695, 999, 1100, 1200, 1300, 1499

Therefore, the mode is 695.
The mode is particularly valuable when seeking the most popular choice or price among a group. It provides insight into common trends or preferences within the set of data, offering a quick way to spot prevalent figures.