Problem 1 - Algebra Form and Function

Question

Find the mean of the data set. $$ 2,4,6,8 $$

Step-by-Step Solution

Answer

Answer: The mean of the data set is 5.

Step 1: Add the values in the data set

Add the numbers 2, 4, 6, and 8 together: $$ 2 + 4 + 6 + 8 = 20 $$ The sum of the values in the set is 20.

Step 2: Divide the sum by the total number of values

The total number of values in the set is 4. Divide the sum (20) by the number of values (4) to calculate the mean: $$ \frac{20}{4} = 5 $$ The mean of the data set is 5.

Key Concepts

Understanding a Data Set

A data set in statistics is simply a collection of numbers or values that you are analyzing. This could be any set of numbers that you're interested in studying. In our example, the data set includes the numbers: 2, 4, 6, and 8.

• The data set provides the raw information that you work with in statistics.
• Each individual number in the set is called an "observational value."
• These values are usually listed in a straightforward way, from smallest to largest, or vice versa, to make them easier to work with.

When you want to know something specific about this collection of numbers, you can perform various calculations. One common calculation is finding the mean, which gives you an average value of your data set.

Calculating the Sum of Values

To find the mean of a data set, the first step is to add together all the numbers in the data set. This process of adding all observational values results in what statisticians call the "sum of the values."Here is how you sum the values in our example:

• Start by taking each individual number from the data set: 2, 4, 6, and 8.
• Add these numbers together: \( 2 + 4 + 6 + 8 \).
• You get a sum total: 20.

The sum is an important component for finding other measurements, such as the mean, variance, or standard deviation. It is simply the aggregate total of all the numbers added together.

Understanding Division in Calculating Mean

Once we've found the sum of the values in our data set, the next step to calculate the mean is division. Division allows us to find out how the sum of values balances out across all the individual data points. For our data set of 2, 4, 6, and 8, we found that the sum is 20. We also know there are four numbers in the data set. Therefore, we:

• Divide the sum of values (20) by the number of data points (4).
• This calculation is represented as \( \frac{20}{4} \).
• The result of this division is the mean: 5

This process of division distributes the total sum evenly across the numbers in the data set and provides a singular value—the mean or average—that represents the central tendency of the data points. Understanding how to perform this division step is essential for accurately calculating the mean in any set of data.