Problem 6 - Algebra Form and Function

Question

Find the mean of the data set. $$ 5,5,5,0,0,0,0,0,5,5 $$

Step-by-Step Solution

Answer

Answer: The mean of the data set is 2.5.

Step 1: Add the numbers in the data set

To find the mean, we first need to find the sum of all numbers in the data set. We can do this by adding all the numbers together: $$ 5+5+5+0+0+0+0+0+5+5 = 25 $$

Step 2: Count the number of items in the data set

Next, we need to know the total number of elements in the data set. We can count the numbers in the given data set: $$ 10\: elements $$

Step 3: Divide the sum by the number of items

Finally, to calculate the mean, we divide the total sum of the numbers in the data set by the total number of elements: $$ \frac{25}{10} = 2.5 $$ Hence, the mean of the data set is 2.5.

Key Concepts

Understanding a Data Set

A **data set** is a collection of numbers or values that you want to analyze or describe in a statistical study. Think of it as a group of friends who you ask the same question to, and their answers form your data set. In our example, the data set consists of these values:

• 5, 5, 5, 0, 0, 0, 0, 0, 5, 5

This data set is like a basket of numbers collected from observations or measurements. It's crucial to first identify the data points correctly before any analysis begins. Whether it comes from a survey, an experiment, or just collected casually, each number in the data set has its own meaning.
For example, if these were scores in a game, each number represents how many points a player achieved each round. Understanding your data set is the foundation for any calculations you will perform later, like finding the mean.

Calculating the Sum of Elements

To find the **mean** of a data set, a critical step is calculating the sum of all the elements. The sum gives us the total when all the individual numbers in the data set are added together. It’s like collecting all of the numbers into one big pile. For our specific case, consider the numbers:

• 5, 5, 5, 0, 0, 0, 0, 0, 5, 5

Adding these together, you perform:\[ 5 + 5 + 5 + 0 + 0 + 0 + 0 + 0 + 5 + 5 = 25 \]Imagine adding up the money you have, each bill or coin is similar to each number in our set. By summing them, you have a clear idea of the total amount. This step is essential because it allows you to proceed to the next step of finding the mean, providing a clear numerical value that will be divided by the number of elements in the data set.

Determining the Number of Elements

The **number of elements** in a data set represents how many data points exist in your collection. Each number you see in the set is considered one element. Counting them gives us the total number of elements, which is crucial for calculating the mean.
For our example, the data set is:

• 5, 5, 5, 0, 0, 0, 0, 0, 5, 5

By counting each individual number, we find there are 10 elements in total. This is similar to counting the number of apples in a basket—you’re determining how many individual items or data points you have.
Knowing the number of elements is essential because it enables us to compute the mean by dividing the total sum by this number. It's a basic, yet fundamental step in statistical analysis, ensuring that our calculations reflect the true average.