Problem 72 - Prealgebra

Question

Find the mean and the range for each set of numbers. $$2,4,4,6,9$$

Step-by-Step Solution

Answer

Mean = 5, Range = 7.

Step 1: Understand the Problem

We are given a set of numbers: \(2, 4, 4, 6, 9\). We need to find the mean (average) and range of this set.

Step 2: Find the Mean

To find the mean, sum all the numbers and then divide by the total count of numbers.Sum: \(2 + 4 + 4 + 6 + 9 = 25\).There are 5 numbers in the set. So, the mean is \(\frac{25}{5} = 5\).

Step 3: Find the Range

To find the range, subtract the smallest number from the largest number in the set.Largest number is \(9\) and smallest number is \(2\).Range: \(9 - 2 = 7\).

Step 4: Conclusion

The mean of the given set of numbers is \(5\) and the range is \(7\).

Key Concepts

Mean Calculation

The mean, often referred to as the average, is a central concept in statistics that gives us an idea of what a typical value in a set of numbers might look like. To arrive at the mean, we follow a simple process:

• Step 1: Add up all the numbers in the set.
• Step 2: Count how many numbers are in the set.
• Step 3: Divide the total sum by the number of values.

For example, if we take the numbers from the set provided, namely, \(2, 4, 4, 6, 9\), the sum of these numbers is \(25\). Since there are \(5\) numbers, we calculate the mean by dividing \(25\) by \(5\), resulting in a mean of \(5\).
This tells us that, on average, each number in the set is \(5\). Understanding mean helps compare and interpret datasets, as it represents a balance point of the distribution.

Range Calculation

The range of a set of numbers gives us a sense of how spread out the numbers are. It's a basic way of measuring the variability in data. Here's how you can determine it:

• Step 1: Identify the smallest number in the set.
• Step 2: Identify the largest number in the set.
• Step 3: Subtract the smallest number from the largest number.

Let's take our example set of numbers: \(2, 4, 4, 6, 9\). The smallest number here is \(2\), and the largest is \(9\). By subtracting the smallest number from the largest, \(9 - 2\), we find that the range is \(7\).
The range helps to quickly recognize the span of data, letting us know how much difference there is between the highest and lowest figures.

Statistics Basics

Statistics is a field that helps us gather, analyze, interpret, and present data. Two foundational concepts in statistics are the mean and the range. They provide insight into the nature of numerical data:

• The mean gives a single value that represents the middle of our data set, helping us to summarize the data with one number.
• The range shows the dispersion or spread of the data, indicating the difference between the maximum and minimum values.

Understanding these basic concepts is crucial, as they form the basis for more advanced statistical techniques.
They allow for a quick insight into the data, guiding initial interpretations and informing decision-making processes.

Prealgebra Concepts

Before diving into algebra, it's essential to have a good grasp of prealgebra concepts, which lay the groundwork for future coursework. Important concepts include:

• Arithmetic Operations: Addition, subtraction, multiplication, and division form the basis for all future calculations in mathematics.
• Understanding Numbers: Recognizing different forms of numbers, such as integers and fractions, and the ability to perform operations on them.
• Basic Statistical Measures: Concepts like mean and range naturally arise even before entering algebra, helping learners make sense of numerical data.

Learning prealgebra helps students develop problem-solving skills and mathematical thinking, crucial for algebra and other higher-level math courses.