Problem 70 - Prealgebra

Question

Find the mean and the range for each set of numbers. $$6,8,10,12$$

Step-by-Step Solution

Answer

The mean is 9 and the range is 6.

Step 1: Calculate the Mean

To find the mean of a set of numbers, add all the numbers together and then divide by the number of numbers in the set. For the numbers 6, 8, 10, and 12:Add the numbers: 6 + 8 + 10 + 12 = 36Divide by the number of numbers in the set, which is 4: \[\text{Mean} = \frac{36}{4} = 9\]

Step 2: Find the Range

To find the range, subtract the smallest number from the largest number in the set. For the numbers 6, 8, 10, and 12:The largest number is 12 and the smallest number is 6.Subtract the smallest from the largest: \[\text{Range} = 12 - 6 = 6\]

Key Concepts

Calculating Mean

To calculate the mean of a set of numbers, follow two simple steps: adding all the numbers in the set and then dividing this sum by the number of elements in the set. The mean provides a central value that represents the data set as a whole.
For example, consider the numbers 6, 8, 10, and 12. The first step is to add these numbers together:

• 6 + 8 + 10 + 12 = 36

Next, you'll need to divide the sum by the count of numbers in this set, which is 4:

• \( \text{Mean} = \frac{36}{4} = 9 \)

The result, 9, is the average value of the numbers in the set. It effectively balances the collection of numbers, giving you an idea of their central tendency.

Finding Range

Finding the range of a set of numbers is another crucial aspect of basic statistics. The range tells you about the spread of the numbers, showing how far apart the smallest and largest numbers are.
To find the range, you subtract the smallest number in the set from the largest. With our example set of numbers: 6, 8, 10, and 12:

• The largest number is 12.
• The smallest number is 6.

Subtract the smallest number from the largest:

• \( \text{Range} = 12 - 6 = 6 \)

The range, 6, indicates the distance between the highest and lowest values in the data set. It's a simple way to understand the variability of numbers in the set.

Basic Statistics

Basic statistics provides fundamental tools like mean and range, which offer insights into numerical data. These tools help summarize large sets of numbers, making them easier to understand and interpret.
Understanding the mean and range allows you to measure central tendency and variability. The mean gives you an idea of the average performance within a data set, while the range informs you about the difference between the maximum and minimum values.

• Mean: Balances the values, offering a central point.
• Range: Measures spread and variability.

Utilizing these concepts helps in making informed decisions based on data. They're integral in various fields, from academics to finance, helping transform raw data into meaningful insights.