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Chapter 14: Problem 53

Find the arithmetic mean for each set of values. $$ 8,11,14,9,7,6,8 $$

Short Answer

Expert verified
The arithmetic mean is 9.

Step by step solution

01

Understand the Arithmetic Mean

The arithmetic mean is the sum of all numbers in a set divided by the number of elements in that set.
02

Sum All Numbers

Add all the given numbers together. Calculate: \[ 8 + 11 + 14 + 9 + 7 + 6 + 8 = 63 \]
03

Count the Numbers

Count how many numbers are in the set. There are 7 numbers in the set.
04

Divide the Sum by the Count

Divide the sum from Step 2 by the count from Step 3. Calculate: \[ \frac{63}{7} = 9 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Arithmetic Mean
Arithmetic mean, often simply called the mean, is a basic concept in statistics. It's used to find the average value of a set of numbers. When you hear 'average,' most people are referring to the arithmetic mean.
To calculate the arithmetic mean, follow these simple steps:
  • Add up all the numbers in the set to get their sum.
  • Count how many numbers are in the set.
  • Divide the sum by the number of items.
This process gives you a single number that represents the central tendency of the set.
Let's explore each part of this calculation in more detail.
Summing Up the Numbers
Summing up the numbers in your data set is the first step to finding the arithmetic mean. In this step, you add all the values together to get a total sum.
For example, consider this set of numbers: \[ 8, 11, 14, 9, 7, 6, 8 \]
Let's add them up:
\[ 8 + 11 + 14 + 9 + 7 + 6 + 8 = 63 \]
The total sum is 63. This sum will be used in the next step to find the arithmetic mean.
Adding numbers is straightforward but essential in understanding how each value contributes to the overall average. It's important to ensure that every number is included in the summation to get an accurate mean.
Division to Find the Mean
After summing up the numbers, the next step is to divide this sum by the number of elements in the set. This will provide the arithmetic mean.
First, count the number of elements. For our example set:
Numbers: 8, 11, 14, 9, 7, 6, 8
Total count: 7
Then, use this count for division. Here's the calculation:
\[ \frac{63}{7} = 9 \]
So, the arithmetic mean is 9.
Division might sometimes be tricky. Make sure you have counted all the elements correctly and divided the total sum by this number for an accurate mean.
Summing up, understanding each step helps clear up how averages work, making them easier and more intuitive.

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Most popular questions from this chapter

In a certain community, the consumption of electricity has increased about \(6 \%\) per yr. (a) If the community uses 1.1 billion units of electricity now, how much will it use 5 yr from now? Round to the nearest tenth. (b) Find the number of years (to the nearest year) it will take for the consumption to double.

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(-6,-10,-14,-18, \ldots\)

Find the number of terms in each arithmetic sequence. \(2, \frac{3}{2}, 1, \frac{1}{2}, \ldots,-5\)

Write each series as a sum of terms and then find the sum. $$ \sum_{i=3}^{7}(i-3)(i+2) $$

Evaluate each expression. $$ 9 ! $$
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