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Chapter 14: Problem 54

Find the arithmetic mean for each set of values. $$ 10,12,8,19,23,12 $$

Short Answer

Expert verified
The arithmetic mean is 14.

Step by step solution

01

Understand the Problem

The arithmetic mean, or average, is found by summing all the values in the set and then dividing by the number of values. The formula for the arithmetic mean is \[\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]
02

Count the number of values

There are 6 values in the given set: 10, 12, 8, 19, 23, and 12.
03

Sum all the values

Add all the values together: \[10 + 12 + 8 + 19 + 23 + 12 = 84 \]
04

Apply the Mean Formula

Now, divide the total sum by the number of values: \[\text{Mean} = \frac{84}{6} = 14 \]

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Key Concepts

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

average
The arithmetic mean, often called the average, is a measure of central tendency. It's a very common statistic used to summarize a set of numbers. In essence, the average tells us what a typical value in the set might be. To find it, you add up all the numbers and then divide by how many numbers there are.

For example, with the numbers 10, 12, 8, 19, 23, and 12, you would add these numbers together first. The average helps to understand the general value among a list of items, making it easier to compare different sets of data.

Why are averages useful?

They help to:
  • Summarize data into a single value
  • Compare different groups
  • Identify trends over time
So always remember: the average/mean gives you a centralized value, reflecting the overall character of the data.
sum of values
The sum of values plays a crucial role in calculating the mean. By adding all values together, you get the total sum, which is an important first step in finding the average.

For example, let's consider our list of numbers: 10, 12, 8, 19, 23, and 12. When you add these, the calculation is:
\(10 + 12 + 8 + 19 + 23 + 12 = 84\)
This number (84) tells you the cumulative total of all individual values in the set.

Why do we sum values?
  • To find the overall total
  • To proceed with further calculations like the mean
  • To compare the combined magnitude with other sets of data
Summing values is simple arithmetic, but important for deeper statistical analysis.
number of values
Knowing the number of values in your dataset is essential for many statistical calculations, including the mean. It's basically a count of how many individual items are in your list.

For our dataset: 10, 12, 8, 19, 23, and 12, we have 6 values in total. This count is what you'll use to divide the sum by when you're calculating the average.

Steps to count the number of values:
  • Identify each individual item in the list
  • Count them carefully
  • Record the total number
It's straightforward but crucial. Without the correct number of values, your calculations could be incorrect.

Number of values helps in:
  • Understanding the size of your dataset
  • Calculating accurate averages
  • Further statistical analysis and comparisons
Getting this number right ensures all your other calculations are on point.

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Most popular questions from this chapter

Solve each application. A tracer dye is injected into a system with an ingestion and an excretion. After \(1 \mathrm{hr}, \frac{2}{3}\) of the dye is left. At the end of the second hour, \(\frac{2}{3}\) of the remaining dye is left, and so on. If one unit of the dye is injected, how much is left after \(6 \mathrm{hr} ?\)

A student incorrectly claimed that the common difference for the arithmetic sequence $$ -15,-10,-5,0,5, \ldots $$ is - 5. WHAT WENT WRONG? Find the correct common difference.

Solve each applied problem by writing the first few terms of a sequence. Horacio borrows \(\$ 1000\) and agrees to pay \(\$ 100\) plus interest of \(1 \%\) on the unpaid balance each month. Find the payments for the first six months and the remaining debt at the end of that period.

Fill in each blank with the correct response. The value of the sum \(\sum_{i=1}^{3}(i+2)\) is ____.

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 1,-3,9,-27,81, \ldots $$
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