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Chapter 1: Problem 79

Joey's first 14 quiz grades in a marking period were $$ \begin{array}{lllllll} \hline 86 & 84 & 91 & 75 & 78 & 80 & 74 \\ 87 & 76 & 96 & 82 & 90 & 98 & 93 \\ \hline \end{array} $$ Calculate the mean. Show your work.

Short Answer

Expert verified
The mean of Joey's quiz grades is 85.

Step by step solution

01

Understand the Problem

To find the mean of Joey's quiz grades, you need to sum all the grades and then divide by the total number of quizzes.
02

Count the Total Number of Grades

Count the number of quiz grades given: there are 14 grades in total.
03

Sum All the Quiz Grades

Add together all the grades: \[ 86 + 84 + 91 + 75 + 78 + 80 + 74 + 87 + 76 + 96 + 82 + 90 + 98 + 93 = 1190 \]
04

Divide the Sum by the Number of Grades

To find the mean, divide the total sum by 14: \[ \frac{1190}{14} = 85 \]
05

Verify the Calculation

Double-check the summation of the grades and the division operation to ensure the mean is calculated accurately. After verifying, the mean is found to be 85.

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Key Concepts

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

Quiz Grades
Understanding what quiz grades represent is foundational for calculating a mean. Quiz grades are numerical scores that reflect a student's performance on quizzes. Joey's 14 quiz grades are individual pieces of data.
  • Each grade tells us how Joey performed in that particular quiz.
  • A series of these grades will help us analyze overall performance over time.
When calculating the mean, these grades show us both individual achievements and their collective significance when averaged. By summing and averaging them, you gain insights into Joey's consistent studying habits and overall grasp of the material.
Summation
Summation is a fundamental mathematical process used to find the total of a sequence of numbers. It involves adding all the numbers in your data set to get one single figure. To determine Joey's quiz grade mean, summation is crucial:
  • Add each individual quiz grade.
  • Make sure not to miss any number or include any extra ones.
  • Use precise arithmetic to avoid errors.
For Joey's grades, the sum is: \[86 + 84 + 91 + 75 + 78 + 80 + 74 + 87 + 76 + 96 + 82 + 90 + 98 + 93 = 1190\]This total means that the sum of all Joey's effort over 14 quizzes is 1190, which is essential for calculating the mean.
Arithmetic Mean
The arithmetic mean, commonly known as the average, is a way to find the central value of a set of numbers. To calculate the mean of Joey's quiz grades, you need both the sum and the count of the grades. Here鈥檚 how it works:
  • First, you find the sum of all the quiz scores, which is 1190.
  • Second, count the number of grades, which in Joey's case is 14.
  • Lastly, divide the sum by the count: \[ \frac{1190}{14} = 85 \]
The arithmetic mean of Joey's grades is 85, indicating his overall performance level. This average helps Joey and his teacher understand typical performance and identify areas for improvement.
Step-by-step Solution
A step-by-step solution is crucial for breaking down the process of calculating the mean into clear, manageable actions. Let's look at how this helps in practice:
  • **Understand the Problem**: Identify that you need a total sum and know the number of grades.
  • **Count** the grades to verify you have all pieces of data.
  • **Sum** the grades using careful arithmetic to avoid mistakes.
  • **Divide** to find the average, performing the division correctly.
  • **Verify** the steps by revisiting each calculation to ensure accuracy and avoid errors.
By following these steps precisely, you can reliably calculate any arithmetic mean while gaining confidence that your conclusion is correct.

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Most popular questions from this chapter

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For each of the following summary statistics, decide (i) whether it could be used to measure center or spread and (ii) whether it is resistant. (a) \(\frac{Q_{1}+Q_{3}}{2}\) (b) \(\frac{\text { Max }-\text { Min }}{2}\)

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