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Chapter 3: Problem 5

Find the mean, median, and mode of the data set 8 2 7 2 6

Short Answer

Expert verified
Mean: 5, Median: 6, Mode: 2.

Step by step solution

01

Organize the Data Set

First, we need to organize the given numbers in ascending order. The data set provided is 8, 2, 7, 2, 6. Arranging these numbers from smallest to largest gives us: 2, 2, 6, 7, 8.
02

Calculate the Mean

To find the mean, sum all the numbers and then divide by the number of data points. The sum of the data set 2, 2, 6, 7, 8 is: \[ 2 + 2 + 6 + 7 + 8 = 25 \] There are 5 numbers in the data set, so the mean is: \[ \text{Mean} = \frac{25}{5} = 5 \]
03

Determine the Median

To find the median, locate the middle number in the ordered data set. Since there are 5 numbers (an odd number of values), the median is the third number when the data is ordered. Our ordered set is 2, 2, 6, 7, 8, so the median is 6.
04

Identify the Mode

The mode is the number that appears most frequently in the data set. Reviewing the numbers 2, 2, 6, 7, 8, we see that 2 appears twice, while all other numbers appear once. Therefore, the mode is 2.

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Key Concepts

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

Mean Calculation
Calculating the mean of a data set is like finding the average. It's a simple process. Here's how you do it:
  • First, you add up all the numbers in your data set.
  • Then, you divide that sum by the total number of data points.
Let's use the example of the data set: 2, 2, 6, 7, 8. We first sum the numbers: \[ 2 + 2 + 6 + 7 + 8 = 25 \].
Next, you count how many numbers there are. In this case, there are 5 numbers.
To find the mean, divide the sum by the count of numbers: \[ \text{Mean} = \frac{25}{5} = 5 \]. And there you have it, the mean, or average, of this data set is 5.
This number gives you an idea of the overall level of your data.
Median Calculation
The median is the middle value of a data set. This value represents the center line of your data when organized in ascending order.
This is how you find it:
  • First, ensure your data set is arranged from lowest to highest.
  • If the number of data points is odd, the median is the middle number.
  • If the count is even, the median is the average of the two middle values.
For our example, the ordered data set is 2, 2, 6, 7, 8. There are 5 numbers, an odd count, so the middle number is 6. Therefore, 6 is the median.
So, in a data set, the median helps to understand what the middle point is, ignoring the extremes.
Mode Calculation
The concept of mode is about frequency. It tells us the most common value in a data set. Finding the mode is straightforward:
  • Check which number appears the most in the data set.
  • If one number appears more than others, that's your mode.
  • If all numbers appear the same amount of times, the data set has no mode.
In our data set: 2, 2, 6, 7, 8 The number 2 appears twice, while others appear only once.
This makes 2 the mode. The mode is particularly useful for understanding what is most common in your data, giving insight into trends or patterns present.

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Most popular questions from this chapter

The town of Butler, Nebraska, decided to give a teacher-competency exam and defined the passing scores to be those in the 70 th percentile or higher. The raw test scores ranged from 0 to \(100 .\) Was a raw score of 82 necessarily a passing score? Explain.

How old are professional football players? The 11th edition of The Pro Football Encyclopedia gave the following information. Random sample of pro football player ages in years: $$\begin{array}{llllllllll}24 & 23 & 25 & 23 & 30 & 29 & 28 & 26 & 33 & 29 \\\24 & 37 & 25 & 23 & 22 & 27 & 28 & 25 & 31 & 29 \\\25 & 22 & 31 & 29 & 22 & 28 & 27 & 26 & 23 & 21 \\\25 & 21 & 25 & 24 & 22 & 26 & 25 & 32 & 26 & 29\end{array}$$ (a) Compute the mean, median, and mode of the ages. (b) Interpretation Compare the averages. Does one seem to represent the age of the pro football players most accurately? Explain.

Grades: Weighted Average In your biology class, your final grade is based on several things: a lab score, scores on two major tests, and your score on the final exam. There are 100 points available for each score. However, the lab score is worth \(25 \%\) of your total grade, each major test is worth \(22.5 \%\) and the final exam is worth \(30 \%\). Compute the weighted average for the following scores: 92 on the lab, 81 on the first major test, 93 on the second major test, and 85 on the final exam.

If a data set has an even number of data, is it true or false that the median is never equal to a value in the data set? Explain.

What is the relationship between the variance and the standard deviation for a sample data set?
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