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Chapter 8: Problem 7

Compute the mean, median, and mode of the data samples \(2.5,-5.4,4.1,-0.1,-0.1\)

Short Answer

Expert verified
For the given data samples, the mean is \(0.2\), the median is \(-0.1\), and the mode is \(-0.1\).

Step by step solution

01

Calculate the Mean

To calculate the mean, we first need to add up all the data samples: \(2.5 + (-5.4) + 4.1 + (-0.1) + (-0.1) = 1\) Next, we divide this sum by the total number of data samples (5): \(Mean = \frac{1}{5} = 0.2\)
02

Calculate the Median

To find the median, we must first arrange the data samples in ascending order: \(-5.4, -0.1, -0.1, 2.5, 4.1\) Since there are 5 data samples, the median is the middle value, which is -0.1.
03

Calculate the Mode

To determine the mode, we must identify the most frequent data sample. In this case, the number -0.1 appears twice, while all other numbers appear only once. Therefore, the mode is -0.1. So, for the given data samples, the mean is 0.2, the median is -0.1, and the mode is -0.1.

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Key Concepts

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

Mean Calculation
In descriptive statistics, the mean or average is a fundamental concept. It's the sum of a group of numbers divided by the count of those numbers. For students, understanding mean calculation helps in many fields including science and mathematics. Here鈥檚 how we calculate it:
  • Add all the numbers together.
  • Divide by how many numbers there are.
For example, consider the data set: \(2.5, -5.4, 4.1, -0.1, -0.1\). Start by adding these numbers:
\[2.5 + (-5.4) + 4.1 + (-0.1) + (-0.1) = 1\]This sum, 1, is then divided by the number of values, 5, to get the mean:
\[Mean = \frac{1}{5} = 0.2\]
  • Mean is sensitive to extreme values (known as outliers).
  • Great in providing a quick snapshot of the data鈥檚 central point.
By following these steps, you can easily find the mean of any set of data.
Median Finding
The median is a measure of central tendency that represents the middle of a data set. It's particularly useful because it isn't affected by outliers or skewed data. Finding the median involves a few clear steps:
  • Order the numbers from smallest to largest.
  • If the number of data points is odd, the median is the middle number.
  • If the number of data points is even, it's the average of the two middle numbers.
Take, for instance, our data set \(2.5, -5.4, 4.1, -0.1, -0.1\).
First, order them:\[-5.4, -0.1, -0.1, 2.5, 4.1\]Since we have 5 numbers (an odd amount), take the middle value. The median here is \(-0.1\).
  • Median is highly reliable as a central measure for skewed distributions.
  • It often provides a better sense of a dataset's central point compared to the mean.
The median gives a true reflection of the central tendency when data has outliers.
Mode Identification
The mode is the number that appears most frequently in a data set. Sometimes, a dataset may have one mode, more than one mode, or no mode at all. Mode identification is simple and ideal for categorical data where mean and median aren't applicable.
To find the mode from the data set \(2.5, -5.4, 4.1, -0.1, -0.1\):
  • List the numbers with their frequencies.
  • The number appearing most frequently is the mode.
In our example:
  • -0.1 appears twice.
  • The other numbers only once.
Thus, the mode is \(-0.1\).
  • Mode can be useful in finding the most common item in non-numeric data sets.
  • It provides insights into data鈥檚 peakedness.
Quick and straightforward, mode gives insightful information on the frequency of data within a set.

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Most popular questions from this chapter

If we define a "rich" household as one whose after-tax income is at least \(1.3\) standard deviations above the mean, what is the household income of a rich family in the United States?

Suppose you take larger and larger samples of a given population. Would you expect the sample and population standard deviations to get closer or further apart? Explain.

Calculate the standard deviation of \(X\) for each probability distribution. (You calculated the expected values in the last exercise set. Round all answers to two decimal places.) $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 10 & 20 & 30 & 40 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & \frac{3}{10} & \frac{2}{5} & \frac{1}{5} & \frac{1}{10} \\ \hline \end{array} $$

\- Find an algebraic formula for the sample standard deviation of a sample \(\\{x, y\\}\) of two scores \((x \leq y)\).

Your grade in a recent midterm was \(80 \%\), but the class median was \(100 \%\). Your score was lower than the average score, right?
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