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Chapter 2: Problem 45

For each set of data in Exercises 2.43 to 2.46: (a) Find the mean \(\bar{x}\). (b) Find the median \(m\). (c) Indicate whether there appear to be any outliers. If so, what are they? \(\begin{array}{llllllll}\mathbf15, & 22, & 12, & 28, & 58, & 18, & 25, & 18\end{array}\)

Short Answer

Expert verified
The mean \(\bar{x}\) of the data set is calculated as 25.75, the median \(m\) is 20, and the data point 58 seems to be an outlier as it is significantly separate from the rest of the data.

Step by step solution

01

Calculate the Mean

Start by finding the mean, \(\bar{x}\), which involves adding up all the data points together and then dividing it by the number of data points. Therefore, calculate \(\bar{x} = \frac{15 + 22 + 12 + 28 + 58 + 18 + 25 + 18}{8}\).
02

Compute the Median

To find the median, \(m\), first sort the data in ascending order which becomes: 12, 15,18, 18, 22, 25, 28, 58. Since there are eight data points, an even number, the median is the average of the middle two numbers i.e. the fourth and the fifth data point. Thus, \(m = \frac{18 + 22}{2}\).
03

Check for Outliers

To identify possible outliers, one can use the Interquartile Range (IQR) method. First, find the lower quartile, Q1 (median of the first half of the data), and the upper quartile, Q3 (median of the second half of data). The IQR is then Q3 - Q1. However, upon initial observation, only one value, 58, appears significantly distant from the rest of the data.

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Key Concepts

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

Understanding Mean Calculation
Calculating the mean is one of the foundational tasks in descriptive statistics. The mean, often referred to as the average, provides a single value that summarizes the entire set of data. It helps to understand the central tendency of the numbers. To calculate the mean, simply follow these steps:
  • Add all the numbers in the dataset together. This is called the sum of the dataset.
  • Count how many numbers are in the dataset, which is referred to as the 'number of data points'.
  • Divide the sum by the number of data points. This result is the mean, denoted as \( \bar{x} \).
For example, with our dataset \(15, 22, 12, 28, 58, 18, 25, 18\), you would calculate:\[ \bar{x} = \frac{15 + 22 + 12 + 28 + 58 + 18 + 25 + 18}{8} = \frac{196}{8} = 24.5 \] The mean in this scenario is 24.5, providing a central point around which the data clusters.
Finding the Median
The median is another key measure in descriptive statistics that identifies the middle point of a data set. Unlike the mean, the median is less affected by extremely large or small values. Here's how you can calculate it:
  • First, organize the dataset in ascending order, which helps in identifying the middle value more easily.
  • If the dataset has an odd number of values, the median is the middle one.
  • If there is an even number of values, as in our dataset, the median is the average of the two middle numbers.
For the dataset \(12, 15, 18, 18, 22, 25, 28, 58\), the two middle numbers are 18 and 22. Thus, the median \( m \) is calculated as:\[ m = \frac{18 + 22}{2} = 20 \]This median splits the data into two equal halves, offering an insightful look at data distribution.
Outliers Identification Basics
In any dataset, outliers are those values that appear significantly different from other observations. They can provide useful insights or may indicate errors. Identifying outliers can be done through several methods; the Interquartile Range (IQR) method is commonly used.Here's how to spot outliers in a straightforward way:
  • Find the first quartile \( Q1 \), the median of the first half of the sorted data.
  • Find the third quartile \( Q3 \), the median of the second half of the sorted data.
  • Compute the IQR: \( IQR = Q3 - Q1 \).
  • Values that are smaller than \( Q1 - 1.5 \times IQR \) or larger than \( Q3 + 1.5 \times IQR \) are considered outliers.
Upon briefly examining the given dataset, the number 58 stands out, appearing far from the cluster of others. While not computed using IQR here, it is identified as an outlier due to its noticeable gap from other values.

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