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Chapter 2: Problem 65

Calculate the mean and the median for the numbers \(1, \quad 1, \quad 1, \quad 1, \quad 1, \quad 1, \quad 2, \quad 5,7, \quad 12\) Which do you think is a better measure of center for this set of values? Why? (There is no right answer, but think about which you would use.)

Short Answer

Expert verified
The mean of the given set of numbers is 3.1, and the median is 1. In this case, the median might be a better measure of central tendency as it鈥檚 less influenced by the outliers, reflecting more the most frequent numbers in the dataset.

Step by step solution

01

Calculation of Mean

To calculate the mean, sum all the numbers and then divide by the count of numbers. So: \( \frac{1+1+1+1+1+1+2+5+7+12}{10} \) which equals \( \frac{31}{10} \) or 3.1. The mean of this set is 3.1.
02

Calculation of Median

To calculate the median, write the numbers in ascending order and identify the middle number. If there is an even number of figures, the median is the mean of the two middle numbers. In this case, the sorted numbers are: 1, 1, 1, 1, 1, 1, 2, 5,7,12. The two middle numbers are the 5th and 6th numbers, which are both 1. Therefore, the median is \( \frac{1+1}{2} = 1 \)
03

Analysis of Results

In this scenario, the mean and median yield significantly different results. The mean is influenced by the higher numbers and thus is larger, while the median reflects the most frequent number in the set, which is 1. Given this distribution, the median might be considered a better measure of central tendency for this set of numbers, as it's less affected by outliers.

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Key Concepts

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

Mean Calculation
The mean, also known as the arithmetic average, provides an indication of the central tendency by taking the sum of all values and dividing it by the number of values.
To calculate the mean for the given numbers, follow these steps:
  • Add all the numbers together. In this case, sum them up: 1 + 1 + 1 + 1 + 1 + 1 + 2 + 5 + 7 + 12.
  • Divide the total by the count of the numbers, which is 10 here.
The calculation looks like this: \[ \text{Mean} = \frac{1+1+1+1+1+1+2+5+7+12}{10} = \frac{31}{10} = 3.1 \]This mean value of 3.1 gives a sense of where data points generally fall on the number line. However, it's important to note that its accuracy as a central measure can be skewed by higher values or outliers in the dataset.
Median Calculation
The median is the middle value that splits a dataset into two halves. This measure is often more reliable in understanding the center of a dataset, especially when outliers are present.
To find the median:
  • Rearrange the data points in ascending order.
  • Identify the middle number. If there are an even number of data points, take the mean of the two central numbers.
For the numbers 1, 1, 1, 1, 1, 1, 2, 5, 7, and 12, rearranging in order is done, and the two middle numbers (since there are 10 numbers) are both 1.
So, the median calculation is: \[ \text{Median} = \frac{1+1}{2} = 1 \]This shows that the median is 1, effectively representing the central tendency without being swayed by the larger numbers at the end of the dataset.
Outliers in Data
Outliers are numbers in a dataset that are significantly higher or lower than most of the other numbers. They can distort statistical measures like the mean.
  • In the given dataset, numbers like 5, 7, and 12 are potentially influencing the mean.
  • This results in the mean being larger compared to the median, which directly reflects frequent smaller numbers.
In many cases, outliers can mislead analysis if the mean is solely relied upon.
Recognizing outliers help in choosing the appropriate measure of central tendency. For this dataset, the median (which is 1) might better serve as a center measure because it reflects the concentration of repeated smaller numbers and is not skewed by the larger, less frequent values.

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Most popular questions from this chapter

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