Here Is A Distribution Consisting Of Six Observations, Sorted In Ascending Order: 3.7, 8.2, 9.9, 11.3, 15.6, 17.3. The Mean For This Distribution Is 11. What Is The Value Of The Sum Of The Differences Between Each Observation And The Mean, Or $\sum_{i=1}^6 (x_i - \bar{x})$?

Jun 18, 2025 by ADMIN 278 views

Understanding the Sum of Deviations from the Mean

[ 3.7, 8.2, 9.9, 11.3, 15.6, 17.3 ]

The mean of this distribution is 11. What is the value of ${\sum_{i=1}^6 (x_i - \bar{x})}$ ?

A. 12 B. 72 C. 6 D. 0

Understanding the Concept of Deviations from the Mean

When analyzing a set of data, a crucial concept to grasp is the deviation from the mean. This refers to the difference between each individual data point and the average value of the dataset. The mean, denoted as ${\bar{x}}$ , serves as the central point around which the data fluctuates. Understanding these deviations helps us to gain insights into the spread and distribution of the data. In essence, we're looking at how far each data point strays from the central tendency of the dataset. Calculating the deviation for each data point involves a simple subtraction: we take the value of the data point, often denoted as ${x_i}$ , and subtract the mean ${\bar{x}}$ from it. This gives us a measure of how much the data point differs from the average. If the result is positive, the data point is above the mean; if it's negative, the data point is below the mean; and if it's zero, the data point is exactly at the mean. This process is fundamental in many statistical analyses, as it lays the groundwork for understanding variance, standard deviation, and other key measures of data dispersion.

Furthermore, the sum of these deviations carries a unique property. When we add up all the individual deviations from the mean, a fascinating phenomenon occurs: the sum always equals zero. This is not a coincidence but a mathematical certainty. The mean, by its very definition, is the balancing point of the dataset. It's the value that minimizes the sum of the squared deviations, a concept used in various optimization problems. Because the mean sits at this equilibrium point, the negative deviations (values below the mean) perfectly cancel out the positive deviations (values above the mean). This fundamental characteristic makes the sum of deviations from the mean a rather uninteresting statistic on its own, as it provides no information about the spread or variability of the data. However, it is a crucial stepping stone in understanding more complex statistical measures. The sum of the squared deviations, for instance, is the basis for variance and standard deviation, which are essential for quantifying the dispersion of data around the mean. Therefore, while the sum of deviations itself might seem trivial, it is a cornerstone concept in the broader landscape of statistical analysis.

Applying the Concept to the Given Data Set

Now, let's apply this understanding to the specific data set provided. We have six observations: 3.7, 8.2, 9.9, 11.3, 15.6, and 17.3. The problem states that the mean of this distribution, ${\bar{x}}$ , is 11. Our task is to calculate the value of the sum of deviations from the mean, represented by the expression ${\sum_{i=1}^6 (x_i - \bar{x})}$ . This means we need to find the difference between each observation and the mean (11), and then sum up these differences. Let's break down the calculation step by step. First, we find the deviation for each observation:

For 3.7, the deviation is 3.7 - 11 = -7.3
For 8.2, the deviation is 8.2 - 11 = -2.8
For 9.9, the deviation is 9.9 - 11 = -1.1
For 11.3, the deviation is 11.3 - 11 = 0.3
For 15.6, the deviation is 15.6 - 11 = 4.6
For 17.3, the deviation is 17.3 - 11 = 6.3

Next, we sum up these deviations: -7.3 + (-2.8) + (-1.1) + 0.3 + 4.6 + 6.3. If we perform this addition, we get a total of 0. This result might seem surprising at first, but it aligns perfectly with the fundamental property we discussed earlier: the sum of deviations from the mean always equals zero. This principle holds true regardless of the specific data points or the distribution's shape. It's a direct consequence of the mean being the balancing point of the data.

This property simplifies our task significantly. Instead of performing the individual calculations and summing them up, we can immediately conclude that the value of ${\sum_{i=1}^6 (x_i - \bar{x})}$ is 0. This understanding not only provides the correct answer but also reinforces the core concept of the mean as the central balancing point of a dataset. Recognizing this property can save time and effort in problem-solving, especially in statistical contexts where this calculation might be a part of a larger analysis.

The Answer and Its Implications

Therefore, the value of ${\sum_{i=1}^6 (x_i - \bar{x})}$ for the given distribution is 0. This corresponds to option D in the provided choices. While this answer might seem straightforward, it underscores a fundamental principle in statistics: the sum of the deviations from the mean is always zero. Understanding this concept is crucial for grasping more advanced statistical ideas. It's a building block for understanding concepts like variance and standard deviation, which measure the spread or dispersion of data around the mean.

Moreover, this principle has practical implications in various fields. For instance, in finance, understanding deviations from the mean is essential for analyzing investment risk. The mean return of an investment provides an average expectation, but the deviations from this mean represent the volatility or risk associated with that investment. Similarly, in quality control, deviations from the mean in manufacturing processes can indicate potential problems or inconsistencies. By monitoring these deviations, businesses can identify and address issues before they escalate, ensuring consistent product quality. In scientific research, understanding deviations from the mean is critical for interpreting experimental results. Researchers often compare the mean of an experimental group to the mean of a control group, but the deviations within each group provide valuable information about the variability of the data and the reliability of the findings.

In conclusion, while the answer to this specific problem is a simple zero, the underlying concept is far-reaching. The fact that the sum of deviations from the mean is always zero is not just a mathematical curiosity; it's a fundamental property that underpins much of statistical analysis and has practical applications across diverse fields. By understanding this principle, we can gain a deeper appreciation for the power and utility of statistical methods in making sense of data and the world around us.

Choosing the Correct Option

Given our understanding that the sum of deviations from the mean is always zero, the correct answer to the question "What is the value of ${\sum_{i=1}^6 (x_i - \bar{x})}$ ?" is D. 0. The other options, A. 12, B. 72, and C. 6, are incorrect because they do not reflect this fundamental property of the mean. Selecting the correct answer not only demonstrates an understanding of the calculation but also an appreciation for the underlying statistical principle. It's a clear indication that the individual comprehends the nature of the mean as the balancing point of a dataset, where positive and negative deviations perfectly offset each other.

The incorrect options might stem from a misunderstanding of how to calculate deviations or a failure to recognize the inherent property of the mean. For example, someone might mistakenly sum the absolute values of the deviations, which would result in a non-zero value. Alternatively, they might perform the subtractions in the wrong order (mean minus observation instead of observation minus mean), which would change the signs of the deviations but still ultimately lead to a sum of zero. Therefore, arriving at the correct answer requires both computational accuracy and a firm grasp of the statistical concept. The ability to identify and avoid these potential errors is a key skill in data analysis and problem-solving in statistics.