Calculating Population Variance A Step-by-Step Guide

Population variance is a crucial concept in statistics that quantifies the spread or dispersion of data points within a population around its mean. In simpler terms, it tells us how much the individual data values deviate from the average value. A high variance indicates that the data points are widely scattered, while a low variance suggests that they are clustered closely around the mean.

In this article, we will dive deep into the concept of population variance and provide a comprehensive, step-by-step solution to Question 21, a typical problem encountered in mathematics and statistics courses. Our approach is designed to not only solve the problem at hand but also to enhance your understanding of the underlying principles. We'll break down the formula for population variance, explain each component, and illustrate its application with clear examples. By the end of this guide, you'll be well-equipped to tackle similar problems and grasp the significance of variance in statistical analysis. So, let's embark on this journey of learning and problem-solving!

Breaking Down the Problem

Let's revisit Question 21, which presents a scenario involving a population with a known mean (μ = 6) and a set of data values. Our objective is to calculate the population variance using the provided information, particularly the table that shows the squared deviations from the mean ((x - μ)²). This table is a crucial piece of the puzzle, as it simplifies the variance calculation process. However, it's important to understand what these squared deviations represent and how they fit into the overall formula for variance.

The squared deviations (x - μ)² represent the square of the difference between each data value (x) and the population mean (μ). Squaring these deviations is a critical step because it eliminates negative values, ensuring that all deviations contribute positively to the overall variance. This prevents underestimation of the spread due to negative deviations canceling out positive ones. The table provides these pre-calculated squared deviations, saving us a significant amount of computation time. However, to fully grasp the concept, we'll also discuss how these deviations are calculated and why they are so important in determining variance.

Understanding the question and the data provided is the first step towards solving it effectively. By carefully analyzing the given information, we can formulate a strategy for calculating the population variance accurately and efficiently. In the following sections, we'll delve into the formula for population variance and apply it to the data at hand.

Understanding the Formula for Population Variance

The population variance, denoted by σ², is a statistical measure that quantifies the dispersion or spread of data points in a population around the population mean (μ). It provides a crucial understanding of how much individual data values deviate from the average value. A higher variance indicates greater variability, meaning the data points are more scattered, while a lower variance suggests that the data points are clustered more closely around the mean.

The formula for population variance is expressed as:

σ² = Σ(xᵢ - μ)² / N

Where:

• σ² represents the population variance.
• Σ (sigma) denotes the sum of the values.
• xᵢ represents each individual data value in the population.
• μ represents the population mean.
• N represents the total number of data values in the population.

Let's break down each component of the formula to understand its role in calculating the variance. The term (xᵢ - μ) represents the deviation of each data value from the population mean. These deviations can be positive or negative, depending on whether the data value is above or below the mean. However, simply averaging these deviations would not give us a true measure of spread, as positive and negative deviations could cancel each other out. This is where the squaring operation comes into play.

The term (xᵢ - μ)² squares each of these deviations, eliminating the negative signs and ensuring that all deviations contribute positively to the overall variance. Squaring also gives more weight to larger deviations, making the variance more sensitive to extreme values in the data set. The sum of these squared deviations, Σ(xᵢ - μ)², represents the total variability in the population.

Finally, dividing the sum of squared deviations by the total number of data values (N) gives us the average squared deviation, which is the population variance (σ²). This normalization step ensures that the variance is comparable across populations of different sizes. Understanding this formula and its components is essential for accurately calculating and interpreting population variance.

Applying the Formula to Solve Question 21

Now that we have a solid understanding of the population variance formula, let's apply it to solve Question 21. We are given the following data values representing a population:

x: 1, 5, 7, 11

The table provides the squared deviations from the mean (μ = 6):

(x - μ)²: 25, 1, 1, 25

Our goal is to calculate the population variance (σ²) using this information. We can follow these steps:

1. Identify the components of the formula: We have the squared deviations (x - μ)² for each data value, and we know the population mean (μ = 6). We also need to determine the total number of data values (N).
2. Determine the total number of data values (N): By counting the number of x values, we find that N = 4.
3. Calculate the sum of the squared deviations (Σ(x - μ)²): We add up the squared deviations provided in the table:25 + 1 + 1 + 25 = 52
4. Apply the formula: We plug the values we've calculated into the population variance formula:σ² = Σ(x - μ)² / N = 52 / 4
5. **Calculate the population variance:**σ² = 13

Therefore, the population variance for the given data is 13. This result indicates the average squared deviation of the data values from the population mean. A variance of 13 suggests a moderate level of dispersion in the data, meaning the values are neither too tightly clustered around the mean nor too widely scattered.

By carefully applying the formula and using the information provided in the table, we have successfully calculated the population variance for Question 21. This step-by-step approach demonstrates how to effectively use the formula and interpret the results in the context of the given data.

Answer and Explanation

Based on our calculations, the population variance for the given data is 13. Therefore, the correct answer to Question 21 is B. 13. This result tells us that, on average, the squared deviation of each data value from the population mean (6) is 13.

To further solidify your understanding, let's delve into why this answer is correct and what it signifies in the context of the data. The variance of 13 represents the average of the squared differences between each data point and the mean. The squaring operation ensures that all deviations contribute positively to the variance, preventing underestimation due to negative values canceling out positive ones.

In this specific case, the data values are 1, 5, 7, and 11, and the mean is 6. We calculated the squared deviations as follows:

• (1 - 6)² = 25
• (5 - 6)² = 1
• (7 - 6)² = 1
• (11 - 6)² = 25

These squared deviations represent the individual contributions to the overall variance. By summing them up (25 + 1 + 1 + 25 = 52) and dividing by the number of data values (4), we arrive at the population variance of 13.

Understanding the magnitude of the variance is crucial for interpreting the spread of the data. A variance of 13 indicates a moderate level of dispersion, suggesting that the data values are not extremely close to the mean but are also not excessively spread out. This provides valuable insights into the variability within the population.

The Significance of Population Variance in Statistics

Population variance is a cornerstone concept in statistics, playing a vital role in various statistical analyses and decision-making processes. Its significance stems from its ability to quantify the spread or dispersion of data points within a population around the mean. This measure of variability is crucial for understanding the characteristics of a dataset and making informed inferences.

One of the primary applications of population variance is in comparing the variability of different datasets. For example, consider two populations with the same mean but different variances. The population with the higher variance has data points that are more spread out, indicating greater variability, while the population with the lower variance has data points clustered more closely around the mean, indicating less variability. This comparison is essential in fields like finance, where understanding the volatility of investments is critical.

Population variance is also a key component in many statistical tests, such as t-tests and ANOVA (Analysis of Variance). These tests are used to compare the means of two or more populations and determine if the differences are statistically significant. Variance plays a crucial role in these tests by providing a measure of the within-group variability, which is used to assess the significance of the between-group differences. A higher within-group variance can make it more difficult to detect significant differences between group means.

In addition to its role in statistical tests, population variance is used in confidence interval estimation. A confidence interval provides a range of values within which the population mean is likely to fall. The width of the confidence interval is influenced by the variance of the population; a higher variance leads to a wider confidence interval, reflecting greater uncertainty in the estimate of the population mean.

Furthermore, population variance is essential in quality control and process monitoring. By tracking the variance of a process, it is possible to identify deviations from the norm and take corrective actions. For instance, in manufacturing, monitoring the variance of product dimensions can help ensure that the products meet specifications. An increase in variance may indicate a problem with the production process that needs to be addressed.

In essence, population variance provides a fundamental understanding of the variability within a dataset and is a crucial tool for statistical analysis and decision-making across a wide range of disciplines. Its ability to quantify spread and dispersion makes it an indispensable concept in statistics.

Conclusion

In conclusion, Question 21 provided an excellent opportunity to delve into the concept of population variance and its calculation. We successfully determined that the population variance for the given data is 13 by applying the formula σ² = Σ(xᵢ - μ)² / N and carefully following the steps outlined. This process not only answered the specific question but also enhanced our understanding of variance as a measure of data dispersion.

Throughout this guide, we have emphasized the importance of understanding the underlying principles behind statistical concepts. Breaking down the formula for population variance, explaining each component, and illustrating its application with a clear example has been our focus. We explored the significance of squared deviations, the role of the population mean, and the interpretation of the resulting variance value.

Population variance is a fundamental concept in statistics with wide-ranging applications. It allows us to quantify the spread of data points within a population and compare the variability of different datasets. Understanding variance is crucial for various statistical analyses, including hypothesis testing, confidence interval estimation, and quality control.

By mastering the calculation and interpretation of population variance, you equip yourself with a powerful tool for analyzing data and making informed decisions. Whether you are a student learning statistics or a professional applying statistical methods in your field, a solid understanding of variance is essential. We hope this guide has provided you with a clear and comprehensive understanding of population variance and its significance in statistics. Keep practicing and exploring statistical concepts, and you'll be well-prepared to tackle more complex problems and gain valuable insights from data.