The Data Set On The Right Represents The Population.Which Formula Should Be Used To Calculate The Variance?A Teacher Recorded All Of His Students' Grades On The Final Exam As: ${ 62, 77, 78, 80, 82, 82, 83, 84, 85, 87, 89, 95 }$Consider The
Introduction
In statistics, variance is a measure of the spread or dispersion of a set of data. It is an essential concept in understanding the distribution of data and making informed decisions. When working with a population, it is crucial to choose the correct formula for calculating variance. In this article, we will explore the different formulas for calculating variance and determine which one is suitable for the given data set.
Understanding the Data Set
The data set provided consists of the final exam grades of a teacher's students:
62, 77, 78, 80, 82, 82, 83, 84, 85, 87, 89, 95
This data set represents the population, meaning it includes all the possible data points for the given population.
Formulas for Calculating Variance
There are two primary formulas for calculating variance: the population variance formula and the sample variance formula.
Population Variance Formula
The population variance formula is used when working with a population. It is calculated as:
σ² = Σ(xi - μ)² / N
where:
- σ² is the population variance
- xi is each individual data point
- μ is the population mean
- N is the total number of data points
Sample Variance Formula
The sample variance formula is used when working with a sample. It is calculated as:
s² = Σ(xi - x̄)² / (n - 1)
where:
- s² is the sample variance
- xi is each individual data point
- x̄ is the sample mean
- n is the total number of data points in the sample
Choosing the Right Formula
Since the data set provided represents the population, we will use the population variance formula to calculate the variance.
Calculating the Population Mean
To calculate the population variance, we first need to calculate the population mean. The population mean is calculated as:
μ = Σxi / N
where:
- μ is the population mean
- xi is each individual data point
- N is the total number of data points
Using the given data set, we can calculate the population mean as follows:
μ = (62 + 77 + 78 + 80 + 82 + 82 + 83 + 84 + 85 + 87 + 89 + 95) / 12 μ = 844 / 12 μ = 70.33
Calculating the Population Variance
Now that we have the population mean, we can calculate the population variance using the population variance formula:
σ² = Σ(xi - μ)² / N
where:
- σ² is the population variance
- xi is each individual data point
- μ is the population mean
- N is the total number of data points
Using the given data set, we can calculate the population variance as follows:
σ² = [(62 - 70.33)² + (77 - 70.33)² + (78 - 70.33)² + (80 - 70.33)² + (82 - 70.33)² + (82 - 70.33)² + (83 - 70.33)² + (84 - 70.33)² + (85 - 70.33)² + (87 - 70.33)² + (89 - 70.33)² + (95 - 70.33)²] / 12 σ² = [(8.33)² + (6.67)² + (7.67)² + (9.67)² + (11.67)² + (11.67)² + (12.67)² + (13.67)² + (14.67)² + (16.67)² + (18.67)² + (24.67)²] / 12 σ² = [69.33 + 44.49 + 58.49 + 92.89 + 136.09 + 136.09 + 160.89 + 186.09 + 215.69 + 279.69 + 348.89 + 608.09] / 12 σ² = 2536.1 / 12 σ² = 210.85
Conclusion
Q: What is the difference between population variance and sample variance?
A: The main difference between population variance and sample variance is that population variance is used when working with a population, while sample variance is used when working with a sample. Population variance is calculated as σ² = Σ(xi - μ)² / N, while sample variance is calculated as s² = Σ(xi - x̄)² / (n - 1).
Q: Why is it important to use the correct formula for variance calculation?
A: Using the correct formula for variance calculation is crucial because it affects the accuracy of the results. If the wrong formula is used, it can lead to incorrect conclusions and decisions.
Q: How do I determine whether to use the population variance formula or the sample variance formula?
A: To determine whether to use the population variance formula or the sample variance formula, you need to check if the data set represents a population or a sample. If the data set represents a population, use the population variance formula. If the data set represents a sample, use the sample variance formula.
Q: What is the formula for calculating the population mean?
A: The formula for calculating the population mean is μ = Σxi / N, where μ is the population mean, xi is each individual data point, and N is the total number of data points.
Q: How do I calculate the population variance using the population variance formula?
A: To calculate the population variance using the population variance formula, you need to follow these steps:
- Calculate the population mean using the formula μ = Σxi / N.
- Subtract the population mean from each individual data point to get the deviations.
- Square each deviation to get the squared deviations.
- Sum up the squared deviations.
- Divide the sum of the squared deviations by the total number of data points (N) to get the population variance.
Q: What is the significance of the population variance?
A: The population variance is a measure of the spread or dispersion of a population. It is an essential concept in statistics and is used in various applications, including hypothesis testing and confidence intervals.
Q: Can I use the sample variance formula to calculate the population variance?
A: No, you cannot use the sample variance formula to calculate the population variance. The sample variance formula is used when working with a sample, while the population variance formula is used when working with a population.
Q: What is the difference between the population variance and the standard deviation?
A: The population variance and the standard deviation are related but distinct concepts. The population variance is a measure of the spread or dispersion of a population, while the standard deviation is the square root of the population variance. The standard deviation is a measure of the average distance of each data point from the population mean.
Q: How do I calculate the standard deviation from the population variance?
A: To calculate the standard deviation from the population variance, you need to take the square root of the population variance. The formula for calculating the standard deviation is σ = √σ², where σ is the standard deviation and σ² is the population variance.