Problem 13 - Basiswissen Statistik: Kompaktkurs für Anwender aus Wirtschaft, Informatik und Technik

Question

Welcher rechnerische Zusammenhang besteht zwischen der Stichprobenvarianz der Summe von zwei Datensätzen und den einzelnen Stichprobenvarianzen?

Step-by-Step Solution

Answer

The variance of the sum of two datasets is the sum of their variances if they are independent; otherwise, add twice their covariance.

Step 1: Understand the Problem

We need to find the relationship between the variance of the sum of two datasets and the individual sample variances.

Step 2: Recall the Formula for Variance

Variance () of a dataset is the average of the squared differences from the mean. For example, for a dataset X with sample variance \(s_X^2\), it is calculated as \(s_X^2 = \frac{1}{n-1}\sum (X_i - \bar{X})^2\).

Step 3: Sum of Two Independent Datasets

For two independent datasets, say X and Y, the variance of their sum \(Z = X + Y\) is \(s_Z^2 = s_X^2 + s_Y^2\). This comes from the property of variances, where the variance of a sum of independent random variables is the sum of their variances.

Step 4: Combine the Variances

For datasets that are not independent, the general formula for the variance of the sum is adjusted for covariance: \(s_Z^2 = s_X^2 + s_Y^2 + 2cov(X,Y)\), where \(cov(X, Y)\) is the covariance between datasets X and Y.

Step 5: Conclusion

The equation  =  +  holds when datasets are independent. If they are not, we must consider their covariance.

Key Concepts

Varianz

Variance is a key concept in statistics, measuring the degree to which each number in a set of data differs from the mean of the set. It indicates how spread out the numbers are around the average. For sample data, the variance is calculated by squaring the differences between each data point and the mean, summing these squared differences, and then dividing by the number of data points minus one. This formula is denoted as:
\[s_X^2 = \frac{1}{n-1} \sum (X_i - \bar{X})^2\]where:

• \(s_X^2\) is the sample variance
• \(X_i\) represents each data point
• \(\bar{X}\) is the mean of the dataset
• \(n\) is the number of data points in the dataset

The concept of variance helps to understand the variability in data, providing insights into how much individual data points deviate from the mean.

Unabhängigkeit

Independence in statistics refers to a scenario where two datasets or random variables do not influence one another. Essentially, the occurrence of one event or the value in one set does not affect the occurrence or value in the other. When two datasets are independent, their statistical properties do not inconvenience each other, making calculations like variance much simpler.
For independent datasets, the variance of their sum equals the sum of their variances. This is an essential principle because it allows us to determine characteristics of combined datasets without intricate interdependencies. Mathematically, the simplicity arising from independence is expressed as the variance of sum \(Z = X + Y\) being:
\[s_Z^2 = s_X^2 + s_Y^2\]Understanding independence is crucial since it directly affects the calculation of variance and other statistical metrics.

Kovarianz

Covariance measures the degree to which two variables change together. If they tend to vary together at the same direction, their covariance is positive, if inversely, the covariance is negative. This metric plays a pivotal role when dealing with datasets that are not independent.
When datasets are dependent, covariance introduces an adjustment in the variance computation. For the sum of datasets \(X\) and \(Y\), where they have some degree of linear dependence, the variance formula modifies to account for covariance:
\[s_Z^2 = s_X^2 + s_Y^2 + 2cov(X,Y)\]Here, \(cov(X, Y)\) represents the covariance of the datasets \(X\) and \(Y\). By understanding and adjusting for covariance, we can accurately compute variances for non-independent datasets, reflecting true variability in combined data metrics.