For two data sets, each with size 5 , the variances are...

For two data sets, each with size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is
(A) $\frac{11}{2}$
(B) 6
(C) $\frac{13}{2}$
(D) $\frac{5}{2}$

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Key Concepts

Variance

Variance is a measure of dispersion that indicates how far the data points are spread out from the mean. It is calculated by averaging the squared differences between each data point and the mean, providing an insight into the variability present within a dataset.

Arithmetic Mean

The arithmetic mean represents the central tendency of a dataset and is computed as the sum of all data points divided by the number of points. It serves as the 'balance point' of the data and is crucial when comparing different datasets.

Combined Variance

When merging two datasets, the overall variance must account for both the variability within each dataset and the differences between their means. This is accomplished by taking a weighted average of the individual variances and incorporating the squared differences of each dataset's mean from the overall mean. This approach ensures that the combined variance accurately reflects all sources of variability from the two groups.

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