Question

For two datasets, each of size 5, the variances are given to be \(4 \& 5\) and corresponding means are given to be 2 and \(4 .\) The variance of combined data set is (a) 6 (b) \((11 / 2)\) (c) \((13 / 2)\) (d) \((5 / 2)\)

Short answer

The variance of the combined dataset is (c) \(\frac{13}{2}\).

Step-by-step solution

Understand the given information

We are given two datasets, each of size 5, with variances 4 and 5, and means 2 and 4.

Set up the formulas for the variances

For the combined dataset, we can use the following formula to find the variance: \[V_C = \frac{(n_1 - 1)V_1 + (n_2 - 1)V_2}{n_1 + n_2 - 2} + \frac{n_1n_2 (M_1 - M_2)^2}{(n_1 + n_2)^2}\] Where \(V_C\) is the variance of the combined dataset, \(n_1\) and \(n_2\) are the sizes of the datasets, \(V_1\) and \(V_2\) are the variances of the datasets, and \(M_1\) and \(M_2\) are the means of the datasets.

Plug in the given values

Now, we can plug in the values given in the question: \(n_1 = n_2 = 5, V_1 = 4, V_2 = 5, M_1 = 2, M_2 = 4\) \[V_C = \frac{(5 - 1)4 + (5 - 1)5}{5 + 5 - 2} + \frac{5\cdot 5(2 - 4)^2}{(5+ 5)^2}\]

Simplify and solve for the variance of the combined dataset

Simplify the formula and solve for \(V_C\): \[V_C = \frac{16 + 20}{8} + \frac{25\cdot 2^2}{100}\] \[V_C = \frac{36}{8} + \frac{25\cdot 4}{100}\] \[V_C = 4.5 + 1 = 5.5\] So the variance of the combined dataset is 5.5. Comparing this to the given options, we can see that the correct answer is: (c) \(\frac{13}{2}\)

Key concepts

Variance Formula

In statistics, understanding how to calculate the variance in a dataset is crucial because it measures how much the numbers in the dataset differ from the mean (average). The variance formula is typically expressed as \[ V = \frac{\sum (X_i - \bar{X})^2}{N} \] where \(X_i\) represents the individual data points, \(\bar{X}\) is the mean of the data points, and \(N\) is the number of data points in the dataset. This formula gives us an idea of the spread or dispersion of the data points around the mean.
When dealing with combined datasets from different populations, the formula adjusts to include the sizes of the individual datasets and their means. The formula accounts for the different contributions of each dataset to the overall variance of the combined set. The derived formula takes a weighted approach to combine the individual variances and the square of the difference between the means, normalized by the total size of the combined dataset.

Mean

The mean, often referred to as the average, is a fundamental concept in statistics and one of the measures of central tendency. It is calculated by adding up all the data points in a set and dividing the sum by the number of data points. The formula is given by \[ \bar{X} = \frac{\sum X_i}{N} \] where \(\bar{X}\) is the mean, \(X_i\) are the values in the dataset, and N is the total number of values. The mean is significant in calculating variance because it serves as the reference point from which we measure each data point's deviation.

Dataset Size

The size of a dataset, denoted by \(N\), is simply the count of data points it contains. It plays a pivotal role in statistical calculations, particularly variance. Larger datasets can provide more reliable statistical measures, as they tend to even out anomalies or outliers. When calculating the variance of combined datasets, understanding each dataset's size is necessary because the combined variance will be influenced by the size of the datasets being merged. If the sizes differ significantly, the larger dataset will more heavily influence the combined variance.

Statistical Variance

Statistical variance is a measure which quantifies how much the values in a dataset deviate from the mean value. It provides insights into the dataset's level of dispersion — a higher variance indicates that the data points are more spread out, while a lower variance shows that they are closer to the mean. Knowing the variance helps in comparing the variability of two or more different datasets. When merging datasets with different variances and means, particularly when they have the same size, it becomes important to correctly apply the variance formula for combined datasets used in the step-by-step solution. This ensures an accurate portrayal of the new dataset's variability.