A Combined Variance Calculator computes the variance of multiple datasets that are combined into a single dataset. This is particularly useful when analyzing aggregated data from different sources. It eliminates the need to recalculate variance for the entire combined dataset manually, offering a streamlined and accurate way to handle data analysis.
By using this calculator, you can assess the spread of the data points within the combined dataset without accessing the raw data of all individual sets. This saves time and ensures precision, especially in statistical, financial, and scientific research scenarios.
Formula of Combined Variance Calculator
The formula for calculating combined variance is as follows:
Combined_variance = [(n₁ – 1) * s₁² + (n₂ – 1) * s₂² + … + (nk – 1) * sk² + Adjustment_term] / (N – k)
Where:
- Combined_variance is the variance of the combined data set.
- n₁, n₂, …, nk are the sizes of each data set.
- s₁², s₂², …, sk² are the variances of each data set.
- N = n₁ + n₂ + … + nk is the total size of all combined data sets.
- k is the number of data sets.
- Adjustment_term adjusts for the differences between the data set means.
Adjustment Term
Adjustment_term = [n₁ * (m₁ – m_combined)² + n₂ * (m₂ – m_combined)² + … + nk * (mk – m_combined)²]
Where:
- m₁, m₂, …, mk are the means of the individual data sets.
- m_combined is the mean of the combined data set.
Mean of the Combined Data Set
m_combined = (n₁ * m₁ + n₂ * m₂ + … + nk * mk) / N
General Terms Table
Here is a table for general terms and their meaning, aiding users in applying the formula effectively:
| Term | Meaning |
|---|---|
| Combined_variance | Variance of the combined dataset |
| nᵢ, nᵣ, …, nk | Sizes of each dataset |
| s₁², s₂², …, sk² | Variances of individual datasets |
| N | Total size of all datasets combined |
| k | Number of datasets |
| Adjustment_term | Adjustment for differences in dataset means |
| m_combined | Mean of the combined dataset |
| m₁, m₂, …, mk | Means of individual datasets |
Example of Combined Variance Calculator
Suppose you have two datasets:
- Dataset 1: n₁ = 5, s₁² = 4, m₁ = 10
- Dataset 2: n₂ = 7, s₂² = 9, m₂ = 14
- Calculate the total size: N = n₁ + n₂ = 5 + 7 = 12
- Find the combined mean: m_combined = (n₁ * m₁ + n₂ * m₂) / N m_combined = (5 * 10 + 7 * 14) / 12 = 12.33 (approximately)
- Compute the adjustment term: Adjustment_term = [n₁ * (m₁ – m_combined)² + n₂ * (m₂ – m_combined)²] Adjustment_term = [5 * (10 – 12.33)² + 7 * (14 – 12.33)²] Adjustment_term = 46.67 (approximately)
- Calculate combined variance: Combined_variance = [(n₁ – 1) * s₁² + (n₂ – 1) * s₂² + Adjustment_term] / (N – k) Combined_variance = [(5 – 1) * 4 + (7 – 1) * 9 + 46.67] / (12 – 2) Combined_variance = 12.87 (approximately)
Most Common FAQs
It simplifies calculating the variance for combined datasets, saving time and reducing computational errors. This is especially important in large-scale data analysis.
Yes, the formula accounts for differences in dataset sizes and adjusts the variance calculation accordingly.
Absolutely. The method is mathematically sound and widely used in statistics, finance, and research.