Variance Calculator
Calculate Variance
Enter your data set below to calculate the mean, sum of squares, and variance.
Understanding the Variance Calculator
What is Variance?
Variance is a statistical measurement that quantifies the spread or dispersion of a set of data points around their average value (the mean). A low variance indicates that the data points tend to be close to the mean, while a high variance indicates that the data points are spread out over a wider range of values. The Variance Calculator helps you determine this spread quickly and accurately.
Statisticians, data analysts, researchers, financial analysts, and anyone working with data sets use variance to understand the variability within their data. For instance, in finance, variance is used to measure the volatility of an investment. Our Variance Calculator is a useful tool for these professionals and students alike.
A common misconception is that variance is the same as standard deviation. While related (standard deviation is the square root of variance), variance is expressed in squared units of the original data, whereas standard deviation is in the original units, making it more intuitive for some interpretations.
Variance Formula and Mathematical Explanation
There are two main formulas for variance, depending on whether you are working with an entire population or a sample from that population.
Population Variance (σ²)
If your data set represents the entire population of interest, you use the population variance formula:
σ² = Σ (xᵢ – μ)² / N
Where:
- σ² is the population variance
- Σ is the summation symbol (sum of)
- xᵢ represents each individual data point
- μ is the population mean
- N is the total number of data points in the population
You sum the squared differences between each data point and the population mean, then divide by the total number of data points.
Sample Variance (s²)
If your data set is a sample taken from a larger population, and you want to estimate the population variance from this sample, you use the sample variance formula:
s² = Σ (xᵢ – x̄)² / (n – 1)
Where:
- s² is the sample variance
- Σ is the summation symbol (sum of)
- xᵢ represents each individual data point in the sample
- x̄ is the sample mean
- n is the total number of data points in the sample
The key difference is dividing by (n – 1) instead of n. This is known as Bessel's correction, which provides a more unbiased estimate of the population variance when using sample data.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| xᵢ | Individual data point | Units of data | Varies with data |
| μ or x̄ | Mean (average) of the data | Units of data | Varies with data |
| N or n | Number of data points | Count (integer) | ≥1 (n-1 requires n>1 for sample) |
| σ² or s² | Variance | Squared units of data | ≥0 |
Practical Examples (Real-World Use Cases)
Example 1: Test Scores
Imagine a teacher wants to analyze the spread of test scores in a small class of 5 students. The scores are: 70, 75, 80, 85, 90. Since this is the entire class (population), we use the population variance formula.
- Calculate the mean (μ): (70 + 75 + 80 + 85 + 90) / 5 = 400 / 5 = 80
- Calculate squared deviations from the mean: (70-80)² = (-10)² = 100 (75-80)² = (-5)² = 25 (80-80)² = 0² = 0 (85-80)² = 5² = 25 (90-80)² = 10² = 100
- Sum of squared deviations: 100 + 25 + 0 + 25 + 100 = 250
- Calculate Population Variance (σ²): 250 / 5 = 50
The variance of the test scores is 50. Using the Variance Calculator with these numbers as input and selecting "Population" would yield this result.
Example 2: Heights of a Sample of Plants
A botanist measures the heights of a sample of 4 plants from a larger field: 10 cm, 12 cm, 15 cm, 13 cm. They want to estimate the variance of heights for all plants in the field using this sample.
- Calculate the sample mean (x̄): (10 + 12 + 15 + 13) / 4 = 50 / 4 = 12.5 cm
- Calculate squared deviations from the mean: (10-12.5)² = (-2.5)² = 6.25 (12-12.5)² = (-0.5)² = 0.25 (15-12.5)² = (2.5)² = 6.25 (13-12.5)² = (0.5)² = 0.25
- Sum of squared deviations: 6.25 + 0.25 + 6.25 + 0.25 = 13
- Calculate Sample Variance (s²): 13 / (4 – 1) = 13 / 3 ≈ 4.33
The sample variance of the plant heights is approximately 4.33 cm². The Variance Calculator, with these inputs and "Sample" selected, provides this value.
How to Use This Variance Calculator
Using our Variance Calculator is straightforward:
- Enter Your Data: Type or paste your numerical data into the "Data Set" input field. Ensure the numbers are separated by commas (e.g., 10, 20, 30) or spaces (e.g., 10 20 30).
- Select Variance Type: Choose whether your data represents an entire "Population" or a "Sample" from a larger population using the radio buttons. This affects the denominator in the variance calculation (N or n-1).
- Calculate: Click the "Calculate" button.
- View Results: The calculator will display:
- The calculated Variance (primary result).
- The number of data points (N or n).
- The Mean (μ or x̄).
- The Sum of Squared Deviations (SS).
- The Standard Deviation (square root of variance).
- The formula used based on your selection.
- A table showing each data point, its deviation, and squared deviation.
- A chart visualizing the data points relative to the mean.
- Reset: Click "Reset" to clear the input and results for a new calculation.
- Copy Results: Click "Copy Results" to copy the main outputs to your clipboard.
Understanding the results: A higher variance suggests greater variability or dispersion in your data, while a lower variance suggests the data points are clustered more closely around the mean.
Key Factors That Affect Variance Results
Several factors influence the calculated variance:
- Data Spread: The more spread out the data points are from the mean, the higher the variance. Conversely, data points clustered close to the mean result in a lower variance.
- Outliers: Extreme values (outliers) can significantly increase the variance because the squared differences for these points will be very large.
- Sample Size (n): For sample variance, the (n-1) denominator means smaller sample sizes can lead to larger variance estimates, especially if there's significant spread. As n increases, the difference between dividing by n and n-1 becomes smaller.
- Measurement Errors: Inaccurate measurements can introduce artificial variability, increasing the calculated variance.
- Nature of the Data: Some datasets are inherently more variable than others (e.g., incomes in a diverse population vs. heights of adults of the same gender).
- Population vs. Sample: Choosing between population and sample variance directly affects the calculation (dividing by N or n-1), with sample variance typically being slightly larger for the same data set.
The Variance Calculator accurately reflects these factors based on the data you provide and the type of variance selected.
Frequently Asked Questions (FAQ)
1. What is variance in simple terms?
Variance measures how far a set of numbers are spread out from their average value. High variance means numbers are very spread out; low variance means they are close together.
2. What is the difference between population variance and sample variance?
Population variance (σ²) is calculated using data from the entire group of interest and divides the sum of squared deviations by N (total number of data points). Sample variance (s²) is calculated from a subset (sample) of the population and divides by n-1 (sample size minus one) to provide an unbiased estimate of the population variance.
3. Why divide by n-1 for sample variance?
Dividing by n-1 (Bessel's correction) corrects the bias that occurs when using a sample mean to estimate the population mean, making the sample variance a better, unbiased estimator of the population variance.
4. What does a variance of 0 mean?
A variance of 0 means all the values in the dataset are identical. There is no spread or dispersion around the mean because all data points are the mean.
5. How is variance related to standard deviation?
Standard deviation is the square root of the variance. While variance gives you a measure in squared units, standard deviation gives you a measure of spread in the original units of the data, which is often easier to interpret. Our Standard Deviation Calculator can also be helpful.
6. Can variance be negative?
No, variance cannot be negative because it is calculated from the sum of squared values, and squares are always non-negative.
7. Is a high or low variance better?
It depends on the context. In manufacturing, low variance is often desired (consistency). In investments, high variance (volatility) might mean higher risk but also potentially higher return. Understanding the context is key to interpreting the Variance Calculator results.
8. What are the units of variance?
The units of variance are the square of the units of the original data. For example, if your data is in meters, the variance will be in meters squared.