Find The Variance of The Following Probability Distribution Calculator

Variance is a fundamental measure in statistics that quantifies the spread of a probability distribution. This calculator helps you find the variance of any given probability distribution by following a straightforward process.

What is Variance?

Variance measures how far each number in a dataset is from the mean (average) of the dataset. A higher variance indicates that the numbers are more spread out, while a lower variance indicates they are closer to the mean.

Variance is calculated by taking the average of the squared differences from the mean. The square root of variance gives you the standard deviation, which is often more intuitive to interpret.

Key points about variance:

Variance is always non-negative
It's measured in the same units as the original data squared
Higher variance means more dispersion in the data
Variance is sensitive to outliers

How to Calculate Variance

The formula for variance (σ²) of a probability distribution is:

σ² = Σ [ (xᵢ - μ)² × P(xᵢ) ]

Where:

xᵢ = each possible value in the distribution
μ = mean of the distribution
P(xᵢ) = probability of each value
Σ = sum over all possible values

For a sample variance (s²), you would divide by (n-1) instead of n, but this calculator assumes a population variance.

Example Calculation

Let's calculate the variance for a simple probability distribution:

Value (xᵢ)	Probability (P(xᵢ))
1	0.2
2	0.3
3	0.5

First, calculate the mean (μ):

μ = Σ (xᵢ × P(xᵢ)) = (1×0.2) + (2×0.3) + (3×0.5) = 0.2 + 0.6 + 1.5 = 2.3

Then calculate the variance:

σ² = [(1-2.3)²×0.2] + [(2-2.3)²×0.3] + [(3-2.3)²×0.5] = [1.21×0.2] + [0.09×0.3] + [0.49×0.5] = 0.242 + 0.027 + 0.245 = 0.514

The variance of this distribution is 0.514.

Interpreting Variance

The variance value itself can be difficult to interpret directly. Here are some practical ways to understand it:

Compare variances between different distributions to see which has more spread
Calculate the standard deviation (√variance) for a more intuitive measure
Use variance to assess the reliability of estimates in statistical models
Variance of zero indicates a deterministic distribution (all values are the same)

Remember that variance is affected by the scale of your data. For example, if you measure height in centimeters vs. meters, the variance will be 10,000 times larger.

FAQ

What's the difference between variance and standard deviation?

Variance is the average of the squared differences from the mean, while standard deviation is the square root of variance. Standard deviation is often more intuitive because it's in the same units as the original data.

How do I know if my variance calculation is correct?

Double-check your calculations using the formula provided. Make sure you're using the correct probabilities and values, and that you've calculated the mean correctly first.

Can variance be negative?

No, variance is always non-negative because it's based on squared differences. The smallest possible variance is zero, which occurs when all values in the distribution are identical.