Variance of Probability Distribution Calculator
Calculate Variance
Enter the values of the random variable (x) and their corresponding probabilities (P(x)). Ensure the sum of probabilities is 1. Leave unused fields blank or probabilities as 0.
Results:
Expected Value (E[X] or μ): 0.00
E[X²]: 0.00
Standard Deviation (σ): 0.00
Sum of Probabilities (Σ P(x)): 0.00
| i | x_i | P(x_i) | x_i * P(x_i) | x_i² * P(x_i) |
|---|---|---|---|---|
| 1 | 0 | 0.1 | 0 | 0 |
| 2 | 1 | 0.2 | 0.2 | 0.2 |
| 3 | 2 | 0.4 | 0.8 | 1.6 |
| 4 | 3 | 0.2 | 0.6 | 1.8 |
| 5 | 4 | 0.1 | 0.4 | 1.6 |
| Sum | – | 1.0 | 2.0 | 5.2 |
What is the Variance of a Probability Distribution?
The variance of a probability distribution is a measure of the spread or dispersion of the values of a random variable around its expected value (mean). It quantifies how much the values of the random variable tend to deviate from the average value. 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.
This Variance of Probability Distribution Calculator helps you compute this value for a discrete probability distribution. It's used by students, statisticians, data analysts, and researchers to understand the variability within a set of outcomes and their associated probabilities.
Common misconceptions include confusing variance with standard deviation (standard deviation is the square root of variance and is in the same units as the random variable) or thinking variance can be negative (it is always non-negative, being an average of squared differences).
Variance of Probability Distribution Formula and Mathematical Explanation
For a discrete random variable X that can take values x₁, x₂, …, xₙ with corresponding probabilities P(x₁), P(x₂), …, P(xₙ), the expected value (or mean) E[X] is calculated as:
E[X] = μ = Σ [xᵢ * P(xᵢ)]
The expected value of X squared, E[X²], is calculated as:
E[X²] = Σ [xᵢ² * P(xᵢ)]
The variance of X, denoted as Var(X) or σ², is then calculated using the formula:
Var(X) = σ² = E[X²] – (E[X])² = Σ [xᵢ² * P(xᵢ)] – (Σ [xᵢ * P(xᵢ)])²
Alternatively, it can be calculated as the expected value of the squared deviations from the mean:
Var(X) = E[(X – μ)²] = Σ [(xᵢ – μ)² * P(xᵢ)]
Our Variance of Probability Distribution Calculator uses the E[X²] – (E[X])² formula for computational efficiency.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| xᵢ | The i-th value of the random variable X | Depends on X | Any real number |
| P(xᵢ) | Probability of xᵢ occurring | Dimensionless | 0 to 1 |
| E[X] (μ) | Expected Value or Mean of X | Same as X | Any real number |
| E[X²] | Expected Value of X squared | (Unit of X)² | Non-negative |
| Var(X) (σ²) | Variance of X | (Unit of X)² | Non-negative |
| σ | Standard Deviation of X | Same as X | Non-negative |
Practical Examples (Real-World Use Cases)
Example 1: Dice Roll Game
Imagine a game where you roll a fair six-sided die. If you roll a 1, 2, or 3, you win $0. If you roll a 4 or 5, you win $2. If you roll a 6, you win $5. Let X be the amount you win.
The values of X are 0, 2, and 5. The probabilities are:
- P(X=0) = 3/6 = 0.5
- P(X=2) = 2/6 ≈ 0.333
- P(X=5) = 1/6 ≈ 0.167
Using the Variance of Probability Distribution Calculator with these values (and others set to 0 probability):
- x1=0, p1=0.5
- x2=2, p2=0.333
- x3=5, p3=0.167
- (other x, p are 0)
E[X] ≈ (0*0.5) + (2*0.333) + (5*0.167) ≈ 0 + 0.666 + 0.835 = 1.501
E[X²] ≈ (0²*0.5) + (2²*0.333) + (5²*0.167) ≈ 0 + (4*0.333) + (25*0.167) ≈ 0 + 1.332 + 4.175 = 5.507
Var(X) ≈ 5.507 – (1.501)² ≈ 5.507 – 2.253 = 3.254
The variance is about 3.254, indicating the spread of winnings around the average expected win of $1.50.
Example 2: Defective Items
A machine produces items, and the number of defective items in a batch of 5 follows a certain probability distribution:
- 0 defects: P(0) = 0.6
- 1 defect: P(1) = 0.25
- 2 defects: P(2) = 0.1
- 3 defects: P(3) = 0.05
- 4 or 5 defects: P(4)=0, P(5)=0
Using the Variance of Probability Distribution Calculator:
- x1=0, p1=0.6
- x2=1, p2=0.25
- x3=2, p3=0.1
- x4=3, p4=0.05
- x5=0, p5=0 (or any other value with p=0)
E[X] = (0*0.6) + (1*0.25) + (2*0.1) + (3*0.05) = 0 + 0.25 + 0.2 + 0.15 = 0.6
E[X²] = (0²*0.6) + (1²*0.25) + (2²*0.1) + (3²*0.05) = 0 + 0.25 + 0.4 + 0.45 = 1.1
Var(X) = 1.1 – (0.6)² = 1.1 – 0.36 = 0.74
The variance in the number of defective items is 0.74.
How to Use This Variance of Probability Distribution Calculator
- Enter Values and Probabilities: Input the values of the random variable (x1, x2, x3, etc.) and their corresponding probabilities (p1, p2, p3, etc.) into the designated fields.
- Check Probabilities: Ensure each probability is between 0 and 1, and that the sum of all probabilities is equal to 1 (or very close, due to rounding). The calculator will show the sum and flag if it's far from 1.
- View Results: The calculator automatically updates the Expected Value (E[X]), E[X²], Variance (Var(X)), and Standard Deviation (σ) as you type.
- Analyze Table and Chart: The table below the results shows the intermediate calculations (x*P(x) and x²*P(x) for each pair), and the chart visualizes the probability distribution and the mean.
- Reset: Use the "Reset" button to clear the inputs to their default values.
- Copy Results: Use the "Copy Results" button to copy the main results and intermediate values to your clipboard.
The primary result, the Variance (σ²), tells you how spread out your distribution is. A higher variance means more spread.
Key Factors That Affect Variance Results
- Range of Values (xᵢ): A wider range of x values, especially those far from the mean, can significantly increase the variance.
- Probabilities of Extreme Values: Higher probabilities associated with values far from the mean will increase the variance more than higher probabilities for values close to the mean.
- Concentration of Probabilities: If most of the probability mass is concentrated around a few values close to the mean, the variance will be low. If it's spread out, the variance will be higher.
- Number of Outcomes: While not directly a factor, having more possible outcomes with non-zero probabilities spread over a wide range can lead to higher variance.
- Symmetry of the Distribution: While not directly affecting the variance value itself compared to an asymmetric one with the same spread, the interpretation in relation to the mean can differ.
- Accuracy of Probabilities: Small errors in the input probabilities, especially if they don't sum to 1, can affect the variance calculation. Our Variance of Probability Distribution Calculator warns if the sum is off.
Frequently Asked Questions (FAQ)
- What is the difference between variance and standard deviation?
- Standard deviation is the square root of the variance. Variance is measured in squared units of the random variable, while standard deviation is in the same units as the random variable, making it more interpretable in many contexts.
- Can variance be negative?
- No, variance cannot be negative because it is the average of squared differences (or calculated as E[X²] – (E[X])², which is always non-negative).
- What does a variance of 0 mean?
- A variance of 0 means all the values of the random variable are the same; there is no spread or variability. The random variable is constant.
- Why does the sum of probabilities have to be 1?
- The sum of probabilities of all possible mutually exclusive outcomes of a random variable must equal 1, representing 100% certainty that one of the outcomes will occur.
- How do I use the calculator if I have more than 5 pairs of x and P(x)?
- This calculator is set up for up to 5 pairs. For more, you would need a more advanced tool or software like Excel, R, or Python, or group less probable outcomes if appropriate.
- What if my probabilities don't sum to exactly 1 due to rounding?
- The calculator checks if the sum is close to 1 (e.g., between 0.999 and 1.001). Small rounding differences are usually acceptable, but large deviations indicate an error in probabilities.
- Is this calculator for discrete or continuous distributions?
- This Variance of Probability Distribution Calculator is designed for discrete probability distributions, where you have specific values xᵢ and their probabilities P(xᵢ).
- What is E[X²]?
- E[X²] is the expected value of the square of the random variable X. It's calculated by summing the products of each squared value of X and its corresponding probability (Σ [xᵢ² * P(xᵢ)]).
Related Tools and Internal Resources
- Expected Value Calculator: Calculate the mean or expected value of a discrete probability distribution.
- Standard Deviation Calculator: Find the standard deviation from a set of data points or a probability distribution.
- Basic Probability Calculator: Calculate simple probabilities of events.
- Confidence Interval Calculator: Estimate a population parameter with a certain confidence level.
- Z-Score Calculator: Find the z-score of a data point.
- P-Value Calculator: Calculate the p-value from a test statistic.