Find Mean Of Probability Distribution Calculator

Enter the possible outcomes (x) and their corresponding probabilities P(x) below. The sum of probabilities should be 1.

Input Data Table

Outcome (x)	Probability P(x)	x * P(x)	x² * P(x)

Table showing the entered outcomes, their probabilities, and intermediate calculations.

What is the Mean of a Probability Distribution?

The mean of a probability distribution, also known as the expected value (E[X]), represents the weighted average of all possible values that a random variable can take, with the weights being the probabilities of those values. It's a fundamental concept in probability and statistics, indicating the long-run average outcome of a random process if it were repeated many times. Unlike the simple arithmetic mean of a dataset, the mean of a probability distribution accounts for the likelihood of each outcome occurring.

Anyone dealing with uncertain outcomes can use it, including investors analyzing potential returns, businesses forecasting sales, scientists modeling random phenomena, and gamblers assessing the fairness of a game. For example, if you know the probabilities of different profit levels for an investment, the mean of the probability distribution gives you the expected profit.

A common misconception is that the mean of the distribution must be one of the possible outcomes. This is not necessarily true. For instance, the expected value when rolling a fair six-sided die is 3.5, which is not a value you can actually roll.

Mean of Probability Distribution Formula and Mathematical Explanation

For a discrete random variable X that can take values x₁, x₂, x₃, …, xₙ with corresponding probabilities P(x₁), P(x₂), P(x₃), …, P(xₙ), the mean (or expected value) E[X] is calculated as:

E[X] = Σ [xᵢ * P(xᵢ)] = x₁P(x₁) + x₂P(x₂) + … + xₙP(xₙ)

Where:

• xᵢ represents the i-th possible outcome of the random variable X.
• P(xᵢ) represents the probability of the i-th outcome xᵢ occurring.
• Σ denotes the summation over all possible outcomes.

The variance Var(X) or σ², which measures the spread of the distribution, is calculated as:

Var(X) = E[(X – E[X])²] = Σ [(xᵢ – E[X])² * P(xᵢ)]

Alternatively, Var(X) can be calculated as:

Var(X) = E[X²] – (E[X])² = (Σ [xᵢ² * P(xᵢ)]) – (E[X])²

The standard deviation σ(X) is the square root of the variance:

σ(X) = √Var(X)

Variables in the Mean of Probability Distribution Calculation

Variable	Meaning	Unit	Typical Range
xᵢ	i-th possible outcome	Varies (e.g., currency, score, units)	Any real number
P(xᵢ)	Probability of outcome xᵢ	Dimensionless	0 to 1
E[X]	Mean or Expected Value	Same as xᵢ	Any real number
Var(X)	Variance	(Unit of xᵢ)²	≥ 0
σ(X)	Standard Deviation	Same as xᵢ	≥ 0

Practical Examples (Real-World Use Cases)

Example 1: Investment Return

An investor is considering an investment with the following potential returns and probabilities after one year:

• Return $1000 with probability 0.2
• Return $500 with probability 0.5
• Return -$200 (loss) with probability 0.3

Using the Mean of Probability Distribution Calculator or formula:

E[X] = (1000 * 0.2) + (500 * 0.5) + (-200 * 0.3) = 200 + 250 – 60 = $390

The expected return (mean) of this investment is $390.

Example 2: Number of Defective Items

A factory produces batches of widgets. The number of defective widgets per batch and their probabilities are:

• 0 defective with probability 0.80
• 1 defective with probability 0.15
• 2 defective with probability 0.04
• 3 defective with probability 0.01

The expected number of defective widgets per batch is:

E[X] = (0 * 0.80) + (1 * 0.15) + (2 * 0.04) + (3 * 0.01) = 0 + 0.15 + 0.08 + 0.03 = 0.26

The mean number of defective widgets per batch is 0.26.

How to Use This Mean of Probability Distribution Calculator

1. Enter Outcomes and Probabilities: In the input fields, enter each possible outcome (x) and its corresponding probability P(x). The calculator starts with three rows, but you can add more using the "Add Another Outcome" button or remove them using the 'X' button next to each row (except the first two).
2. Probabilities Check: Ensure that each probability is between 0 and 1, and the sum of all probabilities is very close to 1. The calculator will show the sum and highlight if it's not 1.
3. Calculate: Click the "Calculate Mean" button.
4. View Results: The calculator will display the Mean (Expected Value), Variance, Standard Deviation, and the Sum of Probabilities.
5. See Table & Chart: A table with your inputs and intermediate calculations, along with a bar chart visualizing the distribution, will appear below the results.
6. Reset: Use the "Reset" button to clear all inputs and start over with default values.
7. Copy: Use the "Copy Results" button to copy the main results and input summary to your clipboard.

The displayed mean gives you the long-term average you can expect if the random process is repeated many times under the same conditions. The standard deviation gives you a measure of the risk or variability around this mean.

Key Factors That Affect Mean of Probability Distribution Results

• Values of Outcomes (x): Higher outcome values, especially those with significant probabilities, will increase the mean.
• Probabilities of Outcomes (P(x)): Outcomes with higher probabilities have a greater influence on the mean. A shift in probability towards higher-value outcomes increases the mean.
• Number of Possible Outcomes: While not directly affecting the formula for a given set of outcomes, the range and distribution of outcomes are crucial.
• Symmetry of the Distribution: For symmetric distributions, the mean coincides with the median. Skewed distributions (with long tails) will have the mean pulled towards the tail.
• Outliers: Extreme outcome values, even with small probabilities, can significantly impact the mean and especially the variance.
• Sum of Probabilities: The sum of P(x) must equal 1 for a valid discrete probability distribution. Our Mean of Probability Distribution Calculator checks this.

Frequently Asked Questions (FAQ)

What is the difference between the mean of a sample and the mean of a probability distribution?

The mean of a sample is the arithmetic average of observed data points. The mean of a probability distribution (expected value) is a theoretical weighted average of all possible outcomes of a random variable, weighted by their probabilities. The sample mean is an estimate of the population mean or the mean of the underlying distribution.

Can the mean of a probability distribution be negative?

Yes, if the possible outcomes include negative values (like losses in an investment), the mean can be negative.

What does a mean of 0 imply?

A mean of 0 implies that, on average, the positive and negative outcomes (weighted by their probabilities) balance each other out. In gambling, a game with an expected value of 0 is considered fair (in the long run).

How is the mean related to variance and standard deviation?

The mean is a measure of central tendency, while variance and standard deviation measure the spread or dispersion of the distribution around the mean.

What if the sum of my probabilities is not 1?

If the sum is not 1 (or very close to 1 due to rounding), it's not a valid discrete probability distribution. You should re-check your probabilities. Our Mean of Probability Distribution Calculator will flag this.

Can I use this calculator for continuous distributions?

No, this calculator is specifically for discrete probability distributions, where the random variable takes on a finite or countably infinite number of values. Continuous distributions require integration to find the mean.

Why is it called "expected value"?

It's called the expected value because it represents the average value we would expect to get if we repeated the random experiment or process a very large number of times.

What are the limitations of using just the mean?

The mean doesn't tell the whole story. Two distributions can have the same mean but very different shapes and risks (variances). It's often useful to consider the standard deviation and the overall shape of the distribution alongside the mean.