Discrete Standard Deviation Calculator
Easily calculate the standard deviation of a discrete random variable. Just input the outcomes and their corresponding probabilities.
Formula: $$ \sigma = \sqrt{\sum_{i=1}^{n} (x_i - \mu)^2 p_i} $$, where $$\mu = \sum_{i=1}^{n} x_i p_i$$
Enter comma-separated numerical values representing the outcomes.
Enter comma-separated probabilities for each outcome. Ensure they sum to 1.
Standard Deviation: SD[X]
Calculation Breakdown
| Outcome (xi) | Probability (pi) | xi * pi | (xi - E[X])2 * pi |
|---|---|---|---|
| Total | 1 |
Understanding Standard Deviation
Standard Deviation (SD) is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. For a discrete random variable, it quantifies the spread of possible outcomes around the expected value.
Formula Explained:
- $$ \sigma = \sqrt{\sum_{i=1}^{n} (x_i - \mu)^2 p_i} $$ is the formula for the standard deviation of a discrete random variable.
- $$x_i$$ represents each possible outcome.
- $$p_i$$ is the probability of each outcome $$x_i$$.
- $$\mu = \sum_{i=1}^{n} x_i p_i$$ is the expected value (mean) of the discrete random variable.
- The calculator computes $$\sigma$$ based on your input outcomes and probabilities.
How to Use This Calculator:
1. Enter the possible outcomes in the 'Outcomes' field, separated by commas (e.g., 1,2,3). 2. Enter the corresponding probabilities in the 'Probabilities' field, also comma-separated (e.g., 0.2,0.3,0.5). 3. Ensure the number of outcomes matches the number of probabilities and that probabilities sum up to 1. 4. Click 'Calculate SD' to compute the standard deviation. 'Reset' will clear all inputs and results.
Learn more about standard deviation on resources like Wikipedia.