Calculator Input
Enter discrete values with probabilities or frequencies. The calculator returns the expected value, variance, standard deviation, and advanced spread checks.
Formula Used
Mean: μ = Σ[x × P(x)]
Expected square: E[X²] = Σ[x² × P(x)]
Variance: σ² = Σ[P(x) × (x − μ)²]
Standard deviation: σ = √σ²
Coefficient of variation: CV = σ ÷ |μ| × 100
For frequency input, each frequency is divided by total frequency first. That converts frequencies into probabilities.
How to Use This Calculator
- Select whether your second column contains probabilities or frequencies.
- Enter each possible value of the random variable in the x column.
- Enter the related probability or frequency beside each value.
- Enable normalization when probabilities do not already total one.
- Choose the decimal precision and chart style.
- Press the calculate button to show results above the form.
- Use the CSV or PDF button to save your report.
Example Data Table
This sample shows a discrete distribution for four possible outcomes.
| Outcome x | Probability P(x) | Meaning |
|---|---|---|
| 0 | 0.10 | Low outcome chance |
| 1 | 0.25 | Moderate low chance |
| 2 | 0.40 | Most likely outcome |
| 3 | 0.25 | Upper outcome chance |
Understanding Probability Distribution Standard Deviation
What It Measures
A probability distribution lists every possible value of a random variable. It also gives the chance for each value. The standard deviation measures how far outcomes usually sit from the mean. A small value means the distribution is tight. A large value means outcomes are more spread out. This makes the measure useful in statistics, finance, quality control, research, and education.
Why the Mean Comes First
The mean is the balance point of the distribution. It is also called the expected value. Each value is multiplied by its probability. Those products are then added together. The result gives the long run average. Standard deviation needs this mean before it can measure spread.
How Variance Supports Deviation
Variance uses squared distances from the mean. Squaring removes negative signs. It also gives larger gaps more weight. Each squared gap is multiplied by its probability. The products are added to form variance. Standard deviation is the square root of variance. This returns the spread to the original unit.
Using Probabilities Correctly
A valid probability distribution should total one. Each probability must be zero or positive. If values are entered as frequencies, they are first converted into probabilities. This calculator can also normalize weights. Normalization is helpful when values are proportional but not already scaled. It protects your answer from simple entry format issues.
Reading the Results
Compare the standard deviation with the mean and range. A high coefficient of variation signals strong relative spread. Skewness shows whether the distribution leans left or right. Kurtosis describes tail behavior. Cumulative probability helps locate the median. The chart gives a visual check. The detailed table also shows each contribution to variance.
Practical Use
Use this tool for homework, model checks, risk summaries, and reports. It is best for discrete distributions. Examples include demand counts, survey scores, defect counts, and simple payoff tables. Always review input values before trusting the result. Clean inputs create reliable statistical summaries.
FAQs
What is standard deviation in a probability distribution?
It measures the typical distance between possible outcomes and the expected value. A higher standard deviation means outcomes are more spread out.
Can I enter frequencies instead of probabilities?
Yes. Select the frequency input option. The calculator converts each frequency into a probability by dividing it by the total frequency.
Should probabilities always add to one?
Yes. A proper probability distribution totals one. If your values are proportional weights, enable normalization before calculating.
What does variance mean here?
Variance is the weighted average of squared distances from the mean. It is the value under the square root in standard deviation.
Why is the standard deviation not negative?
Standard deviation is based on squared distances and a square root. It describes size of spread, so it cannot be negative.
What is coefficient of variation?
It compares standard deviation with the absolute mean. It is useful when you want relative spread as a percentage.
What does skewness show?
Skewness describes distribution direction. Positive skew leans toward larger values. Negative skew leans toward smaller values.
Can I save my results?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a simple report.