Expected Value and Variance Calculator

Advanced calculator for discrete or weighted datasets with probabilities and options. Validate inputs, auto normalize percentages, and flag probability errors in real time. Compute moments, standard deviation, and choose population or sample variance as needed. Export tables and summaries to CSV or PDF now.

Variance type:
Mean CI:
Probability Table
# Value (x) Probability (p) Remove

Results
E[X]
E[X²]
Variance
Std. Dev.
Coefficient of Variation CV = Std. Dev. / Mean
Displayed as ratio and percent when applicable.
Mean CI (95%)
For samples: uses selected z/t method and sample deviation.
Rows 0

Example Data

Discrete distribution (PMF)
xp
00.10
10.20
20.40
30.30
Raw data with weights
xw
23
54
72

Formulas Used

For a discrete distribution with values $x_i$ and probabilities $p_i$: $$E[X] = \sum_i x_i p_i,\quad E[X^2] = \sum_i x_i^2 p_i,\quad \mathrm{Var}(X) = E[X^2] - (E[X])^2.$$

For raw data with values $x_i$ and weights $w_i$ (treated as frequencies): $$\mu = \frac{\sum_i w_i x_i}{\sum_i w_i},\quad E[X^2] = \frac{\sum_i w_i x_i^2}{\sum_i w_i}.$$ Population variance: $\sigma^2 = E[X^2] - \mu^2$. Sample variance (frequency-based): $$s^2 = \frac{\sum_i w_i (x_i-\mu)^2}{\sum_i w_i - 1}\quad \text{when } \sum_i w_i > 1.$$

All computations are performed in your browser using floating-point arithmetic. Small rounding differences can occur.

How to Use

  1. Select the mode: probability table or raw data with weights.
  2. Enter rows. Use “Add row” for more entries as needed.
  3. Toggle “percentages” if probabilities are given from 0–100.
  4. Keep “normalize” on to scale probabilities that don’t sum to 1.
  5. Choose population or sample variance for the variance calculation.
  6. Review results. Resolve warnings about invalid or inconsistent inputs.
  7. Export your inputs and results as CSV or PDF when finished.
  8. Optionally view coefficient of variation and a mean confidence interval.

FAQs

No. If “Normalize” is enabled, the calculator scales probabilities so their total is one before computing expectations and variance.

Enable the “percentages” switch. Values like 25 become 0.25 internally. Normalization optionally rescales them afterward if needed.

Choose sample variance when your table represents a sample from a wider population. Population variance applies when your data include the whole population.

Use the “Raw data with weights” mode. Enter each unique value once and assign a weight or frequency count.

Browsers use binary floating-point. The calculator rounds outputs to a sensible number of decimal places to minimize display artifacts.

This tool focuses on clean entry and export. Paste from spreadsheets into the table fields, then export as CSV or PDF.

This version targets discrete distributions and weighted datasets. For continuous cases, discretize the range or use numerical integration tools.

Reference: Means and Variances of Common Distributions

DistributionParametersE[X]Var(X)
Bernoullippp(1 − p)
Binomialn, pnpnp(1 − p)
Geometric*p1/p(1 − p)/p²
Poissonλλλ
Discrete Uniforma..b (integers)(a + b)/2((b − a + 1)² − 1)/12

*Geometric assumes support {1, 2, 3, …}: trials until first success.

Worked Examples: Expected Value and Variance

Dataset A (PMF)
xp
00.10
10.20
20.40
30.30
  • E[X] = 1.9
  • E[X²] = 4.5
  • Var(X) = 0.89
  • Std. Dev. ≈ 0.9434
Dataset B (Weighted)
xw
23
54
72
  • μ = 40/9 ≈ 4.444444
  • E[X²] = 210/9 ≈ 23.333333
  • Population Var ≈ 3.580247
  • Sample Var ≈ 4.027778
  • Population Std. Dev. ≈ 1.89112

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.