Input Parameters
Formula Used
For a binomial random variable X ~ Binomial(n, p), the mean (expected value) is:
E[X] = μ = n · p
Variance: Var(X) = n · p · (1 − p)
Standard deviation: σ = √(n · p · (1 − p))
Mode (one of): ⌊(n + 1) · p⌋
Skewness: (1 − 2p) / √(n · p · (1 − p)) (defined when variance > 0)
The mean arises from linearity of expectation: X = ∑i=1..n Ii with E[Ii]=p, hence E[X]=∑E[Ii]=n p.
Example Data Table
| n (trials) | p (probability) | Mean (n·p) | Variance | Std Dev |
|---|---|---|---|---|
| 10 | 0.3000 | 3.000000 | 2.100000 | 1.449138 |
| 20 | 0.5000 | 10.000000 | 5.000000 | 2.236068 |
| 15 | 0.1000 | 1.500000 | 1.350000 | 1.161895 |
| 100 | 0.7000 | 70.000000 | 21.000000 | 4.582576 |
| 8 | 0.2500 | 2.000000 | 1.500000 | 1.224745 |
Reference: Mean across varying n (fixed p = 0.30)
Demonstrates linear scaling of the mean with n while p stays constant.
| n (trials) | p | Mean (n·p) | Variance | Std Dev |
|---|---|---|---|---|
| 5 | 0.3000 | 1.500000 | 1.050000 | 1.024695 |
| 10 | 0.3000 | 3.000000 | 2.100000 | 1.449138 |
| 20 | 0.3000 | 6.000000 | 4.200000 | 2.049390 |
| 50 | 0.3000 | 15.000000 | 10.500000 | 3.240370 |
| 100 | 0.3000 | 30.000000 | 21.000000 | 4.582576 |
| 200 | 0.3000 | 60.000000 | 42.000000 | 6.480741 |
Reference: Mean across varying p (fixed n = 20)
Shows how the mean increases linearly with p for a fixed n.
| n (trials) | p | Mean (n·p) | Variance | Std Dev |
|---|---|---|---|---|
| 20 | 0.1000 | 2.000000 | 1.800000 | 1.341641 |
| 20 | 0.2500 | 5.000000 | 3.750000 | 1.936492 |
| 20 | 0.4000 | 8.000000 | 4.800000 | 2.190890 |
| 20 | 0.5000 | 10.000000 | 5.000000 | 2.236068 |
| 20 | 0.7500 | 15.000000 | 3.750000 | 1.936492 |
| 20 | 0.9000 | 18.000000 | 1.800000 | 1.341641 |
Results
Enter inputs and press Compute Mean to view results.
How to Use This Calculator
- Specify n, the number of independent Bernoulli trials.
- Specify p, the success probability per trial (between 0 and 1).
- Click Compute Mean to calculate μ = n·p and related quantities.
- Use the CSV or PDF buttons to export the results table.
- Consult the example table for reference values and sanity checks.
FAQs
The mean μ = n·p is the expected number of successes across n trials with success probability p per trial.
Yes. For the mean, only n and p are needed. Variance and standard deviation are derived from them.
n must be an integer ≥ 0. p must lie in the closed interval [0, 1].
When np(1−p)=0 the variance is zero, making the denominator of the skewness formula zero; skewness is undefined in that case.
The mean is n·p. A mode is any integer ⌊(n+1)p⌋ where the PMF attains its maximum. They coincide only for special parameter values.
Often yes. In practice, p is estimated from data as the probability of success for an individual trial under similar conditions.
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