Normal Distribution Expected Value Calculator

Expected Value Calculator - Normal Distribution (from Sample Data)

Calculate the expected value E(X) = μ by estimating it from sample data. Provide your dataset and the calculator will compute the sample mean as an estimate of the population mean, assuming the data comes from a normal distribution.

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Key Insight: From 10 data points, the estimated E(X) = 100.70. This is your sample mean x̄, which is the best estimate of the population mean μ. We are 95% confident the true μ is between 99.00 and 102.40.

Sample Data

Enter your data (comma, space, or newline separated)

✓ 10 valid data points

Sample size (n):10

Formula:E(X) = μ ≈ x̄ = Σx / n

Range:[97.00, 105.00]

Sample Statistics

Min:97.00

Q1:98.00

Median:100.50

Mean (E(X)):100.70

Q3:103.00

Max:105.00

Estimated Expected Value

E(X) ≈ 100.70

x̄ = Σx / n = 100.70

Sample mean from 10 observations

Estimated Normal Distribution with E(X) = 100.70

Density

x̄ = 100.70

95.2100.7106.2

Note: Bell curve shows estimated normal distribution N(μ=100.70, σ=2.75). Red line shows sample mean (E(X) estimate). Green shaded region shows ±1σ (≈68% of data). Based on 10 sample observations.

📊Sample of 10 observations

Sample mean (E(X) estimate): 100.70
95% confident true mean is in [99.00, 102.40]
Most values fall within 97.9 to 103.5 (±1σ)

Understanding Expected Value Estimation for Normal Distribution

What Does E(X) Mean?

For a normal distribution, the expected value E(X) equals the population mean μ. When we do not know μ, we estimate it from sample data using the sample mean x̄. This x̄ is our best estimate of E(X).

Interpreting Your Result

With E(X) ≈ 100.70, this means:

The average of your 10 observations is 100.70
This estimates the center of the population distribution
Confidence interval quantifies estimation uncertainty

Real-World Examples

Test scores: Sample mean estimates average performance
Heights: Average height from sample estimates population mean
Manufacturing: Sample average part dimension estimates true mean
Response times: Mean time from sample estimates typical response

Why x̄ Estimates μ

The sample mean x̄ is an unbiased estimator of the population mean μ. This means that if we took many samples and computed x̄ for each, the average of all those x̄ values would equal μ. The Central Limit Theorem tells us x̄ follows a normal distribution centered at μ, allowing us to construct confidence intervals.

Understanding Expected Value

Formula and Calculation

Applications

Calculate Expected Value

Use the calculator below to compute the expected value with step-by-step solutions and detailed explanations.