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Sum Distribution Calculator (ΣX)

Free sum distribution calculator. Sampling distribution of ΣX. E(ΣX)=nμ, SD(ΣX)=σ√n. P(ΣX ≤ s), P(ΣX

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Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

Σ
STATISTICSSampling Distributions

ΣX Distribution — Mean nμ, SD σ√n, Probabilities for Sums

Sampling distribution of the sum of n observations. E(ΣX)=nμ, SD(ΣX)=σ√n. P(ΣX ≤ s), P(ΣX ≥ s), P(a ≤ ΣX ≤ b). Step-by-step breakdown.

Normal Probability X̄ →CLT Calculator →

Real-World Scenarios — Click to Load

Input Mode

Probability Query

sum_distribution_results.sh
CALCULATED
$ sum_distribution --mu=2 --sigma=0.3 --n=50
E(ΣX) = nμ
100.0000
SD(ΣX) = σ√n
2.1213
Probability
99.0789%
Z-score
2.3570
Share:
Sum Distribution ΣX
E(ΣX) = nμ
100.00
SD(ΣX) = 2.1213P = 99.08%
numbervibe.com/calculators/statistics/sum-distribution-calculator

ΣX Distribution (Normal) — Shaded Probability Region

Individual X ~ N(μ, σ²)

Sum ΣX ~ N(nμ, nσ²)

SD(ΣX) = σ√n vs Sample Size

Calculation Breakdown

COMPUTATION
E(ΣX) = nμ
100.0000
n × μ = 50 × 2.0000
COMPUTATION
SD(ΣX) = σ√n
2.1213
σ × √n = 0.3000 × √50
Z-score
2.3570
Z = (s − nμ)/(σ√n) = (105 − 100.0000)/2.1213
P(ΣX ≤ s)
99.0789%
Φ(Z)

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • For iid X₁, X₂, ..., Xₙ: E(ΣX) = nμ, SD(ΣX) = σ√n, Var(ΣX) = nσ²
  • By CLT (large n): ΣX ~ N(nμ, nσ²) — the sum is approximately normal
  • Z = (ΣX − nμ) / (σ√n) — standardized sum for probability lookups
  • P(ΣX ≤ s) = Φ((s − nμ)/(σ√n)); P(ΣX ≥ s) = 1 − P(ΣX ≤ s)
  • P(a ≤ ΣX ≤ b) = Φ((b−nμ)/(σ√n)) − Φ((a−nμ)/(σ√n))
  • Use population parameters (μ, σ) or derive from a discrete P(x) table

Did You Know?

📦Total weight of n packages: if each has mean μ and SD σ, total has mean nμ and SD σ√n.Source: NIST
🎲Sum of 100 dice: mean 350, SD ≈ 17.1. P(340 ≤ sum ≤ 360) ≈ 45%.Source: Khan Academy
💰Total sales over n days: sum distribution enables revenue forecasting and risk assessment.Source: Finance
⏱️Total service time of n customers: used in queueing theory and capacity planning.Source: Operations
📐The sum distribution is the foundation of the Central Limit Theorem.Source: OpenIntro
🏭Quality control: total output of n machines follows sum distribution when machines are independent.Source: NIST

Expert Tips

From discrete table

If you have x and P(x), compute μ = Σx·P(x) and σ² = Σ(x−μ)²·P(x).

Small n

For small n, normal approximation may be poor. Use exact distribution if known (e.g., sum of dice).

Independence

Formulas assume iid. Correlated observations require different variance formulas.

Units

Keep units consistent: if μ is in kg, ΣX is in kg; σ√n has same units as σ.

Frequently Asked Questions

What is the sum distribution?

The sampling distribution of ΣX = X₁+...+Xₙ. For iid observations, E(ΣX)=nμ and SD(ΣX)=σ√n. By CLT, it is approximately normal for large n.

When can I use the normal approximation?

Rule of thumb: n≥30 for many populations. For skewed distributions, larger n may be needed.

How do I get μ and σ from a P(x) table?

μ = Σ x·P(x), σ² = Σ (x−μ)²·P(x). Ensure probabilities sum to 1.

What is the difference between sum and mean distribution?

Sum: mean nμ, SD σ√n. Mean X̄: mean μ, SD σ/√n. They are related: ΣX = n·X̄.

Can I use this for non-normal populations?

Yes. The CLT says the sum approaches normal regardless of population shape, for large n.

What if my probabilities do not sum to 1?

The discrete P(x) table must have ΣP(x)=1. Normalize by dividing each P(x) by the total before computing μ and σ.

How accurate is the normal approximation for small n?

For n<30, the approximation can be poor for skewed populations. Consider simulation or exact methods.

Relationship to sample mean?

ΣX = n·X̄. So E(ΣX)=nμ and SD(ΣX)=n·(σ/√n)=σ√n.

Formulas at a Glance

nμ
Mean of ΣX
σ√n
SD of ΣX
nσ²
Variance of ΣX
Z
(ΣX−nμ)/(σ√n)

Worked Example

Example: Packages have μ=2 kg, σ=0.3 kg. For n=50, find P(total weight ≤ 105 kg).

Step 1: E(ΣX) = nμ = 50 × 2 = 100 kg

Step 2: SD(ΣX) = σ√n = 0.3 × √50 ≈ 2.12 kg

Step 3: Z = (105 − 100) / 2.12 ≈ 2.36

Step 4: P(ΣX ≤ 105) = Φ(2.36) ≈ 0.991 (99.1%)

Disclaimer: This calculator uses the normal approximation (CLT) for the sum distribution. For small n or highly skewed populations, results may be approximate. For exact probabilities (e.g., sum of dice), consider convolution or simulation.

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