Normal Probability Calculator for Sampling Distributions

Calculate probabilities for sample means using the central limit theorem and sampling distributions

Population and Sample Parameters

Population Mean (μ)

True population mean

Population Std (σ)

Population standard deviation

Sample Size (n)

Number of observations in sample

What probability do you want?

X̄ < XX̄ > XX₁ < X̄ < X₂

Lower Bound (X₁)

Upper Bound (X₂)

Sampling Distribution Results

49.5015%

P(98 < X̄ < 102)

3.0000

Standard Error (σ/√n)

-0.6667

Z-Score (Lower)

0.6667

Z-Score (Upper)

Key Formula

Standard Error: σ_X̄ = σ / √n = 15 / √25 = 3.0000

Z-Score: z = (X̄ - μ) / (σ/√n) = (X̄ - 100) / 3.0000

Confidence Intervals for Sample Mean

90% CI: [95.06, 104.94]±4.94

95% CI: [94.12, 105.88]±5.88

99% CI: [92.27, 107.73]±7.73

Example: American Women's Height

Problem Setup

Population: American women aged 20+

Population mean height (μ): 161.3 cm

Population standard deviation (σ): 7.1 cm

Sample size (n): 7 women

Question: What's the probability that the average height falls below 160 cm?

Solution

1. Standard Error: σ_X̄ = 7.1 / √7 = 2.684

2. Z-Score: z = (160 - 161.3) / 2.684 = -0.484

3. P(X̄ < 160) = P(Z < -0.484) = 0.314

Answer: 31.4% probability

Central Limit Theorem

•

Sample means approach normal distribution as n increases

•

Mean of sampling distribution equals population mean

•

Standard error decreases with larger samples

•

Works for any population distribution (n ≥ 30)

Key Formulas

Standard Error

σ_X̄ = σ / √n

Z-Score for Sample Mean

z = (X̄ - μ) / (σ/√n)

Sampling Distribution

X̄ ~ N(μ, σ²/n)

Understanding Sampling Distributions

What is a Sampling Distribution?

A sampling distribution is the probability distribution of a sample statistic (like the sample mean) obtained from all possible samples of a fixed size from a population. It tells us how sample means vary across different samples.

Key Properties

•Mean equals population mean (μ)
•Standard deviation = σ/√n (standard error)
•Shape approaches normal for large n

When to Use This Calculator

Quality Control

Monitor if sample means fall within expected ranges

Research Studies

Calculate probability of observing certain sample means

Hypothesis Testing

Determine if observed means are statistically significant

Sample Size Guidelines

• n ≥ 30: CLT applies for any population distribution
• n < 30: Population should be approximately normal
• Larger n: More precise estimates (smaller standard error)

Related Statistics Calculators

Normal Distribution Calculator

Calculate normal distribution probabilities

Central Limit Theorem Calculator

Demonstrate central limit theorem principles

Standard Error Calculator

Calculate standard error of the mean