Central Limit Theorem

Central Limit Theorem (CLT)

• The theorem stating that the distribution of sample means

• approaches a normal distribution as the sample size increases

• regardless of the shape of population distribution

• provided the samples are independent and identically distributed.

Importance of CLT

• Enables the use of normal probability models for analyzing sample means and proportions

• even when the population distribution is not normal

• In quality testing, average defect rates can be analyzed using z-scores

• even if individual product data is skewed.

Sampling Distribution of the Mean

• The probability distribution of the means of all possible random samples of a given size n from a population.

• X̄ ~ N (μ, σ/((n)^1/2))

• Used in manufacturing to determine how consistent sample averages are across batches

Samples must be random and independent

First condition for CLT

Sample size n must be sufficiently large (usually n ≥ 30)

Second condition for CLT

The population should have a finite mean μ and variance σ^2

Third condition for CLT

analyzing average sensor readings or testing the strength of materials.

CLT is required when…

Mean of Sampling Distribution

• The mean of Sampling Distribution equals the Population mean.

• E(X̄) = μ

• If an average battery life is 10 hours, the average of all sample means will also center around 10.

Standard Deviation of Sampling Distribution (Standard Error)

• σ_X̄ = σ/(n^1/2)

• Represents how much sample means vary from the population mean.

• Engineers use standard error to estimate how much variability in product quality is due to sampling, not process flaws.

Standard Error v/s Standard Deviation

• The standard deviation measures variability of individual observations

• The standard error measures the variability of sample means

• Smaller standard error indicates more reliable average measurements in repeated experiments.

Approximation to Normality

• As n → ∞

• The sampling distribution of X̄

• Approaches a normal distribution

• Even if the population distribution is skewed or irregular

• Used when approximating waiting times or

• Voltage readings that are not originally normal

Standardization of Sample Mean

• z = (X̄ - μ)/(σ/(n^1/2))

• Allows using the standard normal table to calculate probabilities related to sample means

• Used to compute the probability that the average of 10 machine measurment exceeds a specified limit.

Law of Large Numbers (LLN) v/s CLT

• LLN states that as n increases, the sample mean approaches the population mean

• CLT describes the distribution of sample means and their convergence to normality.

• LLN ensures accuracy in estimates

• CLT allows probability predictions about averages.

Practical Rule for Non-Normal Populations

• If the population is strongly skewed

• a sample size of at least 30 is usually sufficient for CLT to hold.

• In financial return modeling,

• 30-day average returns often approximate a normal curve even if daily returns are skewed.

Application in Quality Control

• The CLT allows quality engineers to monitor the average output of a process using control charts based on normal distributions.

• Used to determine if the mean weight of packaged goods deviates from the standard target.

Application in Engineering Measurements

• CLT justifies assuming normality when analyzing sensor data voltage fluctuations, or repeated resistance measurements.

• If each sensor has a random error, the average of 50 sensor yields a near-normal error distribution.

Application in Biomedical Sciences

• Biological measurements, such as blood pressure or enzyme activity,

• tend to follow near-normal sampling distributions due to CLT

• Researchers can use z-tests even if the underlying data are skewed.

Finite Population Correction

• When sampling without replacement from a finite population of size N:

• σ_X̄ = (σ/(n^1/2))(((N - n)/(N - 1))^1/2)

• Applies when analyzing quality from a limited production batch rather than an infinite process.

Confidence in Estimation

• CLT enables constructing intervals for population means and proportions using normal-based z-values

• Used in reliability studies to report the average lifespan of components with a 95% confidence interval.

Relationship to Population Shape

• The more skewed the population,

• the larger the sample size is

• required for the sampling distribution to appear normal.

• A population of component lifetimes (right-skewed) may need

• n > 50 for CLT to apply accurately

Battery Life Testing

• Even if individual battery lifespans are skewed,

• the average life from many samples will approximate a normal distribution by CLT

• Allows calculating probabilities of average performance within warranty limits

Signal Processing

• Random noise from multiple independent sources tends to follow a normal distribution because of the CLT

• Forms the foundation of Gaussian noise models used in communication systems.

Manufacturing Variation

• The sum of many small independent errors

• e.g. machining, material, environmental

• creates a normal distribution of product measurements

• Used in Six Sigma analysis and process capability studies.

CLT and Simulation

• Monte Carlo simulations rely on the CLT

• to approximate results of complex random processes with normal distributions

• Used in reliability engineering and risk analysis.

Visual Interpretation

• As n increases,

• the histogram of sample means becomes smoother and more symmetric resembling the bell curve

• Shown in experiments where random samples are repeatedly drawn from non-normal populations (e.g. uniform or exponential)