Normal Distribution

Normal Distribution

• A continuous probability distribution

• characterized by a symmetric, bell-shaped curve

• centered around the mean μ,

• defined by two parameters: the mean μ and standard deviation σ

Probability Density Function (PDF)

• f(x) = (1/(σ((2π)^1/2))e^(-(1/2)((x-μ)/σ)²)

• Describes how probabilities are distributed across continuous values of x

• Used in manufacturing to model the distribution of component dimensions around a target mean.

Mean (μ)

• The central value or “expected value” of the distribution

• It determines the location of the peak

• Represents the average battery life of average measurement of the production process.

Standard Deviation (σ)

• A measure of spread or dispersion

• Larger σ indicates data points are more spread out around the mean

• A small σ in voltage output means the consistent performance across devices.

Variance (σ^2)

• The square of the standard deviation.

• Represents the average squared deviation from the mean.

• Used to quantify measurement noise in electronic sensors

Shape Characteristics

• The curve is symmetric about μ

• Mean = Median = Mode

• Approximately 68% of values lie within 1σ

• Approximately 95% of values lie within 2σ

• Approximately 99.7% of values lie within 3σ

• Quality Control Engineers use these bounds to identify defective products

Standard Normal Distribution

• A special case of normal distribution with μ = 0 and σ = 1

• Φ(z) = (1/((2π)^1/2))e^-(z²/2)

• Used when converting any normal variable to its z-score for statistical tables

Z-Score (Standard Score)

• z = (x - μ)/σ

• Represents how many standard deviations a data point is from the mean

• In circuit testing, a resistor with z = 2 lies two standard deviations above the mean resistance

Cumulative Distribution Function (CDF)

• P(X ≤ x) = Φ(z) = ∫(from -∞ to z) (1/((2π)^1/2))e^-t²/2 dt

• Gives the probability that a variable is less than or equal to a certain value.

• Used yo find the probability that voltage fluctuation stays below a threshold.

Complementary Probability

• P(X > x) = 1 - Φ(z)

• In reliability testing, gives the probability that a system will last beyond a certain time.

Empirical Rule (68-95-99.7 Rule)

• 68% of the data lies within ± 1 σ

• 95% of the data lies within ± 2 σ

• 99.7% of the data lies within ± 3 σ

• Helps quality inspectors quickly estimate the defect rates in production batches

Linear transformation of Normal Variables

• If X ~ N(μ, σ²)

• then Y = aX + b ~ N(aμ + b, a²σ²)

• Used when converting sensor outputs or scaling measurement units (e.g. Celsius to Fahrenheit)

Sum of Independent Normals

• If X_1 ~ N(μ, σ²) and X_2 ~ N(μ_2, ((σ_2)²))

• Then X_1 + X_2 ~ N(μ_1 + μ_2, (σ_1)² + (σ_2)²)

• Total error in combined measurements follows a normal distribution

Normal Approximation to Binomial Distribution

• For large n and probability p not too close to 0 or 1

• X ~ Binomial(n, p) = N(np, np(1 - p))

• Used to approximate defect counts in large-scale manufacturing

Continuity Correction

• When using a normal approximation for discrete data

• Adjust by ± 0.5

• P(X ≤ k) = P(Y ≤ k + 0.5)

• Applied when estimating the probability of getting fewer than k defective chips in a large batch.

Standardization Process

• Converting any normal variable to a standard normal variable by

• Z = (X - μ)/σ

• Enables the use of z-tables to calculate probabilities in quality or reliability testing.

Inverse CDF (Percentile Function)

• The value of x such that P(X ≤ x) = p

• Denoted x = Φ^-1 (p)

• Used to find critical values (e.g., 95th percentile tolerance level for mechanical part sizes)

Engineering Measurement Example

• If voltage readings follow N(12.0, 0.04),

• Then P(V > 12.1) = 1 - Φ(2.5) = 0.0062

• Less than 1% of reading exceed 12.1 V, showing high stability

Communication System Example

• Noise amplitude in many channels follows a normal distribution centered around zero.

• Used in signal processing to determine error probabilities in Gaussian noise environments.

Manufacturing Process Example

• Product diameters often follow a normal distribution due to small random variations.

• Engineers compute P(Diameter within spec) using z-scores to ensure tolerance compliance

Biomedical Example

• Human biological variables (e.g., heart rate, blood pressure) are approximately normal across populations

• Used to determine “normal ranges” for medical testing and diagnosis.

Central Limit Theorem Connection

• The normal distribution arises as the limit of the sum (or mean) of any independent random variables, regardless of their individual distributions

• Underlies why measurement errors, environmental variations, and sensor outputs tend to be normally distributed.

Area Under the Normal Curve

• The total area under the curve equals 1, representing the total probability

• Probability of all possible sensor readings equals 1; ensuring model completeness.

Skewness and Kurtosis in Normal Distribution

• Skewness = 0

• Kurtosis = 3 (meskurtic)

• Used in process control to test whether data deviates from normality