Normal Distribution
The normal distribution is a continuous probability distribution with a symmetric, bell-shaped probability density function. It is completely characterized by its mean μ and standard deviation σ. The mean determines the center of the distribution, while the standard deviation determines its spread. Its probability density function is f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²)). Probabilities are obtained from the area under the density curve, so P(X = x) = 0 for any single exact value and interval probabilities are calculated from areas between limits. The standard normal distribution has μ = 0 and σ = 1. Any normally distributed variable can be standardized using Z = (X - μ)/σ, allowing standard normal probability tables or the standard normal cumulative distribution function to be used. For a standard normal variable, approximately 68.27% of observations lie within ±1σ, 95.45% within ±2σ, and 99.73% within ±3σ of the mean. The Central Limit Theorem states that, under appropriate conditions, the sampling distribution of the sample mean approaches a normal distribution as sample size increases, even when the original population is not normal. Normal models are widely used for measurement variation, manufacturing dimensions, errors, and statistical quality control, but they may be inappropriate for strongly skewed, bounded, or heavy-tailed data.
Knowledge Check
1.What determines the center and spread of a normal distribution?
2.Scores on a test are normally distributed with mean 70 and standard deviation 10. What is the Z-score for a student who scored 85?
3.In a standard normal distribution, what is P(Z < 1.96)?
4.According to the empirical rule, approximately what percentage of observations lie within ±2 standard deviations of the mean?
5.A population has mean 100 and standard deviation 15. What is P(X > 130)? Assume P(Z > 2) = 0.0228.
6.A component diameter is normally distributed with μ = 20 mm and σ = 0.10 mm. What is the Z-score for a diameter of 19.8 mm?
7.If Z = -1.5, what is P(Z > -1.5)? Assume P(Z < -1.5) = 0.0668.
8.A product's weight is normally distributed with mean 50 g and standard deviation 2 g. What weight corresponds approximately to the 84th percentile? Use Z ≈ 1.0.
9.What is the standard deviation of the standard normal distribution?
10.Why is the probability at exactly one value zero for a continuous normal random variable?
11.In a normal distribution, what is the relationship between the mean, median, and mode?
12.What happens to a normal distribution when its standard deviation increases while the mean remains unchanged?
13.What is the main purpose of standardizing a normally distributed variable?
14.What does the Central Limit Theorem describe?
15.A manufacturing process has a normally distributed dimension with μ = 10 mm and σ = 0.05 mm. Approximately what percentage of parts fall between 9.90 mm and 10.10 mm?