4 - Normal Model and Measurement Error

Flashcards for reviewing lecture notes on the Normal Model and Measurement Error.

Normal Curve

A statistical model that approximates many natural phenomena and can model data resulting from a large number of independent variables.

Probability Density Function (PDF)

A special function which describes the chance associated with a continuous variable X, for all its possible values x, characterized by population mean μ and population SD σ.

Notation for Normal Distribution

If a variable X is Normally distributed with mean μ and standard deviation σ, it is expressed as X ∼ N(μ, σ^2), where σ^2 is the variance.

Standard Normal Curve

A normal curve with a mean of 0 and a standard deviation of 1, denoted as Z ∼ N(0, 1).

General Normal Curve

A normal curve with any mean and any standard deviation, denoted as X ∼ N(μ, σ^2).

Cumulative Distribution Function (CDF)

F(x) = P(X < x) is the area under the Normal curve from −∞ up to a point x, representing the chance of those particular values of X occurring.

68%-95%-99.7% Rule

Approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

Standard Units (Z-score)

For any point x on X ∼ N(μ, σ^2), the z-score is z = (x−μ) / σ, indicating how many standard deviations the point is above or below the mean

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Measurement Error

The difference between an individual measurement and the exact value, composed of chance error and bias: Individual measurement = exact value + chance error + bias.

Chance Error

Random, unpredictable variations in measurements, best estimated by replicating the measurement under the same conditions and calculating the standard deviation (SD).

Bias (Systematic Error)

A constant amount added to or subtracted from each measurement, which can be deliberate or accidental, and cannot be estimated by replicating measurements.

Calibration

Comparing the measurement of an instrument to a working standard with a known measurement to reduce measurement errors when selling by weight.