Notes on Normal Distribution

Example Calculations

• To calculate probabilities in a normal distribution, we follow a structured approach:

  • Suppose we want to find (P(Z ≤ 1.53)). This requires us to refer to the standard normal table, which provides cumulative probabilities for various Z-values.

  • If we need to compute (P(Z > 1.53)), we use the complementary rule:

    • (P(Z > 1.53) = 1 - P(Z ≤ 1.53)) resulting in approximately 0.0630. This means that there's a 6.30% chance of a random variable being greater than 1.53.

Probability of Intervals

• For finding probabilities across an interval within a normal distribution, we can utilize the following formula:

  • (P(a ≤ Z ≤ b) = P(Z ≤ b) - P(Z ≤ a) = Φ(b) - Φ(a)). This calculates the probability that the Z-value falls between two points, a and b on the Z-distribution.

Distribution N(µ, σ²)

• In the context of a normal distribution characterized by mean (µ) and variance (σ²), the process of standardization is essential for comparison purposes:

  • The transformation to the standard normal distribution is managed by the formula: (Z = \frac{X - µ}{σ}). This standardizes the value X to its Z-score.

  • In this case, the random variable (X) follows a normal distribution represented as (X ∼ N(µ, σ²)). The key attributes are:

    • Mean = µ; Variance = σ²; Standard Deviation = σ.

Practical Applications

• Example: One practical application of these calculations is assessing the probabilities of real-world data, such as standardized test scores or heights in a population.

  • For instance, consider the SAT scores, which typically have a Mean = 500 and a Standard Deviation = 100:

    • To find the probability of scores falling between 550 and 650, we standardize and reference the z-table:

      • P(550 < X < 650) can be calculated as P(0.5 < Z < 1.5). This computation results in an approximate probability of 0.2417, signifying that around 24.17% of students achieve scores in this range.

Percentiles

• Percentiles are crucial in statistics as they indicate the relative standing of a value within a dataset. The k-th percentile of a normal distribution is determined at:

  • (P(X ≤ x_k) = k
    ange{100}). For example, the 90th percentile represents a score that surpasses 90% of the data points in the distribution.

  • To find specific percentiles, one typically employs the inverse of the cumulative distribution function (CDF) associated with the normal distribution.

Example: Top 3% Criterion

• To illustrate a practical application of percentiles, consider the requirement to be in the top 3% of SAT scores:

  • We need to first find the Z-value that corresponds to the 97th percentile, which typically is Z = 1.88. This value indicates that only 3% of the overall statistical population scores higher.

  • To convert this Z-value back to an actual SAT score, we employ the transformation:

    • (X = 500 + 100(1.88)). This calculation indicates that a score of 688 is necessary to be positioned in the top 3 % of test-takers.