GO

Normal Distribution and Z Score

Normal Distribution

  • Source context: Biostatistics for Evidence Based Practice (NURS 8004), Module 2, Session 1. Slides cover basics of the Normal Distribution, forms of kurtosis, and Z scores with a worked example.

I

Bell Curve we mean,median, and mode are all Equal at the peak.

  • Key ideas:

    • Normal distribution characteristics depend on two parameters:

    • μ (mu) = population mean

    • σ (sigma) = population standard deviation

    • The distribution is bell-shaped and symmetric around μ.

    • Different σ values change the spread of the curve while μ shifts its center.

    • Examples shown for illustration:

    • Curve with μ = 52, σ = 6

    • Curve with μ = 52, σ = 12

    • Two reference axis marks observed around μ = 52 in the example plots, illustrating how increasing σ widens the curve.

  • Relevance:

    • The Normal Distribution serves as a baseline model for many biological and health-related measurements.

    • Standardization (converting to Z scores) allows comparison across scales and samples.

Forms of Kurtosis (General Forms)

  • Kurtosis describes the "peakedness" of a distribution relative to a Normal distribution.

    • Leptokurtic: positive kurtosis; distribution is more peaked than normal.

    • Mesokurtic: kurtosis similar to normal (often considered the baseline, i.e., normal distribution has kurtosis close to 0 in excess kurtosis terminology).

    • Platykurtic: negative kurtosis; distribution is flatter than normal.

  • Notation and signs:

    • Leptokurtic (+)

    • Mesokurtic (0)

    • Platykurtic (-)

These terms describe the shape of the distribution in relation to the normal distribution, with leptokurtic indicating a sharper peak and heavier tails, mesokurtic being similar to a standard normal distribution, and platykurtic showing a flatter peak and lighter tails.

  • The slides imply a visualization using Curve 1 and Curve 2 to contrast shapes, and labeled Regions around a Normal Distribution.

  • ‘ Practical implications:

    • Kurtosis affects tail heaviness and peak height, which influences statistical inferences, outlier behavior, and performance of tests that assume normality.

    • In practice, non-normal kurtosis can indicate the need for data transformation or non-parametric methods.

Z Score

  • Purpose:

    • Used to compare an individual observation to the population mean by standardizing the difference relative to the standard deviation.

  • Definition:

    • The Z score for an observation x is given by:

    • Z = rac{x - bc}{\sigma}

    • Here, \u03bc is the population mean and \sigma is the population standard deviation.

  • Population vs sample language in slides:

    • Z = (x – μ) / σ (used to compare a single score to the population distribution)

  • Interpretation:

    • A higher Z score means the observation is several standard deviations above the mean; a negative Z score means below the mean.

  • Related concept:

    • The Standard Normal Distribution is the special case where μ = 0 and σ = 1, i.e., Z follows a standard normal distribution.

Worked Example: Z Score Calculation

  • Given:

    • A student scores x = 95 on a Module 2 statistics quiz.

    • The average (population mean) is μ = 80.

    • The standard deviation is σ = 5.

  • Calculation steps:

    • Numerator: 95 − 80 = 15

    • Z score: Z = \frac{95 - 80}{5} = \frac{15}{5} = 3

  • Result:

    • The student’s Z score is Z = 3, i.e., 3 standard deviations above the mean.

  • Interpretation:

    • A Z score of 3 indicates the score is unusually high relative to the distribution of quiz scores.

  • Expanded context (not in transcript but standard interpretation):

    • In a standard normal distribution, a Z score of 3 corresponds to a percentile around the 99.7th percentile (depending on exact tables/approximation).