Rest of ch 6 lecture (previous recording is also ch 6)

Standardized Distributions and Calculating Probabilities

  • Standardized distributions are crucial for calculating proportions and probabilities within different regions of a distribution curve.

  • The area under the region of the curve indicates the probability of corresponding scores falling within that region.

Standard Normal Distribution

  • When discussing standardized distributions, the focus is often on the normal distribution which has:

    • A mean of 0

    • A standard deviation of 1

  • To facilitate probability calculations, scores are transformed into z-scores using a formula that standardizes these values.

Areas Under the Curve in Standard Normal Distribution

  • Understanding the area under the curve helps in evaluating probabilities:

    • 34.13% of the distribution falls between the mean and 1 standard deviation above the mean.

    • 13.59% falls between 1 and 2 standard deviations above the mean.

    • 2.28% falls above 2 standard deviations above the mean.

    • The distribution is symmetrical, allowing reverse application of these percentages for scores below the mean.

Important Properties of Normal Distribution

  • A normal distribution allows for applying calculus principles to find areas under the curve, albeit not directly in class.

  • The normal distribution can beautifully summarize any distribution into a standard format using z-scores:

    • Approximately 68% fall within 1 standard deviation of the mean.

    • About 95% fall within 2 standard deviations.

    • More than 99% fall within 3 standard deviations.

  • Those scoring more than 3 standard deviations from the mean are considered rare (less than 1% of the distribution).

Z-Scores and Their Calculation

  • Z-scores enable the transformation of raw scores in a distribution to standardized scores, which leads to easier interpretation of where other scores lie in terms of the mean and standard deviation.

  • The z-score formula is used broadly, which allows for the solving of different questions related to calculating proportions or probabilities based on transformed data.

Real-World Application of Z-Scores

  • In real-world application, IQ scores are often standardized to have a mean of 100 and a standard deviation of 15.

    • Example: A score of 100 represents the average; z-scores help to identify how far a particular IQ score deviates from this average.

Proportions and Population

  • Calculating proportions can be approached in several steps:

    1. Transform the raw score (x) to a z-score.

    2. Use the unit normal table to obtain the probability associated with that z-score.

  • Understand the terminology:

    • Body of Distribution refers to the larger slice when the distribution is split.

    • Tail of Distribution refers to the smaller slice that extends towards the extremes of the distribution.

Utilizing Statistical Tables

  • The unit normal table is essential for identifying probabilities corresponding to z-scores:

    • It lists z-scores and provides the proportion of scores that lie below them, helping to visualize the body and tail of the distribution.

  • Clarity of where a z-score lies (body vs. tail) is crucial for proper interpretation of the table.

  • When dealing with the tails, both positive and negative z-scores equally reflect their respective proportion due to the symmetry of the normal distribution.

Miscellaneous

  • Be mindful that in different regions or countries (e.g., Canada), terminology may differ (e.g., 'zed scores').

  • Practice with unit normal tables will be essential, particularly for exam preparation.

  • Engagement in practical applications through classroom exercises and homework will reinforce understanding.