Z-Scores and the Normal Distribution

CHARACTERISTICS OF NORMAL DISTRIBUTIONS

  • Symmetrical with mirror-image sides.

  • Tails never touch the x-axis, extending to infinity.

  • Mean, median, and mode are equal (located at center).

  • Bell-shaped; shape depends on variability.

  • Spread is influenced by variation in the sample.

  • Two parameters: mean and variance.

NORMAL DISTRIBUTION DATA POINTS

  • 68% of data falls within 1 standard deviation (SD) of the mean.

  • 95% within 2 SD.

  • 99% within 3 SD.

  • Calculated using area under the curve.

STANDARDISED NORMAL DISTRIBUTION

  • Mean = 0, SD = 1.

  • Standardized scores are called z-scores.

AREA UNDER THE CURVE (AUC)

  • 68% AUC is between +/- 1 SD from mean (0).

  • 95% AUC is between +/- 2 SDs from mean (0).

  • 99.7% AUC is between +/- 3 SDs from mean (0).

Z-SCORES OR STANDARD SCORES

  • Enables comparison of scores on different variables.

  • Indicates how many SDs a score is from the mean.

IMPORTANCE OF Z-SCORES

  • Calculates the probability of a score within a standard normal distribution.

  • Facilitates comparison of scores from different samples.

CALCULATION OF Z-SCORE

  • Formula: z = \frac{(X - M)}{SD}

EXAMPLE OF Z-SCORE

  • If X = 76 , M = 70 , and SD = 3 :

    • z = \frac{(76 - 70)}{3} = 2 (2 SDs above mean).

SUMMARY OF EXCEL Z-SCORE CALCULATION

  • Procedures for calculating z-scores from exam results.

  • Information on mean, standard deviation, etc., can be summarized via Excel formulas and results.

STANDARDISED NORMAL DISTRIBUTION SUMMARY

  • Similar to normal distribution; allows for score comparisons using z-scores.