Understanding Normal Distribution and Z-Scores
Normal Distribution
Introduction to Normal Distribution
Shape: The normal distribution is bell-shaped.
Most of the area under the curve falls in the middle.
The tails of the distribution (the ends) approach the X-axis but do not touch it (asymptotic).
Symmetry: Bilateral symmetry is a key characteristic of the normal distribution.
When folded in half, both sides are mirror images.
Half of all scores lay on either side of the mean.
Total Area Under the Curve: The entirety of the curve equals 1.00 or 100%.
The Y-axis represents frequency, indicating that when discussing sections under the curve, we refer to proportions of total observations.
Assumptions of Normal Distribution
A large sample size is assumed, typically greater than 25 (N > 25).
Under normal distribution:
Roughly 66.7% of scores fall within one standard deviation of the mean.
A minority, about 5%, fall more than two standard deviations away from the mean.
Standard Deviations and Spread
The area under the curve remains equal to 1.00 or 100% regardless of its width.
The distribution may be narrow or wide based on the standard deviation (SD).
Changes in SD affect the height of the curve but not the area under the curve at specified distances from the mean.
Z-Scores
Definition and Purpose of Z-Scores
Z-Score: A standardized score that indicates how many standard deviations a specific score is above or below the mean of its distribution.
Z-scores provide the precise location of a score within a distribution.
They transform raw scores into relative scores.
Z-scores should be used for data that is interval or ratio in nature and follows a normal, symmetric distribution.
Z-Distribution Characteristics
The standard normal distribution is defined with a mean (μ) of 0 and a standard deviation (σ) of 1.
Illustrated with z-values ranging from -3 to +3 along the axis.
Applications of Z-Scores
Utility: Z-scores enable comparisons between scores from different distributions and are particularly useful when matching dissimilar metrics such as:
SAT scores vs. GPA (1600 vs. 4.0)
Blood pressure vs. heart rates (systolic vs. beats per minute)
Car dealer sales figures vs. performance ratings (thousands of dollars vs. rating scale).
Properties of Z-Scores
Z-scores possess both magnitude and direction:
Magnitude: Indicates the number of standard deviation units away from the mean (values rarely exceed |4|).
Direction: The sign (+ or -) indicates whether the z-score is above or below the mean; the mean itself has a z-score of 0.
Distribution Characteristics
Different areas under the normal curve correspond to different z-scores, maintaining the total area at 1.00.
Probabilities Related to Z-Scores:
Probability that the sample mean is within one standard deviation (σ) of the mean (m) is $P ext{( ext{within } 1 ext{ σ})} = 0.68$.
Probability is $P ext{( ext{within } 2 ext{ σ})} = 0.95$.
The area can be halved: $P=0.50$.
Converting Raw Scores to Z-Scores
Process of Conversion
Raw scores, which are collected data units, must be converted to z-scores for analysis utilizing the z-distribution:
Formula: Z = \frac{x - \mu}{\sigma} where:
x = raw score,
μ = mean,
σ = standard deviation.
Importance of the Z-Table: To find the proportion in the area under the curve:
Column B of the Z-table provides proportions between the mean and the selected z-score.
Column C provides the proportion of the area in the tail beyond the selected z-score.
It's noted that while the notation uses μ and σ for population parameters, in psychology and other sciences, z-scores are often calculated based on sample data due to the unavailability of entire populations.
Example Calculations
Example 1: Mary’s Z-Score Calculation
Standardized test administered to 85 psychology students.
Mean (μ) = 78, Standard Deviation (σ) = 8.64.
Mary scores x = 56.
Calculate z-score:
Z = \frac{56 - 78}{8.64} = \frac{-22}{8.64} = -2.55.
This z-score indicates Mary performed worse than her peers:
Based on the z-table for z = -2.55, Column C gives $p = 0.0054$, indicating that only 0.54% of participants performed worse than Mary.
Example 2: Comparing Bill and Ted’s Scores
Standardized test context remains the same (N = 85, μ = 78, σ = 8.64).
Bill's score: x1 = 74 →
Z_1 = \frac{74 - 78}{8.64} = -0.46.Ted's score: x2 = 88 →
Z_2 = \frac{88 - 78}{8.64} = 1.16.
Z-table proportions extracted:
$p = 0.1772$ for z = -0.46 (Column B).
$p = 0.3770$ for z = 1.16 (Column B).
Total proportion of students scoring between Bill and Ted:
0.1772 + 0.3770 = 0.5542 or approximately 55% of students.
Example 3: Jerry’s Performance Analysis
Jerry has a z-score of +1.44:
Comparison with Ted's z-score (+1.16)
Jerry scores higher than Ted.
Determine Jerry's percentile.
Using Column C, for z = 1.44, subtract from 1 for percentile:
If $p_{C} = 0.0749$, then percentile = $1.00 - 0.0749 = 0.9251$ or 93rd percentile.
Calculate Jerry's raw score.
x = \mu + Z \cdot \sigma = 78 + 1.44(8.64) = 90.44.
Determining percentage of scores between Ted and Jerry:
Ted’s z = 1.16 (Column B: $p = 0.3770$).
Jerry's z = 1.44 (Column B: $p = 0.4251$).
The difference gives the proportion:
0.4251 - 0.3770 = 0.0481 or 4.81% of scores fall between Ted and Jerry's performance.
Example 4: Test Score Comparisons
Comparison of Jake's GPA and SAT scores with Mike's:
Jake:
GPA = 2.86, Score = -2.95, SD = 0.32.
Z-score:
Z_J = \frac{-2.95 - 2.86}{0.32} = 0.09.Mike:
SAT = 1521, Score = 1632, SD = 128.
Z-score:
Z_M = \frac{1632 - 1521}{128} = 0.87.
The evaluation demonstrates that Mike performed better on their respective tests based on the calculated z-scores.