5-2 Z-Scores

Z-Scores and the Standard Normal Distribution

Overview of Z-Scores

  • Definition: A Z-score (or standard score) represents the distance between a raw score and the population mean in units of the standard deviation.

  • The Standard Normal Distribution: When all scores in a normal distribution are converted to z-scores, they form the standard normal distribution, which always has a mean (μ) of 0 and a standard deviation (σ) of 1.

  • Symmetry and Area: The normal distribution is perfectly symmetrical. The total area under the curve is exactly 1.00 (or 100%), meaning the area to the left of the mean is 0.50, and the area to the right is 0.50.

The Empirical Rule (68-95-99.7 Rule)

To understand the spread of data in a normal distribution relative to z-scores:

  • Approximately 68.26% of the data falls within 1 standard deviation of the mean (z=±1).

  • Approximately 95.44% of the data falls within 2 standard deviations of the mean (z=±2).

  • Approximately 99.74% of the data falls within 3 standard deviations of the mean (z=±3).

Z-Score Conversions

  • Converting raw scores to standard scores uses the formula: z=X−μσ\​

    • X: The individual raw score.

    • μ: The population mean.

    • σ: The population standard deviation.

  • Interpretation:

    • A positive z-score indicates the score is above the mean.

    • A negative z-score indicates the score is below the mean.

    • A z-score of 0 indicates the score is exactly equal to the mean.

Finding Areas and Proportions in the Z-Table

Most Z-tables provide three primary columns:

  1. Column A: The z-score itself (the distance from the mean).

  2. Column B (The Body): The area between the mean and the z-score.

  3. Column C (The Tail): The area from the z-score toward the extreme end of the distribution.

  • Finding Area Above a Positive Z-Score:

    • Look up the z-score in Column A and find the value in Column C.

  • Finding Area Below a Positive Z-Score:

    • Subtract the Column C tail area from 1.00: P(Z<z)=1−Column C.

  • Finding Area Between Two Z-Scores:

    • If both are on the same side of the mean: Subtract the smaller area from the larger area.

    • If they are on opposite sides: Add the areas between each z-score and the mean (Column B values).

  • Finding Area Below a Negative Z-Score:

    • Use the positive equivalent: Take the absolute value of your negative Z-score (e.g., if z = -1.65, use 1.65).

    • Look up the Z-score: Locate this positive value in Column A of the Z-table.

    • Locate the Tail area: Find the value in Column C (The Tail). This column specifically represents the area from the score toward the end of the distribution.

    • Final Result: The value from Column C for the positive Z-score is equal to the area below your negative Z-score.

  • Finding Area Above a Negative Z-Score

    • Take the absolute value of the negative Z-score to obtain its positive counterpart.

    • Locate this positive value in Column A of the Z-table and identify the corresponding tail area in Column C.

    • Calculate the area above the original negative Z-score using the formula: 1.00−Column C

Calculating Specific Proportions:

Example 1: Area Beyond z=1.65

  • A z-score of 1.65 corresponds to a tail area (Column C) of 0.0495.

  • This means approximately 4.95% of the population exceeds this value.

Example 2: Area Above a Negative Z-score (z=−1.65)

  • Because of symmetry, the area above −1.65 is the same as the area below +1.65+1.65.

  • 1−0.0495=0.9505. Thus, 95.05% of scores are above −1.65.

Determining Raw Scores from Z-Scores

  • To find an unknown raw score (X) when you have the z-score, use the algebraic rearrangement: X=μ+(z⋅σ)

Example: Top 5% Distribution Analysis

  • To find the cutoff for the top 5% (0.05):

    1. Find the proportion of 0.05in Column C of the Z-table.

    2. The closest value is 0.0495, which gives z = +1.65.

    3. If μ=80 and σ=5:
      X=80+(1.65⋅5)=80+8.25=88.25

Understanding Percentiles and Percentile Ranks

  • Percentile: The value below which a specific percentage of data falls.

  • 50th Percentile: The median of the distribution (z=0).

  • 80th Percentile: To find the raw score for the 80th percentile, look for an area of 0.80 below the score.

    • This means looking for 0.20 in the tail (Column C).

    • A tail area of 0.1977 correlates to z≈0.85.

    • If μ=90 and σ=10: X=90+(0.85⋅10)=98.5.

Critical Z-Values in Inferential Statistics

There are specific z-scores commonly used in hypothesis testing and confidence intervals:

  • 90% Confidence:z=1.645

  • 95% Confidence:z=1.96

  • 99% Confidence:z=2.575

Summary of Step-by-Step Problem Solving

  1. Identify the Given Values: Note the X, μ, and σ.

  2. Sketch the Curve: Visually represent the area you are trying to find (above, below, or between).

  3. Calculate Z: Use the formula z=(X−μ)/σ.

  4. Consult Table: Find the proportion associated with the z-score.

  5. Final Calculation: Adjust the proportion based on whether you need the body, the tail, or the area between scores.