Comprehensive Guide to Normal Distribution: Z-scores and Proportions

The primary focus of this study guide is the transition between Z-scores and proportions, and vice versa using the unit normal table.
Interchangeable Terms: In the context of normal distributions, "proportion," "probability," and "shaded area" are used interchangeably.
Mandatory Visual Component: It is established that one must never attempt these problems without a drawing of the normal distribution. Drawing ensures the analyst knows exactly which portion of the curve is being calculated.
Probability Notation: Proportions can be written using probability notation, such as $P(z > -0.25)$ . This is read as the probability or proportion of Z-scores greater than a specific value.

Every diagram representing a statistical problem on the horizontal axis involves three critical pieces of information:
1. The sign of the Z-score: Whether the score is positive (to the right of the mean) or negative (to the left of the mean).
2. Direction of shading: Whether the area of interest is to the right (above) or to the left (below) of the Z-score.
3. The size of the proportion (Body vs. Tail): Whether the shaded area represents more than half of the distribution (Body) or less than half (Tail).
To solve any problem, you typically provide two of these components to the drawing to derive the third.

Column B (Body): Represents the larger portion of the normal distribution (shaded area $> 0.5000$ ).
Column C (Tail): Represents the smaller portion of the normal distribution (shaded area $< 0.5000$ ).
The Dotted Line Balance: The center of the distribution (mean) always acts as the 50% marker. If you shade from a negative Z-score to the right, you are shading the entire right half of the distribution ( $50\%$ ) plus an additional segment, which automatically identifies the area as the "Body."
Symmetry of the Table: The unit normal table only lists positive Z-scores. Because the distribution is symmetrical, a Z-score of $0.25$ has the same body and tail proportions as a Z-score of $-0.25$ . Only the directional orientation (right vs. left) changes.
Universal Rule: Every score in the distribution must fall either to the right or to the left of a specific point. Therefore: $\text{Body Proportion} + \text{Tail Proportion} = 1.00$ (or $100\%$ ).

Identifying the correct direction to shade is essential for determining Body vs. Tail status:
- Shade to the Right (Greater Than): Clues include words like "above," "greater than," "top," or the "more than" symbol ( $>$ ).
- Shade to the Left (Less Than): Clues include words like "below," "less than," "bottom," or the "less than" symbol ( $<$ ).

Problem: What proportion of scores are above $z = -0.25$ ?
Step 1: Locate Z: Since the Z-score is negative, place it to the left of the mean. Because it is small ( $0.25$ ), it is closer to the center than to one standard deviation away.
Step 2: Shade: The word "above" indicates shading to the right.
Step 3: Define Area: The shading covers the entire right half and a portion of the left half. This is the Body.
Step 4: Table Lookup: Look up the positive value $z = 0.25$ in Column B.
Result: The proportion is $0.5987$ (or $59.87\%$ ).
Validation: Does a value of nearly $60\%$ make sense for a shaded area that is "half plus a little change"? Yes. If you needed the area below, you would look at the tail ( $0.4013$ or $40.13\%$ ).

In a continuous distribution, the probability of selecting an exact score is effectively zero (e.g., exactly $-0.250000...$ with infinite precision).
Because of this, the calculation focus is always on intervals (at, above, or below a certain point) rather than precise singular point values.

Word problems often provide raw scores ( $X$ ), which must be converted to Z-scores before using the table.
Requirement: Use the mean ( $\mu$ ) and standard deviation ( $\sigma$ ) to perform the conversion using the formula: $z = \frac{X - \mu}{\sigma}$ .

Calculation: $z = \frac{105 - 100}{10} = \frac{5}{10} = 0.5$ .
Visualizing: Locate $z = 0.5$ (positive side). Shade "above" (to the right).
Body/Tail Identification: The shaded area is less than half, making it a Tail.
Lookup: Table Column C for $z = 0.5$ is approximately $0.3085$ .
Reasonableness Check: The result of roughly $30\%$ to $40\%$ matches a visual estimate for half a standard deviation above the mean.

Calculation: $z = \frac{45 - 50}{5} = \frac{-5}{5} = -1.0$ .
Visualizing: Locate $z = -1.0$ (left side). If the problem asks for scores "less than" 45, shade to the left.
Body/Tail Identification: This is a small area, identifying it as a Tail.
Lookup: Table Column C for $z = 1.0$ is $0.1587$ or $15.87\%$

Numerical Overlap: Z-scores and proportions can look similar because both can range between $0$ and $1$ (e.g., $z = 0.5$ and $p = 0.5$ ).
Differences:
- Proportions: Only exist between $0$ and $1$ .
- Z-scores: Can be negative, zero, or positive, and frequently exceed the range of $1$ (e.g., $-3, 2, 1.5$ ).
Critical Warning: Students often confuse these two values. A Z-score of $0.5$ does not mean $50\%$ . It means half of one standard deviation above the mean. Always label values clearly as "z" or "p" in notes and drawings to avoid calculation errors.