Normal Distribution
From Raw Score to Probability – The Full Process
🎯 The Full Process
Raw score (x)
Z-score (Z)
Probability/percentage
Step 1: Compute the Z-score: \(Z = \frac{x - \bar{x}}{S}\)
Step 2: Look up the probability in the Z-table
Step 3: Adjust if needed (complement, subtract areas)
✏️ Example 1: P(X < a) – less than a value
Question: Test scores are normally distributed with mean 75 and SD 10.
What is the probability that a student scores below 85?
Step 1 – Computing Z:
\(Z = \frac{85 - 75}{10} = \frac{10}{10} = 1\)
Step 2 – Reading the table:
P(Z ≤ 1) = 0.8413
Answer: 84.13% of students score below 85
✏️ Example 2: P(X > a) – greater than a value
Question: Women's height is normally distributed with mean 165 cm and SD 6 cm.
What is the probability that a woman is taller than 175 cm?
Step 1 – Computing Z:
\(Z = \frac{175 - 165}{6} = \frac{10}{6} \approx 1.67\)
Step 2 – Reading the table:
P(Z ≤ 1.67) = 0.9525
Step 3 – Complementary (because we ask "greater than"):
P(Z > 1.67) = 1 - 0.9525 = 0.0475
Answer: 4.75% of women are taller than 175 cm
✏️ Example 3: P(a < X < b) – between two values
Question: Baby weight is normally distributed with mean 3.2 kg and SD 0.5 kg.
What is the probability that a baby weighs between 2.8 and 3.5 kg?
Step 1 – Computing Z for both values:
\(Z_1 = \frac{2.8 - 3.2}{0.5} = \frac{-0.4}{0.5} = -0.8\)
\(Z_2 = \frac{3.5 - 3.2}{0.5} = \frac{0.3}{0.5} = 0.6\)
Step 2 – Reading the table:
P(Z ≤ -0.8) = 0.2119
P(Z ≤ 0.6) = 0.7257
Step 3 – Subtracting areas:
P(-0.8 < Z < 0.6) = 0.7257 - 0.2119 = 0.5138
Answer: 51.38% of babies weigh between 2.8 and 3.5 kg
✏️ Example 4: Symmetric area about the mean
Question: IQ scores are normally distributed with mean 100 and SD 15.
What is the probability that a person has an IQ between 85 and 115?
💡 Note: The range is symmetric about the mean! (100±15)
Step 1 – Computing Z:
\(Z_1 = \frac{85 - 100}{15} = -1\)
\(Z_2 = \frac{115 - 100}{15} = 1\)
Step 2 – Reading the table:
P(Z ≤ 1) = 0.8413
P(Z ≤ -1) = 0.1587
Step 3 – Subtract:
P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826
Answer: approx. 68% of people have an IQ between 85 and 115
(This is the 68-95-99.7 rule!)
📋 Summary Table – Question Types
| Question type | Method |
|---|---|
| P(X < a) | Compute Z, read from table |
| P(X > a) | Compute Z, read from table, compute 1 − (table value) |
| P(a < X < b) | Compute two Z-scores, look up both, subtract |
📝 Summary
Raw score → Z-score → table → probability
"Greater than" = complementary percentage
"Between" = subtract areas