Normal Distribution – Z-Table | Grade 11 Math

Normal Distribution

Z-Table and Finding Areas

📋 What Is the Z-Table?

The Z-table links a Z-score to an area/probability.

💡 What does the table give?

For each Z-score, the table gives the area to the left of that score.

That is: the probability of a value ≤ the given Z-score.

Z P(Z) Green area = the table value

📌 Notation: \(P(Z \leq z) = \) the table value

📖 How to Read the Table?

✏️ Example: Find P(Z ≤ −1.53)

Step 1: Split −1.53 = −1.5 + 0.03

Step 2: Find −1.5 in the first column

Step 3: Find 0.03 in the top row

Step 4: The intersection cell is the answer

P(Z ≤ -1.53) = 0.0630

✏️ Example: Find P(Z ≤ −0.72)

-0.72 = -0.7 + 0.02

P(Z ≤ -0.72) = 0.2360

🔢 Converting Between Decimal and Percentage

Table value × 100 = percentage

💡 Examples:

  • 0.0630 → 6.30%
  • 0.2360 → 23.60%
  • 0.8413 → 84.13%
  • 0.5000 → 50%

📌 Note: All four terms below are identical:

Area = percentage = probability = chance

⭐ Important Values to Remember

Z-score Table value Percentage
Z = 0 0.5000 50%
Z = 1 0.8413 84.13%
Z = -1 0.1587 15.87%
Z = 2 0.9772 97.72%
Z = -2 0.0228 2.28%

🔄 Finding Area to the Right (Complementary)

Z 1 - P(Z)

\(P(Z > z) = 1 - P(Z \leq z)\)

✏️ Example: Find P(Z > 1.6)

From the table: P(Z ≤ 1.6) = 0.9452

P(Z > 1.6) = 1 - 0.9452 = 0.0548 (or 5.48%)

📊 Area Between Two Z-Scores

Z₁ Z₂

\(P(Z_1 < Z < Z_2) = P(Z \leq Z_2) - P(Z \leq Z_1)\)

✏️ Example: Find P(−0.44 < Z < 0.74)

From the table: P(Z ≤ 0.74) = 0.7704

From the table: P(Z ≤ −0.44) = 0.3300

P(-0.44 < Z < 0.74) = 0.7704 - 0.3300 = 0.4404 (or 44.04%)

🔍 Finding Z from Percentage (Inverse)

💡 Idea: Sometimes a percentage is given and we need Z!

✏️ Example: Find Z such that 77% of values are below it.

Step 1: Convert to decimal: 77% = 0.77

Step 2: Look up 0.77 in the table

Step 3: Find the Z from the row and column

Z ≈ 0.74

✏️ Example: Find Z such that 33% of values are above it.

Step 1: Complement: 100% − 33% = 67% = 0.67

Step 2: Look up 0.67 in the table

Z ≈ 0.44

📋 Summary Table – Question Types

Question type Solution method
P(Z ≤ z) Read directly from the table
P(Z > z) 1 − (table value)
P(Z₁ < Z < Z₂) P(Z ≤ Z₂) - P(Z ≤ Z₁)
Finding Z from percentage Search for the percentage within the table

📝 Summary

The Z-table gives the area to the left of the Z-score

Area to the right = 1 − area to the left

Area between = large area − small area