Normal Distribution
Inverse Problems – From Probability to Raw Score
🔄 The Reverse Process
Probability/percentage
Z-score (Z)
Raw score (x)
Step 1: Adjust percentage to left-area (if needed)
Step 2: Find the corresponding Z-score in the Z-table
Step 3: Compute the raw score: \(x = \bar{x} + Z \cdot S\)
⭐ The Inverse Formula
\(x = \bar{x} + Z \cdot S\)
💡 Explanation:
Raw score = mean + (number of SDs × size of SD)
✏️ Example 1: Finding a Value from Its Percentile Below
Question: Test scores are normally distributed with mean 70 and SD 8.
What score has 90% of students below it?
Step 1: 90% = 0.90 (already left-area)
Step 2: Look up 0.90 in the table
Z ≈ 1.28
Step 3: Compute the raw score
\(x = 70 + 1.28 \cdot 8 = 70 + 10.24 = 80.24\)
Answer: the score is approx. 80.24
✏️ Example 2: Finding a Value from Its Percentile Above
Question: Salaries are normally distributed with mean ₪15,000 and SD ₪4,000.
10% of employees earn above a certain salary. What is that salary?
Step 1: Complement (since "above" is given)
100% − 10% = 90% = 0.90 (left-area)
Step 2: Look up 0.90 in the table
Z ≈ 1.28
Step 3: Compute the raw score
\(x = 15000 + 1.28 \cdot 4000 = 15000 + 5120 = 20120\)
Answer: the salary is approx. ₪20,120
✏️ Example 3: Finding a Percentile
Question: Men's height is normally distributed with mean 175 cm and SD 7 cm.
What is the 25th percentile (the height below which 25% of men fall)?
Step 1: 25% = 0.25
Step 2: Look up 0.25 in the table
Z ≈ -0.67
(negative because it is below the mean!)
Step 3: Compute the raw score
\(x = 175 + (-0.67) \cdot 7 = 175 - 4.69 = 170.31\)
Answer: the 25th percentile is approx. 170.3 cm
🔍 Finding the Mean or Standard Deviation
💡 Sometimes the mean or SD is missing and needs to be computed!
✏️ Example: Salaries are normally distributed with median ₪15,000.
84.4% of employees earn more than ₪10,960. What is the SD?
Step 1: Median = mean (in normal distribution), so \(\bar{x} = 15000\)
Step 2: 84.4% above 10,960 → 15.6% below it
Look up 0.156 in table → Z ≈ −1.01
Step 3: Substitute into the formula
\(Z = \frac{x - \bar{x}}{S}\)
\(-1.01 = \frac{10960 - 15000}{S}\)
\(-1.01 = \frac{-4040}{S}\)
\(S = \frac{-4040}{-1.01} = 4000\)
Answer: the SD is ₪4,000
📋 Summary Table – Inverse Problems
| Given | Find | Method |
|---|---|---|
| Percentage below | Raw score | Find Z from table, compute x |
| Percentage above | Raw score | Compute complement, find Z, compute x |
| Score and percentage | Standard deviation | Find Z, substitute, solve for S |
| Score and percentage | Mean | Find Z, substitute, solve for mean |
📝 Summary
\(x = \bar{x} + Z \cdot S\)
Probability → Z from table → raw score
Note the direction: "below" or "above"