Lesson 3: Finding Probabilities with the Normal Distribution
Master using z-tables to calculate probabilities and percentiles
Area Under the Curve = Probability
One of the most powerful features of the normal distribution is that the area under the curve represents probability. This allows us to calculate the probability that a randomly selected value falls within any range.
Key Concept: Area = Probability
For a continuous distribution like the normal distribution:
- The total area under the entire curve = 1 (or 100%)
- The area between two values = probability of getting a value in that range
- The area to the left of a value = probability of getting a value less than it
- The area to the right of a value = probability of getting a value greater than it
Using z-tables to Find Probabilities
The standard normal table (z-table) gives us P(Z ≤ z) — the probability that a standard normal variable is less than or equal to a particular z-score.
How to Read a z-table
Standard normal tables typically show:
- Rows: The z-score to one decimal place (e.g., 1.2, 1.3)
- Columns: The second decimal place (e.g., .00, .01, .02)
- Body: The cumulative probability P(Z ≤ z)
To find P(Z ≤ 1.25): Look at row 1.2, column .05 → probability ≈ 0.8944
Common z-table Values (Memorize These!)
- P(Z ≤ 0) = 0.5000 (50% are below the mean)
- P(Z ≤ 1) = 0.8413 (84.13% are below z = 1)
- P(Z ≤ 2) = 0.9772 (97.72% are below z = 2)
- P(Z ≤ -1) = 0.1587 (15.87% are below z = -1)
- P(Z ≤ -2) = 0.0228 (2.28% are below z = -2)
Finding P(X < a): Left-Tail Probabilities
This is the most straightforward case — z-tables directly give us left-tail probabilities.
Steps to Find P(X < a):
- Convert x to a z-score: z = (x - μ) / σ
- Look up the z-score in the z-table
- The table value is your answer!
Example 1: Left-Tail Probability
Scenario: SAT scores are normally distributed with μ = 1050 and σ = 100.
Question: What is the probability a randomly selected student scores less than 1200?
Solution:
Step 1: Calculate the z-score
- z = (1200 - 1050) / 100 = 150 / 100 = 1.5
Step 2: Look up z = 1.5 in the table
- P(Z ≤ 1.5) = 0.9332
Answer: P(X < 1200) = 0.9332, or about 93.32% of students score below 1200.
Finding P(X > a): Right-Tail Probabilities
For right-tail probabilities (greater than), we use the complement rule:
Complement Rule:
P(X > a) = 1 - P(X ≤ a)
Since the total probability is 1, the probability of being above a value equals
1 minus the probability of being at or below that value.
Steps to Find P(X > a):
- Convert x to a z-score: z = (x - μ) / σ
- Look up the z-score in the z-table to find P(Z ≤ z)
- Subtract from 1: P(Z > z) = 1 - P(Z ≤ z)
Example 2: Right-Tail Probability
Scenario: SAT scores: μ = 1050, σ = 100
Question: What percentage of students score above 1200?
Solution:
Step 1: Calculate z-score
- z = (1200 - 1050) / 100 = 1.5
Step 2: Find P(Z ≤ 1.5) from table
- P(Z ≤ 1.5) = 0.9332
Step 3: Use complement rule
- P(X > 1200) = P(Z > 1.5) = 1 - 0.9332 = 0.0668
Answer: About 6.68% of students score above 1200.
Finding P(a < X < b): Between Two Values
To find the probability that X falls between two values, we use subtraction:
Probability Between Two Values:
P(a < X < b) = P(X < b) - P(X < a)
Find the area to the left of b, then subtract the area to the left of a
Steps to Find P(a < X < b):
- Convert both values to z-scores
- Look up both z-scores in the table
- Subtract: P(z₁ < Z < z₂) = P(Z ≤ z₂) - P(Z ≤ z₁)
Example 3: Probability Between Two Values
Scenario: SAT scores: μ = 1050, σ = 100
Question: What percentage of students score between 950 and 1200?
Solution:
Step 1: Calculate z-scores
- z₁ = (950 - 1050) / 100 = -1.0
- z₂ = (1200 - 1050) / 100 = 1.5
Step 2: Look up both z-scores
- P(Z ≤ -1.0) = 0.1587
- P(Z ≤ 1.5) = 0.9332
Step 3: Subtract
- P(950 < X < 1200) = 0.9332 - 0.1587 = 0.7745
Answer: About 77.45% of students score between 950 and 1200.
Working Backwards: Finding Values from Percentiles
Sometimes we know the probability and need to find the corresponding value. This is called finding a percentile.
What is a Percentile?
The kth percentile is the value below which k% of the data falls.
Examples:
- 50th percentile = median (50% below, 50% above)
- 90th percentile = 90% of values are below this point
- 25th percentile = first quartile (Q1)
Steps to Find a Percentile:
- Look up the probability in the body of the z-table
- Find the corresponding z-score from the row/column
- Convert back to original value: x = μ + z·σ
Example 4: Finding a Percentile
Scenario: SAT scores: μ = 1050, σ = 100
Question: What score represents the 90th percentile?
Solution:
Step 1: Find z-score for 90th percentile (probability = 0.90)
- Look in z-table body for value closest to 0.9000
- We find 0.8997 at z = 1.28
- So the 90th percentile is approximately z = 1.28
Step 2: Convert to original scale
- x = μ + z·σ = 1050 + (1.28)(100) = 1050 + 128 = 1178
Answer: A score of about 1178 is the 90th percentile. 90% of students score below 1178.
Example 5: Finding a Lower Percentile
Scenario: Test scores: μ = 75, σ = 8
Question: Below what score do the bottom 10% of students fall?
Solution:
Step 1: Find z-score for 10th percentile (probability = 0.10)
- Look for 0.1000 in z-table body
- Closest value is 0.1003 at z = -1.28
- Note: Negative z because we're below the mean
Step 2: Convert to original scale
- x = μ + z·σ = 75 + (-1.28)(8) = 75 - 10.24 = 64.76
Answer: The bottom 10% score below about 65.
Check Your Understanding
Try these questions to test what you've learned in this lesson.
Question 1: Heights of women are N(65, 2.5) inches. What is the probability a randomly selected woman is shorter than 67.5 inches? (Note: z = 1.0 gives P(Z ≤ 1) = 0.8413)
Answer: 0.8413 or 84.13%
Solution:
- z = (67.5 - 65) / 2.5 = 2.5 / 2.5 = 1.0
- P(Z ≤ 1.0) = 0.8413
Question 2: Using the same distribution, what percentage of women are taller than 67.5 inches?
Answer: 15.87%
Solution:
- From Question 1: P(X < 67.5) = 0.8413
- P(X > 67.5) = 1 - 0.8413 = 0.1587
Question 3: IQ scores are N(100, 15). Find P(IQ < 85). (Hint: z = -1 gives P(Z ≤ -1) = 0.1587)
Answer: 0.1587 or 15.87%
Solution:
- z = (85 - 100) / 15 = -15 / 15 = -1.0
- P(Z ≤ -1) = 0.1587
Question 4: Battery life is N(500, 50) hours. What's the probability a battery lasts between 450 and 600 hours? (z = -1: 0.1587, z = 2: 0.9772)
Answer: 0.8185 or 81.85%
Solution:
- z₁ = (450 - 500) / 50 = -1, P(Z ≤ -1) = 0.1587
- z₂ = (600 - 500) / 50 = 2, P(Z ≤ 2) = 0.9772
- P(450 < X < 600) = 0.9772 - 0.1587 = 0.8185
Question 5: Test scores are N(72, 10). What score represents the 75th percentile? (z = 0.67 gives P(Z ≤ 0.67) ≈ 0.75)
Answer: 78.7 (or approximately 79)
Solution:
- For 75th percentile, we need z where P(Z ≤ z) = 0.75
- From table: z ≈ 0.67
- x = μ + z·σ = 72 + (0.67)(10) = 72 + 6.7 = 78.7
Key Takeaways from Lesson 3
- Area under the curve = probability for continuous distributions
- z-tables give P(Z ≤ z) — cumulative probabilities
- Left-tail: P(X < a) — directly from z-table
- Right-tail: P(X > a) = 1 - P(X ≤ a) — use complement
- Between values: P(a < X < b) = P(X < b) - P(X < a)
- Percentiles: Look up probability in table body, find z, convert to x
- Always convert to z-scores first: z = (x - μ) / σ