Lesson 2: Standard Normal Distribution & z-scores
Learn how to standardize any normal distribution and interpret z-scores
What is the Standard Normal Distribution?
While normal distributions can have any mean and standard deviation, there's one special normal distribution that is particularly useful: the standard normal distribution.
Definition: Standard Normal Distribution
The standard normal distribution is a normal distribution with:
- Mean (μ) = 0
- Standard deviation (σ) = 1
We often use the variable Z to represent a standard normal random variable, written as: Z ~ N(0, 1)
The standard normal distribution serves as a reference distribution. By converting any normal distribution to the standard normal, we can use standard tables and techniques to find probabilities.
The z-score: Converting to Standard Normal
A z-score (also called a standard score) tells us how many standard deviations a particular value is from the mean.
The z-score Formula:
z = (x - μ) / σ
where:
x = the original value
μ = the mean of the distribution
σ = the standard deviation
Understanding the z-score
The z-score tells us:
- How far the value is from the mean (in standard deviation units)
- Which direction:
- Positive z-score: value is above the mean
- Negative z-score: value is below the mean
- z-score of 0: value equals the mean
Example 1: Calculating a z-score
Scenario: SAT scores are normally distributed with μ = 1050 and σ = 100. Sarah scored 1200.
Question: What is Sarah's z-score?
Solution:
- x = 1200 (Sarah's score)
- μ = 1050
- σ = 100
- z = (x - μ) / σ = (1200 - 1050) / 100 = 150 / 100 = 1.5
Interpretation: Sarah's score is 1.5 standard deviations above the mean.
Example 2: Negative z-score
Scenario: Using the same SAT distribution (μ = 1050, σ = 100), Mark scored 900.
Question: What is Mark's z-score?
Solution:
- x = 900
- z = (900 - 1050) / 100 = -150 / 100 = -1.5
Interpretation: Mark's score is 1.5 standard deviations below the mean.
Note: Sarah and Mark are equally far from the mean, just in opposite directions!
Interpreting z-scores
z-scores provide a standardized way to compare values from different normal distributions.
Common z-score Interpretations
- z = 0: The value equals the mean (right at average)
- z = 1: One standard deviation above the mean (better than average)
- z = -1: One standard deviation below the mean (worse than average)
- z = 2: Two standard deviations above the mean (much better than average)
- z = -2: Two standard deviations below the mean (much worse than average)
- z = 3 or more: Very unusual (more than 3 SDs from mean)
Example 3: Comparing Across Different Tests
Scenario: Maria took two different placement tests:
- Math Test: Scored 85, where μ = 75, σ = 5
- English Test: Scored 92, where μ = 80, σ = 8
Question: On which test did Maria perform better relative to other students?
Solution:
Math z-score:
- z = (85 - 75) / 5 = 10 / 5 = 2.0
- Maria scored 2 standard deviations above the mean
English z-score:
- z = (92 - 80) / 8 = 12 / 8 = 1.5
- Maria scored 1.5 standard deviations above the mean
Answer: Even though her raw score was higher in English (92 vs 85), Maria performed better on the Math test relative to other students because her z-score was higher (2.0 vs 1.5).
Working Backwards: From z-score to Original Value
Sometimes we know the z-score and need to find the original value x. We can rearrange the z-score formula:
Reverse Formula:
x = μ + z·σ
This tells us the original value when we know the z-score.
Example 4: Finding the Original Value
Scenario: IQ scores are normally distributed with μ = 100 and σ = 15.
Question: What IQ score corresponds to a z-score of 1.8?
Solution:
- z = 1.8
- μ = 100
- σ = 15
- x = μ + z·σ = 100 + (1.8)(15) = 100 + 27 = 127
Answer: An IQ score of 127 is 1.8 standard deviations above the mean.
Introduction to z-tables
A z-table (also called the standard normal table) shows the cumulative probability for different z-scores. In the next lesson, we'll use these tables extensively to find probabilities.
What does a z-table tell us?
For a given z-score, the z-table gives the probability that Z is less than or equal to that value.
In notation: The table shows P(Z ≤ z)
For example:
- P(Z ≤ 0) = 0.5000 (50% of values are below the mean)
- P(Z ≤ 1) ≈ 0.8413 (about 84% are below z = 1)
- P(Z ≤ -1) ≈ 0.1587 (about 16% are below z = -1)
Check Your Understanding
Try these questions to test what you've learned in this lesson.
Question 1: A value has a z-score of 0. Where is this value relative to the mean?
Answer: The value is exactly at the mean.
Explanation: z = (x - μ) / σ = 0 means x - μ = 0, so x = μ.
Question 2: Heights of adult men are N(70, 3) inches. John is 76 inches tall. What is his z-score?
Answer: z = 2.0
Solution: z = (x - μ) / σ = (76 - 70) / 3 = 6 / 3 = 2.0
Interpretation: John is 2 standard deviations taller than average.
Question 3: A student scored z = -0.5 on a test. Did they score above or below the mean?
Answer: Below the mean
Explanation: Negative z-scores indicate values below the mean. This student scored 0.5 standard deviations below average.
Question 4: Test scores are N(500, 100). What score corresponds to z = 1.5?
Answer: 650
Solution: x = μ + z·σ = 500 + (1.5)(100) = 500 + 150 = 650
Question 5: Sarah scored 88 on Test A (μ = 80, σ = 4) and 92 on Test B (μ = 85, σ = 5). On which test did she perform better relative to her peers?
Answer: Test A
Solution:
- Test A: z = (88 - 80) / 4 = 2.0
- Test B: z = (92 - 85) / 5 = 1.4
Sarah's z-score was higher on Test A, meaning she performed better relative to other students, even though her raw score was lower.
Key Takeaways from Lesson 2
- The standard normal distribution has μ = 0 and σ = 1
- z-score formula: z = (x - μ) / σ tells how many SDs from the mean
- Positive z-scores: above the mean; negative z-scores: below the mean
- Reverse formula: x = μ + z·σ finds original value from z-score
- z-scores allow comparison across different distributions
- Most z-scores fall between -3 and +3
- z-tables show cumulative probabilities for the standard normal distribution