Z-Scores and Standard Scores
Learn to calculate z-scores, compare data across different distributions, and use z-tables to find probabilities for normally distributed data.
What is a Z-Score?
A z-score (or standard score) tells us how many standard deviations a data value is above or below the mean. It standardises values from any normal distribution to allow meaningful comparisons.
z = x - μ/σ
where x = data value, μ = mean, σ = standard deviation
z > 0
Value is above the mean
z = 0
Value equals the mean
z < 0
Value is below the mean
Comparing Data Across Different Distributions
Z-scores allow us to compare values from different data sets that may have different means and standard deviations. The higher the z-score, the better the relative performance.
Example: Comparing Exam Results
| Subject | Score | Mean (μ) | Std Dev (σ) | Z-Score |
|---|---|---|---|---|
| Maths | 82 | 70 | 8 | +1.50 |
| English | 78 | 65 | 6 | +2.17 |
Despite scoring lower in English (78 vs 82), the z-score is higher for English (2.17 vs 1.50), meaning the English result was relatively better compared to the class.
Using Z-Tables to Find Probabilities
A z-table gives the probability that a value from a standard normal distribution (mean = 0, σ = 1) falls below a given z-score. This is written as P(Z < z).
Common Z-Score Probabilities
| Z-Score | P(Z < z) | Meaning |
|---|---|---|
| -2.0 | 0.0228 | 2.28% below |
| -1.0 | 0.1587 | 15.87% below |
| 0.0 | 0.5000 | 50% below (the mean) |
| +1.0 | 0.8413 | 84.13% below |
| +2.0 | 0.9772 | 97.72% below |
Key Vocabulary
Z-Score
The number of standard deviations a value is above or below the mean: z = (x - μ) / σ.
Standard Normal Distribution
A normal distribution with mean = 0 and standard deviation = 1. Z-scores convert any normal distribution to this form.
Z-Table
A reference table giving the cumulative probability P(Z < z) for the standard normal distribution.
Standardise
The process of converting a raw score to a z-score, enabling comparison across different distributions.
Worked Examples
A student scores 85 on a test with μ = 72 and σ = 8. Find the z-score.
Step 1: z = (x - μ) / σ = (85 - 72) / 8.
Step 2: z = 13/8 = 1.625.
Answer: The z-score is 1.625, meaning the student scored 1.625 standard deviations above the mean.
IQ scores have μ = 100 and σ = 15. What IQ corresponds to a z-score of +2?
Step 1: Rearrange: x = μ + zσ = 100 + 2(15).
Step 2: x = 100 + 30 = 130.
Answer: An IQ of 130 corresponds to a z-score of +2.
Find the percentage of values below z = 1.5 using a z-table.
Step 1: Look up z = 1.50 in the z-table.
Step 2: The table gives P(Z < 1.50) = 0.9332.
Answer: Approximately 93.32% of values fall below z = 1.5.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
A value has a z-score of -1.5. This means the value is:
Question 2
Heights have μ = 170 cm and σ = 10 cm. What is the z-score for a height of 155 cm?
Question 3
If P(Z < 1.0) = 0.8413, what is P(Z > 1.0)?
Question 4
Alice scored 90 in Maths (μ=75, σ=10) and 85 in Science (μ=70, σ=5). In which subject did she perform relatively better?
Question 5
A z-score of 0 means the data value:
Key Concepts Summary
- ●The z-score formula is z = (x - μ) / σ.
- ●Positive z = above the mean; negative z = below the mean; z = 0 means at the mean.
- ●Z-scores allow fair comparison of values from different distributions.
- ●The z-table gives cumulative probabilities P(Z < z) for the standard normal distribution.
- ●To find P(Z > z), use the complement: P(Z > z) = 1 - P(Z < z).