Normal Distribution
Z-Score – Introduction, Formula, and Meaning
🎯 Why Do We Need a Z-Score?
💡 The problem: A student scored 65 on a chemistry test. Is this good or poor?
The answer depends on the class! If the average was 53, a score of 65 is excellent!
🔍 Example: Arik scored 65 in chemistry and 65 in physics. In both the mean was 53.
Did he perform equally well in both subjects?
Chemistry
Mean: 53
Standard deviation: 2
Arik stands out greatly!
Physics
Mean: 53
Standard deviation: 12
Arik is good, but not exceptional
📌 Conclusion: To know how much a value stands out, compare it to:
- the mean of the distribution
- the spread (standard deviation) of all observations around the mean
⭐ Z-Score Formula
\(Z_x = \frac{x - \bar{x}}{S_x}\)
| \(Z_x\) | The Z-score of value x |
| \(x\) | The raw (actual) score |
| \(\bar{x}\) | the mean of the distribution |
| \(S_x\) | The standard deviation of the distribution |
💡 Meaning of the formula:
The Z-score measures distance from the mean in standard deviation units
✏️ Computing Arik's Z-Score
Chemistry
Arik's score: x = 65
Mean: \(\bar{x}\) = 53
SD: S = 2
\(Z = \frac{65-53}{2} = \frac{12}{2} = 6\)
Arik is 6 standard deviations above the mean!
Physics
Arik's score: x = 65
Mean: \(\bar{x}\) = 53
SD: S = 12
\(Z = \frac{65-53}{12} = \frac{12}{12} = 1\)
Arik is only 1 standard deviation above the mean!
Conclusion: Although Arik scored 65 in both, he performed much better in chemistry (Z=6) than in physics (Z=1)
📊 Meaning of the Sign of the Z-Score
| Case | Sign of Z | Meaning |
|---|---|---|
| \(x > \bar{x}\) | Z > 0 | Score is above the mean |
| \(x = \bar{x}\) | Z = 0 | Score equals the mean exactly |
| \(x < \bar{x}\) | Z < 0 | Score is below the mean |
💡 The larger |Z|, the farther the score is from the mean!
Z = 3 (very high above mean), Z = −2.5 (very low below mean)
📌 Important Properties of the Z-Score
1️⃣ The Z-score is a pure number – it has no units!
Whether it is scores, weight in kg, or height in cm — the Z-score has no units
2️⃣ The mean of Z-scores is always 0
\(\bar{Z} = 0\)
3️⃣ The standard deviation of Z-scores is always 1
\(S_Z^2 = 1\)
🔄 Inverse Formula – From Z-Score to Raw Score
\(x = \bar{x} + Z \cdot S_x\)
✏️ Example: In a distribution with mean 80 and SD 5, what is the raw score for Z = 1.5?
\(x = 80 + 1.5 \cdot 5 = 80 + 7.5 = 87.5\)
🔧 Transformations and the Z-Score
A linear transformation does not change the Z-score!
💡 What is a linear transformation?
Adding/subtracting/multiplying/dividing all values by a constant
✏️ Example: Arik's chemistry Z-score is Z = 6.
The teacher added a 10-point bonus to all scores. What is Arik's new Z-score?
Arik's Z-score remains Z = 6!
Even adding 10%, doubling, etc. — the Z-score does not change
⚠️ When does the Z-score change?
When the operation is not applied equally to all values (e.g. only scores below 55 are raised)
📝 Summary
\(Z = \frac{x - \bar{x}}{S}\)
Z-score = distance from the mean in units of standard deviation
Z > 0 above mean | Z = 0 equals mean | Z < 0 below mean