📐 Linear Transformation
What happens to the mean and standard deviation when a constant is added or multiplied?
🎯 Why Does This Matter?
A linear transformation applies the same change to every data value. Examples:
- Adding a bonus: every employee receives a 500 pay increase
- Currency conversion: all amounts converted from dollars to shekels (×3.6)
- Temperature conversion: Celsius to Fahrenheit: \(F = 1.8C + 32\)
- Score update: teacher added 5 points to every student
The central question: How does such a change affect the Mean, Median, SD and Variance?
📚 Definition: What Is a Linear Transformation?
If the original data are \(x_1, x_2, \ldots, x_n\)
The new data: \(y_i = a + b \cdot x_i\)
💡 Important special cases:
- If \(b = 1\): adding a constant only → \(y = a + x\)
- If \(a = 0\): multiplying by a constant only → \(y = b \cdot x\)
⭐ The Central Rule: Transformation Formulas
If \(y = a + b \cdot x\), then:
⚠️ The most important rule to remember:
Adding a constant (a) does not change the spread!
Standard deviation and variance are unaffected by adding a constant
🧠 Why Does Adding a Constant Not Change the Spread?
Think of it simply:
Imagine 5 people sting in a line.
If they all jump 3 metres to the right — the distance between any two does not change!
They all moved by the same amount, so the spread remains the same.
Before the shift:
📍2 📍5 📍8 📍11 📍14
Mean = 8, SD = 4.47
After adding 10 to everyone:
📍12 📍15 📍18 📍21 📍24
Mean = 18, SD = 4.47 ✅
In contrast, when multiplying by a constant — the distances between people grow!
Before multiplication:
📍2 📍5 📍8 📍11 📍14
Mean = 8, SD = 4.47
After ×3 for everyone:
📍6 📍15 📍24 📍33 📍42
Mean = 24, SD = 13.42 (= 3 × 4.47)
📝 Example 1: Adding 5 Points to Every Score
Original scores: 70, 80, 90, 60, 100
Transformation: \(y = 5 + x\) (i.e. \(a = 5, \; b = 1\))
| Measure | Before (x) | Calculation | After (y) |
|---|---|---|---|
| Mean | 80 | \(5 + 1 \times 80\) | 85 ✅ |
| Median | 80 | \(5 + 1 \times 80\) | 85 ✅ |
| SD | 14.14 | \(|1| \times 14.14\) | 14.14 (unchanged!) |
| Variance | 200 | \(1^2 \times 200\) | 200 (unchanged!) |
📝 Example 2: Converting Salaries from Dollars to Shekels (×3.6)
Salaries in dollars: 2000, 3000, 4000, 5000, 6000
Transformation: \(y = 3.6 \cdot x\) (i.e. \(a = 0, \; b = 3.6\))
| Measure | In dollars (x) | Calculation | In shekels (y) |
|---|---|---|---|
| Mean | $4,000 | \(0 + 3.6 \times 4000\) | ₪14,400 |
| Median | $4,000 | \(0 + 3.6 \times 4000\) | ₪14,400 |
| SD | $1,581 | \(|3.6| \times 1581\) | ₪5,692 |
| Variance | $²2,500,000 | \(3.6^2 \times 2500000\) | ₪²32,400,000 |
📝 Example 3: Converting Celsius to Fahrenheit
Temperatures in Celsius: 10°, 20°, 30°, 40°, 50°
Transformation: \(F = 32 + 1.8 \cdot C\) (i.e. \(a = 32, \; b = 1.8\))
| Measure | Celsius (x) | Calculation | Fahrenheit (y) |
|---|---|---|---|
| Mean | 30°C | \(32 + 1.8 \times 30 = 32 + 54\) | 86°F |
| SD | 15.81°C | \(|1.8| \times 15.81\) | 28.46°F |
| Variance | 250 °C² | \(1.8^2 \times 250 = 3.24 \times 250\) | 810 °F² |
💡 Note: The 32 (=a) changed the Mean but not the SD. Only the 1.8 (=b) changed the SD!
📋 Summary Table: What Changes When?
🎓 Typical Exam Question
Question: The class mean is 72 and the SD is 8.
The teacher applies a transformation: new score = 10 + 1.2 × old score.
Find the mean and SD of the new scores.
Solution:
Given: \(\bar{x} = 72, \; s_x = 8, \; a = 10, \; b = 1.2\)
New mean:
\(\bar{y} = a + b \cdot \bar{x} = 10 + 1.2 \times 72 = 10 + 86.4 = 96.4\)
New SD:
\(s_y = |b| \cdot s_x = |1.2| \times 8 = 9.6\)
Answer: new mean = 96.4, new SD = 9.6
⚠️ Common Mistakes
📝 Summary
Linear transformation: \(y = a + b \cdot x\)
Measures of centre (Mean, Median) — affected bya andb
Measures of dispersion (SD, Variance) — affected only by|b|
🔑 Adding a constant does not change dispersion!