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Effect of Change in Origin and Scale

Introduction

In data analysis, values are often transformed by adding or multiplying constants - for example, converting temperatures from Celsius to Fahrenheit or marks from one scale to another. The Effect of Change in Origin and Scale pattern helps you understand how such changes affect the mean and standard deviation (SD).

This concept is essential because many real-life datasets are expressed in different units or reference points, and understanding these effects lets you compare data correctly.

Pattern: Effect of Change in Origin and Scale

Pattern

The key concept: Adding or subtracting a constant shifts all data points but does not affect the spread, while multiplying or dividing by a constant changes the spread proportionally.

Let the original data be x, and new data be y = a + b × x.
Then,
New Mean = a + b × (Old Mean)
New Standard Deviation (SD) = |b| × (Old SD)

Step-by-Step Example

Question

The mean and standard deviation of marks of 50 students are 40 and 10 respectively. Each student’s mark is increased by 5 and then multiplied by 2. Find the new mean and standard deviation.

Solution

  1. Step 1: Identify given values

    Old Mean = 40, Old SD = 10
    Added constant (a) = 5, Multiplied constant (b) = 2

  2. Step 2: Apply transformation formula

    New data are formed using y = a + b × x.

  3. Step 3: Find the new mean

    New Mean = a + b × (Old Mean)
    = 5 + 2 × 40
    = 5 + 80 = 85

  4. Step 4: Find the new standard deviation

    New SD = |b| × (Old SD)
    = 2 × 10 = 20

  5. Final Answer:

    New Mean = 85, New SD = 20

  6. Quick Check:

    Only multiplication affects SD; addition affects only the mean ✅

Quick Variations

1. Adding/subtracting a constant → Mean changes, SD unchanged.

2. Multiplying/dividing by a constant → Both Mean and SD change by that factor.

3. Common in unit conversion problems like °C ↔ °F, cm ↔ m, marks scaling.

Trick to Always Use

  • Step 1: If a constant is added/subtracted → affects only mean.
  • Step 2: If a constant is multiplied/divided → affects both mean and SD.
  • Step 3: Always multiply SD by the absolute value of the scale factor.

Summary

Summary

In the Effect of Change in Origin and Scale pattern:

  • Adding or subtracting a constant (change in origin) → affects only the mean.
  • Multiplying or dividing by a constant (change in scale) → affects both mean and SD.
  • Formula:
    New Mean = a + b × (Old Mean)
    New SD = |b| × (Old SD)
  • Useful for conversions and scaled data transformations.
  • Remember: SD remains unchanged by addition or subtraction.

Practice

(1/5)
1. The mean and SD of 40 students’ marks are 60 and 8 respectively. If 5 marks are added to each student’s score, find the new mean and new SD.
easy
A. Mean=65, SD=8
B. Mean=55, SD=8
C. Mean=65, SD=13
D. Mean=65, SD=10

Solution

  1. Step 1: Identify given values

    Old Mean = 60, Old SD = 8; added constant a = 5.

  2. Step 2: Apply origin rule

    Adding a constant increases the mean by that constant; SD remains unchanged.

  3. Step 3: Compute new mean and SD

    New Mean = 60 + 5 = 65; New SD = 8.

  4. Final Answer:

    Mean = 65, SD = 8 → Option A.

  5. Quick Check:

    Addition shifts mean only; SD stays same ✅
Hint: Addition/subtraction affects only the mean, not SD.
Common Mistakes: Incorrectly scaling the SD when a constant is added.
2. The mean temperature of a city is 25°C with SD 4°C. Convert the data to Fahrenheit (F = 32 + 1.8×C). Find the new SD.
easy
A. 32.7
B. 7.2
C. 45.5
D. 4.89

Solution

  1. Step 1: Identify constants

    Conversion: F = 32 + 1.8 × C → a = 32 (origin), b = 1.8 (scale).

  2. Step 2: Apply scale rule

    SD scales by |b| only (addition does not affect SD).

  3. Step 3: Compute new SD

    New SD = |1.8| × 4 = 7.2.

  4. Final Answer:

    New SD = 7.2 → Option B.

  5. Quick Check:

    Multiplying temperatures by 1.8 increases SD by 1.8× ✅
Hint: Only the multiplier changes SD; ignore additive constants for SD.
Common Mistakes: Adding the constant 32 to SD or forgetting the multiplier.
3. A dataset has Mean = 50 and SD = 5. If all values are multiplied by 3, find the new mean and SD.
easy
A. Mean=150, SD=5
B. Mean=53, SD=8
C. Mean=150, SD=15
D. Mean=50, SD=15

Solution

  1. Step 1: Identify transformation

    Multiplication by 3 → a = 0, b = 3.

  2. Step 2: Apply scale rules

    New Mean = a + b × Old Mean = 0 + 3 × 50 = 150.
    New SD = |b| × Old SD = 3 × 5 = 15.

  3. Final Answer:

    Mean = 150, SD = 15 → Option C.

  4. Quick Check:

    Both mean and SD scale by factor 3 ✅
Hint: Multiplying data by k multiplies mean and SD by |k|.
Common Mistakes: Failing to scale both mean and SD when data are multiplied.
4. The mean and SD of salaries are ₹20,000 and ₹2,000 respectively. If all salaries are reduced by ₹1,000 and then doubled, find the new mean and SD.
medium
A. Mean=₹42,000, SD=₹4,000
B. Mean=₹39,000, SD=₹3,000
C. Mean=₹37,000, SD=₹2,000
D. Mean=₹38,000, SD=₹4,000

Solution

  1. Step 1: Write transformation

    y = 2 × (x - 1000) = 2x - 2000, so a = -2000, b = 2.

  2. Step 2: Apply formulas

    New Mean = a + b × Old Mean = -2000 + 2 × 20000 = -2000 + 40000 = ₹38,000.
    New SD = |b| × Old SD = 2 × 2000 = ₹4,000.

  3. Final Answer:

    Mean = ₹38,000, SD = ₹4,000 → Option D.

  4. Quick Check:

    Subtract shifts mean down by 1000, doubling scales mean and SD by 2 ✅
Hint: Combine subtraction and scaling into y = a + b×x before applying rules.
Common Mistakes: Ignoring the subtract-before-scale order when forming a and b.
5. A dataset has Mean = 40 and SD = 5. If every value is first increased by 10 and then divided by 5, find the new mean and SD.
medium
A. Mean=10, SD=1
B. Mean=12, SD=1
C. Mean=11, SD=1
D. Mean=9, SD=1

Solution

  1. Step 1: Write transformation

    y = (x + 10) ÷ 5 = (1/5)×x + 2, so a = 2 and b = 1/5 = 0.2.

  2. Step 2: Apply formulas

    New Mean = a + b × Old Mean = 2 + 0.2 × 40 = 2 + 8 = 10.
    New SD = |b| × Old SD = 0.2 × 5 = 1.

  3. Final Answer:

    Mean = 10, SD = 1 → Option A.

  4. Quick Check:

    Addition shifts mean by +10 then division scales values by 1/5; SD scales to 1 ✅
Hint: Handle addition first to get a, then apply division to determine b for SD.
Common Mistakes: Reversing the order of operations when computing a and b.

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