What is the Effect of Change of Origin on SD? | Simple Explanation for Class 12

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What is the Effect of Change of Origin on Standard Deviation?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Change of origin means adding or subtracting a constant value from every data point in a dataset. The standard deviation, which measures how spread out the data points are from the mean, remains unchanged when the origin is shifted.

Simple Example

Quick Example

Imagine 5 friends score marks in a science test: 10, 12, 14, 16, 18. If the teacher decides to add 5 bonus marks to everyone's score, their new marks become 15, 17, 19, 21, 23. The spread or variability of their marks (how far they are from each other) remains exactly the same, even though all marks are higher.

Worked Example

Step-by-Step

Let's find the standard deviation for two datasets.

Dataset 1: {2, 4, 6}
---Step 1: Calculate the mean (average). Mean = (2 + 4 + 6) / 3 = 12 / 3 = 4.
---Step 2: Calculate the deviations from the mean. (2-4)=-2, (4-4)=0, (6-4)=2.
---Step 3: Square the deviations. (-2)^2=4, (0)^2=0, (2)^2=4.
---Step 4: Sum the squared deviations. 4 + 0 + 4 = 8.
---Step 5: Divide by the number of data points (n). 8 / 3 = 2.67 (approximately).
---Step 6: Take the square root. Standard Deviation = sqrt(2.67) = 1.63 (approximately).

Now, let's change the origin by adding 10 to each data point.

Dataset 2: {12, 14, 16} (adding 10 to each value from Dataset 1)
---Step 1: Calculate the mean. Mean = (12 + 14 + 16) / 3 = 42 / 3 = 14.
---Step 2: Calculate the deviations from the mean. (12-14)=-2, (14-14)=0, (16-14)=2.
---Step 3: Square the deviations. (-2)^2=4, (0)^2=0, (2)^2=4.
---Step 4: Sum the squared deviations. 4 + 0 + 4 = 8.
---Step 5: Divide by the number of data points (n). 8 / 3 = 2.67 (approximately).
---Step 6: Take the square root. Standard Deviation = sqrt(2.67) = 1.63 (approximately).

Answer: The standard deviation for both datasets is approximately 1.63, showing no change.

Why It Matters

Understanding this helps scientists and engineers simplify calculations in many fields. For example, in AI/ML, data scientists often 'normalize' data by shifting its origin, knowing it won't affect the spread. In FinTech, financial analysts use this to compare stock price volatility without worrying about the absolute price level, focusing only on risk.

Common Mistakes

MISTAKE: Assuming standard deviation changes when a constant is added or subtracted. | CORRECTION: Remember that standard deviation only measures spread, and adding/subtracting a constant shifts the entire dataset without changing its spread.

MISTAKE: Confusing change of origin with change of scale. | CORRECTION: Change of origin is adding/subtracting; change of scale is multiplying/dividing. Standard deviation is affected by change of scale.

MISTAKE: Incorrectly calculating the mean after changing the origin. | CORRECTION: Always recalculate the mean for the new dataset, or simply add/subtract the constant to the original mean to get the new mean.

Practice Questions

Try It Yourself

QUESTION: A dataset of daily temperatures in Delhi has a standard deviation of 3 degrees Celsius. If all temperatures are increased by 5 degrees (e.g., due to a sensor recalibration), what will be the new standard deviation? | ANSWER: 3 degrees Celsius

QUESTION: For the data {5, 7, 9, 11}, the standard deviation is sqrt(5). If we subtract 2 from each data point, what will be the new standard deviation? | ANSWER: sqrt(5)

QUESTION: A cricketer's runs in 5 matches are {20, 30, 40, 50, 60}. If each score is increased by 10 runs, and then divided by 2, how will the standard deviation change? (Hint: First consider change of origin, then change of scale). | ANSWER: The original standard deviation is sqrt(200). After adding 10, it's still sqrt(200). After dividing by 2, it becomes sqrt(200)/2 = sqrt(50).

MCQ

Quick Quiz

If a constant 'k' is added to every observation in a dataset, how does the standard deviation (SD) change?

SD increases by k

SD decreases by k

SD remains unchanged

SD is multiplied by k

The Correct Answer Is:

Adding a constant to all data points shifts the entire dataset but does not change how spread out the data points are from each other or from the new mean. Hence, the standard deviation remains unchanged.

Real World Connection

In the Real World

When analyzing mobile network signal strengths in different areas, engineers might calibrate their readings by adding a constant offset to account for equipment variations. They know that this offset won't change the variability (standard deviation) of signal strength within an area, which is crucial for understanding network reliability. Similarly, ISRO scientists might adjust sensor data by a constant value without affecting the spread of their measurements.

Key Vocabulary

Key Terms

STANDARD DEVIATION: A measure of how spread out numbers are from their average (mean). | CHANGE OF ORIGIN: Adding or subtracting a constant value from every data point. | MEAN: The average of a set of numbers. | VARIABILITY: The extent to which data points in a statistical distribution or dataset tend to differ from each other.

What's Next

What to Learn Next

Next, you should explore 'What is the Effect of Change of Scale on Standard Deviation?'. This will teach you how multiplying or dividing data by a constant *does* affect the standard deviation, building on your understanding of data transformations.