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

S7-SA3-0339

Grade Level:

Class 12

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

Definition

What is it?

Changing the scale of data means multiplying or dividing all data points by a constant number. When you change the scale, the standard deviation also changes by the same factor. This means if you multiply all data by 'k', the standard deviation also gets multiplied by 'k'.

Simple Example

Quick Example

Imagine you have the heights of your friends in meters. If you convert all these heights to centimeters (by multiplying each height by 100), the standard deviation of their heights will also become 100 times larger. The spread relative to the new unit remains the same.

Worked Example

Step-by-Step

Let's say we have the daily pocket money (in Rupees) of 3 friends: 10, 20, 30.

Step 1: Calculate the mean (average) of the original data.
Mean = (10 + 20 + 30) / 3 = 60 / 3 = 20 Rupees.

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Step 2: Calculate the deviations from the mean.
(10 - 20) = -10
(20 - 20) = 0
(30 - 20) = 10

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Step 3: Square the deviations.
(-10)^2 = 100
(0)^2 = 0
(10)^2 = 100

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Step 4: Calculate the sum of squared deviations.
Sum = 100 + 0 + 100 = 200.

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Step 5: Calculate the variance (sum of squared deviations / number of data points).
Variance = 200 / 3 = 66.67 (approximately).

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Step 6: Calculate the standard deviation (sqrt of variance).
Standard Deviation = sqrt(66.67) = 8.16 (approximately).

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Now, let's change the scale by multiplying each pocket money value by 2 (e.g., they got a raise!): 20, 40, 60.

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Step 7: Calculate the new mean.
New Mean = (20 + 40 + 60) / 3 = 120 / 3 = 40 Rupees.

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Step 8: Calculate the new standard deviation using the same steps as above for the new data.
Deviations: (20-40)=-20, (40-40)=0, (60-40)=20
Squared Deviations: 400, 0, 400
Sum of Squared Deviations: 800
New Variance = 800 / 3 = 266.67
New Standard Deviation = sqrt(266.67) = 16.33 (approximately).

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Notice: Original SD = 8.16, New SD = 16.33. Since we multiplied the data by 2, the standard deviation also got multiplied by 2 (8.16 * 2 = 16.32). This confirms the rule.

Answer: When the data was multiplied by 2, the standard deviation also multiplied by 2.

Why It Matters

Understanding how standard deviation changes with scale is crucial for data scientists and engineers. In fields like AI/ML, you often 'normalize' data, which involves scaling, and need to know how this affects variability. Climate scientists use this to compare temperature variations across different units (Celsius vs. Fahrenheit) accurately, ensuring their models are correct.

Common Mistakes

MISTAKE: Thinking that adding a constant to data also changes the standard deviation. | CORRECTION: Adding or subtracting a constant to all data points does NOT change the standard deviation. It only shifts the entire dataset.

MISTAKE: Forgetting to apply the scale change to the standard deviation itself. | CORRECTION: If you multiply all data by 'k', the standard deviation is also multiplied by 'k'. If you divide by 'k', it's divided by 'k'.

MISTAKE: Confusing the effect of scale on variance with standard deviation. | CORRECTION: If data is multiplied by 'k', the standard deviation is multiplied by 'k', but the variance is multiplied by 'k^2' (k squared).

Practice Questions

Try It Yourself

QUESTION: The standard deviation of marks in a test is 5. If all marks are doubled, what will be the new standard deviation? | ANSWER: New standard deviation = 5 * 2 = 10.

QUESTION: A dataset has a standard deviation of 12. If each data point is divided by 3, what will be the new standard deviation? | ANSWER: New standard deviation = 12 / 3 = 4.

QUESTION: The average daily temperature in a city over a week has a standard deviation of 4 degrees Celsius. If these temperatures were converted to a different unit by multiplying each by 1.8 and then adding 32, what would be the standard deviation in the new unit? | ANSWER: Only the multiplication affects the standard deviation. So, 4 * 1.8 = 7.2. The addition of 32 does not change the standard deviation.

MCQ

Quick Quiz

If the standard deviation of a set of values is 8, and each value in the set is multiplied by 5, what will be the new standard deviation?

The Correct Answer Is:

When each value in a set is multiplied by a constant 'k', the standard deviation is also multiplied by 'k'. Here, k=5, so the new standard deviation is 8 * 5 = 40.

Real World Connection

In the Real World

Imagine a drone delivering packages in a city. The standard deviation of delivery times tells us how consistent the service is. If the drone software is updated to make it fly twice as fast (scaling the speed), the standard deviation of delivery times would also be halved, indicating much tighter, more predictable delivery windows for customers using apps like Zepto or Swiggy.

Key Vocabulary

Key Terms

STANDARD DEVIATION: A measure of how spread out numbers are from the average. | SCALE: Multiplying or dividing all data points by a constant. | VARIANCE: The average of the squared differences from the mean (standard deviation squared). | CONSTANT: A fixed value that does not change.

What's Next

What to Learn Next

Next, you should explore 'What is the Effect of Change of Origin on Standard Deviation?'. This will help you understand the full picture of how different transformations affect data spread, which is super important for advanced statistics and data analysis.