What is Effect of Change of Scale on Dispersion? | Simple Explanation for Class 12

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What is the Effect of Change of Scale on Measures of Dispersion?

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. This action directly affects measures of dispersion like range, variance, and standard deviation by scaling them by the same constant (or its square for variance). However, adding or subtracting a constant to all data points does NOT change measures of dispersion.

Simple Example

Quick Example

Imagine you have the daily pocket money (in Rupees) of 5 friends: 10, 12, 14, 16, 18. The range is 18 - 10 = 8. If their parents double their pocket money, the new amounts are 20, 24, 28, 32, 36. The new range is 36 - 20 = 16. Notice the range also doubled (8 x 2 = 16).

Worked Example

Step-by-Step

Let's find the standard deviation for a set of marks and then see how it changes if marks are scaled.

DATA: 5, 7, 9, 11, 13

Step 1: Calculate the mean (average). Mean = (5 + 7 + 9 + 11 + 13) / 5 = 45 / 5 = 9.

---Step 2: Calculate the variance. Variance = [(5-9)^2 + (7-9)^2 + (9-9)^2 + (11-9)^2 + (13-9)^2] / 5
= [(-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2] / 5
= [16 + 4 + 0 + 4 + 16] / 5
= 40 / 5 = 8.

---Step 3: Calculate the standard deviation. Standard Deviation = sqrt(Variance) = sqrt(8) approx 2.83.

---Step 4: Now, let's double each mark (multiply by 2). New DATA: 10, 14, 18, 22, 26.

---Step 5: Calculate the new mean. New Mean = (10 + 14 + 18 + 22 + 26) / 5 = 90 / 5 = 18.

---Step 6: Calculate the new variance. New Variance = [(10-18)^2 + (14-18)^2 + (18-18)^2 + (22-18)^2 + (26-18)^2] / 5
= [(-8)^2 + (-4)^2 + 0^2 + 4^2 + 8^2] / 5
= [64 + 16 + 0 + 16 + 64] / 5
= 160 / 5 = 32.

---Step 7: Calculate the new standard deviation. New Standard Deviation = sqrt(32) approx 5.66.

---Answer: The original standard deviation was 2.83. The new standard deviation is 5.66. Notice that 5.66 is approximately 2 times 2.83. This shows that when data is scaled by a factor of 'k', the standard deviation also scales by 'k'. The variance scales by 'k^2' (8 x 2^2 = 8 x 4 = 32).

Why It Matters

Understanding how data scales is super important in fields like AI/ML and Physics. Data scientists use this to normalize data for better model performance, and engineers in FinTech analyze scaled financial data to understand risk. It helps professionals like data analysts and researchers make sense of numbers that come in different units or magnitudes.

Common Mistakes

MISTAKE: Thinking that adding a constant to all data points changes the standard deviation. | CORRECTION: Adding or subtracting a constant only shifts the data, it does not spread it out more or less. So, measures of dispersion like standard deviation remain unchanged.

MISTAKE: Believing that if data is multiplied by 'k', the variance also multiplies by 'k'. | CORRECTION: If data is multiplied by 'k', the variance multiplies by 'k^2' (k squared), not just 'k'. Remember, variance involves squared differences from the mean.

MISTAKE: Applying the scaling rule to all statistical measures, including mean. | CORRECTION: While measures of dispersion are affected by change of scale, measures of central tendency (like mean and median) are affected by both change of origin (addition/subtraction) AND change of scale (multiplication/division).

Practice Questions

Try It Yourself

QUESTION: A data set has a range of 15. If each data point is multiplied by 3, what will be the new range? | ANSWER: 45

QUESTION: The standard deviation of a series of daily temperatures is 4 degrees Celsius. If each temperature reading is converted from Celsius to Fahrenheit using the formula F = (9/5)C + 32, what will be the new standard deviation in Fahrenheit? (Hint: Only the multiplication factor affects SD.) | ANSWER: 7.2 degrees Fahrenheit (4 * 9/5)

QUESTION: A dataset has a variance of 25. If each observation is first increased by 10, and then multiplied by 2, what will be the new variance? | ANSWER: 100 (Increasing by 10 does not change variance. Multiplying by 2 means variance scales by 2^2 = 4. So, 25 * 4 = 100)

MCQ

Quick Quiz

If the standard deviation of a dataset is 6, and each data point is divided by 3, what will be the new standard deviation?

The Correct Answer Is:

When data points are divided by a constant 'k', the standard deviation is also divided by 'k'. Here, 6 / 3 = 2. Options A, C, D are incorrect because they do not follow this scaling rule.

Real World Connection

In the Real World

In cricket analytics, if a player's runs per match are analyzed, and then all scores are converted from 'runs' to 'boundary equivalents' (e.g., each 4 is 1 unit, each 6 is 1.5 units), the variability (dispersion) of their scores would scale accordingly. This helps coaches and analysts compare player consistency across different scoring metrics. Similarly, in financial markets, when stock prices are converted from rupees to paise, the dispersion of prices changes by a factor of 100.

Key Vocabulary

Key Terms

MEASURES OF DISPERSION: Statistics that describe how spread out data points are, like range, variance, standard deviation. | CHANGE OF SCALE: Multiplying or dividing all data points by a constant. | CHANGE OF ORIGIN: Adding or subtracting a constant to all data points. | STANDARD DEVIATION: A measure of how much data points typically deviate from the mean. | VARIANCE: The average of the squared differences from the mean.

What's Next

What to Learn Next

Now that you understand how scaling affects dispersion, you can explore 'Skewness and Kurtosis'. These concepts tell us about the shape and 'tailedness' of a data distribution, giving you an even deeper insight into your data beyond just its spread.