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

S7-SA3-0337

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. The effect of this on Mean Deviation is that the Mean Deviation also gets multiplied or divided by the same constant number. It directly scales with the data.

Simple Example

Quick Example

Imagine you have marks for 5 students: 10, 20, 30, 40, 50. Their Mean Deviation is 12. If you double everyone's marks (change the scale by multiplying by 2) to 20, 40, 60, 80, 100, the new Mean Deviation will also double to 24.

Worked Example

Step-by-Step

Let's find the Mean Deviation for a set of scores and then see how it changes if we multiply each score by 3.

Data Set 1: 5, 10, 15, 20, 25

Step 1: Calculate the Mean (x-bar).
Mean = (5 + 10 + 15 + 20 + 25) / 5 = 75 / 5 = 15

---Step 2: Calculate the absolute deviations from the Mean (|x - x-bar|).
|5 - 15| = 10
|10 - 15| = 5
|15 - 15| = 0
|20 - 15| = 5
|25 - 15| = 10

---Step 3: Calculate the sum of absolute deviations.
Sum = 10 + 5 + 0 + 5 + 10 = 30

---Step 4: Calculate the Mean Deviation (MD).
MD = Sum / Number of data points = 30 / 5 = 6

---Now, let's change the scale by multiplying each data point by 3.
New Data Set: (5*3), (10*3), (15*3), (20*3), (25*3) = 15, 30, 45, 60, 75

---Step 5: Calculate the new Mean.
New Mean = (15 + 30 + 45 + 60 + 75) / 5 = 225 / 5 = 45

---Step 6: Calculate the absolute deviations from the new Mean.
|15 - 45| = 30
|30 - 45| = 15
|45 - 45| = 0
|60 - 45| = 15
|75 - 45| = 30

---Step 7: Calculate the new sum of absolute deviations.
New Sum = 30 + 15 + 0 + 15 + 30 = 90

---Step 8: Calculate the new Mean Deviation.
New MD = New Sum / Number of data points = 90 / 5 = 18

Answer: The original Mean Deviation was 6. After multiplying each data point by 3, the new Mean Deviation is 18. Notice that 18 is 3 times 6. This shows that the Mean Deviation also gets multiplied by the same scaling factor (3).

Why It Matters

Understanding how scale changes affect Mean Deviation is crucial in fields like AI/ML, where data is often normalized or scaled for better model performance. In FinTech, comparing financial data scaled to different currencies requires this knowledge. Engineers use it to understand how measurement errors scale up or down.

Common Mistakes

MISTAKE: Thinking Mean Deviation is unaffected by change of scale. | CORRECTION: Remember that if you multiply or divide all data points by a number, the Mean Deviation also gets multiplied or divided by that same number.

MISTAKE: Confusing change of scale (multiplication/division) with change of origin (addition/subtraction). | CORRECTION: Change of scale affects Mean Deviation, but change of origin (adding/subtracting a constant to all data) does NOT affect Mean Deviation.

MISTAKE: Forgetting to apply the scaling factor to the Mean Deviation after scaling the data. | CORRECTION: Always multiply or divide the original Mean Deviation by the exact same constant that was used to scale the data points.

Practice Questions

Try It Yourself

QUESTION: A data set has a Mean Deviation of 8. If each data point in the set is multiplied by 5, what will be the new Mean Deviation? | ANSWER: 40

QUESTION: The Mean Deviation of daily temperatures in a city for a week was 3 degrees Celsius. If all temperatures were converted from Celsius to a new scale by multiplying by 2 and then adding 5, what would be the Mean Deviation in the new scale? | ANSWER: 6 (The addition of 5, a change of origin, does not affect Mean Deviation. Only the multiplication by 2, a change of scale, affects it.)

QUESTION: For a data set {2, 4, 6, 8}, calculate the Mean Deviation. Then, if each data point is divided by 2, what is the new Mean Deviation? Show your working. | ANSWER: Original Mean Deviation: 2. New Mean Deviation: 1. (Original Mean = 5. Deviations: 3, 1, 1, 3. Sum = 8. MD = 8/4 = 2. New data: {1, 2, 3, 4}. New Mean = 2.5. Deviations: 1.5, 0.5, 0.5, 1.5. Sum = 4. MD = 4/4 = 1. This is 2/2 = 1.)

MCQ

Quick Quiz

If the Mean Deviation of a data set is 15, and each observation in the set is multiplied by 0.5, what will be the new Mean Deviation?

7.5

15.5

The Correct Answer Is:

When data points are multiplied by a constant, the Mean Deviation is also multiplied by the same constant. So, 15 * 0.5 = 7.5. Change of scale directly affects Mean Deviation.

Real World Connection

In the Real World

In cricket analytics, if a player's scores are initially in 'runs' and then converted to a 'points' system where each run is worth 1.5 points, the Mean Deviation of their scores in the points system would be 1.5 times the Mean Deviation in runs. This helps coaches understand consistency across different scoring metrics.

Key Vocabulary

Key Terms

Mean Deviation: The average of the absolute differences between each data point and the mean of the data set. | Change of Scale: Multiplying or dividing all data points by a constant number. | Absolute Deviation: The distance of a data point from the mean, ignoring its sign. | 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 Mean Deviation?' This will help you understand the difference between adding/subtracting values and multiplying/dividing values, giving you a complete picture of how data transformations affect statistical measures.