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

S7-SA3-0336

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 origin (adding or subtracting a constant value) to every data point in a dataset does NOT change the Mean Deviation. The Mean Deviation measures how spread out the data is from the mean, and this spread remains the same even if all values are shifted uniformly.

Simple Example

Quick Example

Imagine you have marks of 3 students: 10, 20, 30. Their mean is 20. If we add 5 marks to everyone (maybe for attendance), the new marks are 15, 25, 35. The mean is now 25. Notice how the distances of marks from their respective means (10-20=-10, 20-20=0, 30-20=10 for old; 15-25=-10, 25-25=0, 35-25=10 for new) are still the same. So, the Mean Deviation will also be the same.

Worked Example

Step-by-Step

Let's find the Mean Deviation for a dataset and then see what happens when we change the origin.

DATASET: 2, 4, 6, 8

Step 1: Calculate the Mean (x̄) of the original dataset.
x̄ = (2 + 4 + 6 + 8) / 4 = 20 / 4 = 5

--- Step 2: Calculate the absolute deviations from the mean for the original dataset.
|2 - 5| = 3
|4 - 5| = 1
|6 - 5| = 1
|8 - 5| = 3

--- Step 3: Calculate the Mean Deviation (MD) for the original dataset.
MD = (3 + 1 + 1 + 3) / 4 = 8 / 4 = 2

--- Step 4: Change the origin by adding a constant (let's add 10) to each data point.
NEW DATASET: 2+10=12, 4+10=14, 6+10=16, 8+10=18

--- Step 5: Calculate the Mean (x̄_new) of the new dataset.
x̄_new = (12 + 14 + 16 + 18) / 4 = 60 / 4 = 15

--- Step 6: Calculate the absolute deviations from the new mean for the new dataset.
|12 - 15| = 3
|14 - 15| = 1
|16 - 15| = 1
|18 - 15| = 3

--- Step 7: Calculate the Mean Deviation (MD_new) for the new dataset.
MD_new = (3 + 1 + 1 + 3) / 4 = 8 / 4 = 2

--- Answer: The Mean Deviation for both the original and the new dataset is 2. This shows that changing the origin does not affect the Mean Deviation.

Why It Matters

Understanding this concept is crucial in fields like AI/ML and Climate Science. Data scientists often 'normalize' data by shifting it (changing origin) to make calculations easier, but they need to know which statistical measures remain unchanged. This helps engineers in FinTech analyze market trends without being misled by simple shifts in currency values, and helps physicists understand data from experiments regardless of calibration shifts.

Common Mistakes

MISTAKE: Thinking that if the mean changes, the Mean Deviation must also change. | CORRECTION: While the mean does change when the origin changes, the *spread* of the data points around their *new* mean remains the same, hence Mean Deviation doesn't change.

MISTAKE: Confusing change of origin with change of scale. | CORRECTION: Change of origin means adding/subtracting a constant. Change of scale means multiplying/dividing by a constant. Mean Deviation *is* affected by change of scale.

MISTAKE: Not taking absolute values when calculating deviations. | CORRECTION: Mean Deviation uses the absolute difference between each data point and the mean. Forgetting absolute values will lead to an incorrect Mean Deviation (often zero).

Practice Questions

Try It Yourself

QUESTION: A set of cricket scores are 25, 30, 35. If 10 runs are added to each score, what will be the effect on the Mean Deviation? | ANSWER: The Mean Deviation will remain the same.

QUESTION: The Mean Deviation of a dataset {10, 12, 14, 16, 18} is 2. If we subtract 5 from each number, what will be the new Mean Deviation? | ANSWER: The new Mean Deviation will still be 2.

QUESTION: For the data set {5, 10, 15}, calculate the Mean Deviation. Then, add 2 to each data point and recalculate the Mean Deviation. Show your steps. | ANSWER: Original Mean = (5+10+15)/3 = 10. Deviations: |5-10|=5, |10-10|=0, |15-10|=5. MD = (5+0+5)/3 = 10/3. New data (after adding 2): {7, 12, 17}. New Mean = (7+12+17)/3 = 12. Deviations: |7-12|=5, |12-12|=0, |17-12|=5. New MD = (5+0+5)/3 = 10/3. Both are 10/3.

MCQ

Quick Quiz

If each value in a dataset is increased by 7, what happens to its Mean Deviation?

It increases by 7

It decreases by 7

It remains unchanged

It gets multiplied by 7

The Correct Answer Is:

Adding a constant to every data point (change of origin) shifts the entire dataset, including the mean, by the same amount. The relative distances of data points from the mean, which Mean Deviation measures, remain unaffected.

Real World Connection

In the Real World

Imagine a sensor in a smart city project, like those used to measure air quality in Delhi. If the sensor is slightly miscalibrated and consistently reads 5 units higher than actual, this is a 'change of origin'. While the average reading will be 5 units higher, the *variability* or *spread* of the air quality readings (how much they change day-to-day) would still be accurately captured by the Mean Deviation, even with the offset.

Key Vocabulary

Key Terms

Mean Deviation: The average of the absolute differences between each data point and the mean | Origin: The reference point or starting value in a dataset | Constant: A value that does not change | Absolute Deviation: The distance of a data point from the mean, always positive | Dataset: A collection of related data points

What's Next

What to Learn Next

Next, explore 'What is the Effect of Change of Scale on Mean Deviation?'. This will help you understand how multiplying or dividing data points affects Mean Deviation, which is a different scenario from just adding or subtracting.