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What is the Effect of Change of Origin 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?
The 'origin' in data refers to the starting point or zero value. Changing the origin means adding or subtracting a constant value from every data point. The key effect is that measures of dispersion (like range, variance, and standard deviation) remain unchanged because they focus on the spread between data points, not their absolute position.
Simple Example
Quick Example
Imagine you have the daily pocket money of five friends: ₹20, ₹25, ₹30, ₹35, ₹40. The difference between the highest and lowest (range) is ₹20. Now, if everyone gets an extra ₹10 from their parents (new data: ₹30, ₹35, ₹40, ₹45, ₹50), the range is still ₹20. Adding a constant amount doesn't change how spread out the numbers are, just where they are located on the number line.
Worked Example
Step-by-Step
Let's calculate the standard deviation for a set of marks and then see what happens when we change the origin.
DATA SET A (Marks): 10, 12, 14, 16, 18
1. Calculate the Mean (x̄): (10+12+14+16+18) / 5 = 70 / 5 = 14
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2. Calculate the squared differences from the Mean:
(10-14)^2 = (-4)^2 = 16
(12-14)^2 = (-2)^2 = 4
(14-14)^2 = (0)^2 = 0
(16-14)^2 = (2)^2 = 4
(18-14)^2 = (4)^2 = 16
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3. Sum of squared differences: 16 + 4 + 0 + 4 + 16 = 40
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4. Calculate Variance (σ^2): Sum of squared differences / N = 40 / 5 = 8
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5. Calculate Standard Deviation (σ): sqrt(Variance) = sqrt(8) ≈ 2.83
Now, let's change the origin by adding 5 to each mark (DATA SET B):
DATA SET B (Marks + 5): 15, 17, 19, 21, 23
1. Calculate the Mean (x̄): (15+17+19+21+23) / 5 = 95 / 5 = 19
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2. Calculate the squared differences from the Mean:
(15-19)^2 = (-4)^2 = 16
(17-19)^2 = (-2)^2 = 4
(19-19)^2 = (0)^2 = 0
(21-19)^2 = (2)^2 = 4
(23-19)^2 = (4)^2 = 16
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3. Sum of squared differences: 16 + 4 + 0 + 4 + 16 = 40
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4. Calculate Variance (σ^2): Sum of squared differences / N = 40 / 5 = 8
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5. Calculate Standard Deviation (σ): sqrt(Variance) = sqrt(8) ≈ 2.83
Answer: The standard deviation remains approximately 2.83 in both cases, showing that changing the origin does not affect measures of dispersion.
Why It Matters
Understanding this concept is crucial in fields like AI/ML, where data is often 'normalized' by shifting its values without changing its spread. In Physics, when you shift the zero point of a scale (like temperature or height), the differences between readings remain the same. This knowledge helps engineers design robust systems and helps economists analyze market volatility accurately.
Common Mistakes
MISTAKE: Assuming all statistical measures are affected by changing the origin. | CORRECTION: Remember that measures of central tendency (mean, median, mode) change, but measures of dispersion (range, variance, standard deviation, quartile deviation) do not.
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, which DOES affect measures of dispersion.
MISTAKE: Recalculating all dispersion measures from scratch after adding a constant. | CORRECTION: If you know the original dispersion, you can confidently state it remains the same after a change of origin, saving time and effort.
Practice Questions
Try It Yourself
QUESTION: A dataset has a standard deviation of 5. If 10 is added to every data point, what will be the new standard deviation? | ANSWER: 5
QUESTION: The range of daily temperatures in Delhi for a week was 12 degrees Celsius. If the average temperature for that week increased by 3 degrees due to a heatwave, what would be the new range? | ANSWER: 12 degrees Celsius
QUESTION: A student's test scores are 60, 70, 80, 90. Calculate the variance. Now, if the teacher adds 5 bonus marks to each score, what is the new variance? Show your working for both. | ANSWER: Original Variance = 125. New Variance = 125. (Working: Original mean = 75, sum of (x-mean)^2 = (60-75)^2 + (70-75)^2 + (80-75)^2 + (90-75)^2 = 225 + 25 + 25 + 225 = 500. Variance = 500/4 = 125. New scores = 65, 75, 85, 95. New mean = 80. Sum of (x-mean)^2 = (65-80)^2 + (75-80)^2 + (85-80)^2 + (95-80)^2 = 225 + 25 + 25 + 225 = 500. New Variance = 500/4 = 125.)
MCQ
Quick Quiz
Which of the following measures of dispersion is NOT affected by a change of origin?
Mean Deviation
Standard Deviation
Range
All of the above
The Correct Answer Is:
D
All measures of dispersion (Range, Quartile Deviation, Mean Deviation, Variance, Standard Deviation) only care about the spread or differences between data points. Adding or subtracting a constant from every data point shifts the entire dataset but does not change the distances between the points, hence their dispersion remains unchanged.
Real World Connection
In the Real World
When weather scientists analyze temperature data, they might convert Celsius to Kelvin (by adding 273.15). While the actual temperature numbers change, the 'spread' or variation in daily temperatures (e.g., how much temperature changes from morning to evening) remains the same, which is crucial for climate modeling and predicting extreme weather events.
Key Vocabulary
Key Terms
ORIGIN: The starting point or zero value of a dataset. | DISPERSION: How spread out or varied data points are. | CONSTANT: A fixed value that does not change. | VARIANCE: The average of the squared differences from the mean. | STANDARD DEVIATION: The square root of the variance, showing typical deviation from the mean.
What's Next
What to Learn Next
Now that you understand how change of origin affects dispersion, your next step should be to learn about the 'Effect of Change of Scale on Measures of Dispersion'. This will complete your understanding of how basic transformations impact statistical measures, which is vital for data analysis.
