All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks

Q: Wait, if the mean increases by 10, why doesn't the variance also increase? Shouldn't the whole distribution change?

That is a fantastic question! Think of variance not as the position of the distribution, but as its 'shape' or 'width'. When you add 10 to every student's marks, you are shifting the entire distribution to the right on the number line—this is a 'change of origin'. However, the relative distances between the students' marks remain exactly the same. Since variance measures the spread (the relative distances), it remains invariant to this shift.

Q: What if the teacher had multiplied the marks by 10 instead of adding 10? Would the variance still be the same?

Great intuition! If the teacher multiplied the marks by 10, that would be a 'change of scale'. In that case, the variance would indeed change—it would be multiplied by $10^2$ (or 100). Variance is independent of the change of origin (addition/subtraction), but it is definitely affected by the change of scale (multiplication/division).

Q: Does this mean standard deviation also stays the same?

Absolutely! Since standard deviation is just the square root of the variance ($\sigma = \sqrt{Var}$), and the variance remains unchanged by the addition of a constant, the standard deviation also remains perfectly constant. Both measures are 'robust' against changes of origin.

Question

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

Options:

(A) mean

(B) median

(C) mode

(D) variance

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a classroom. The teacher has just handed back the math exams, and the results are, well, not great. To lift everyone's spirits, the teacher makes a generous decision: every single student will receive a grace mark of $10$.

We have a set of original marks, $x_1, x_2, \dots, x_n$. These marks form a distribution—a beautiful, bell-shaped curve representing the class's performance. The center of this curve is the mean, defined as:

$$\bar{x} = \frac{\sum x_i}{n}$$

### The Change of Origin

When the teacher adds $10$ to every student's score, we create a new set of marks, $y_i$, where $y_i = x_i + 10$. In the world of statistics, this is known as a **Change of Origin**.

Visually, imagine the entire bell curve picking itself up and sliding $10$ units to the right on the number line. The shape of the curve—its height, its width, and its spread—remains identical. It has simply moved to a new neighborhood.

### Why the Mean Follows

It is intuitive that the mean, which represents the center of the data, must move with the data. If every student gets $10$ more marks, the average score of the class must also increase by $10$.

Mathematically, the new mean $\bar{y}$ is calculated as:

$$\bar{y} = \frac{\sum (x_i + 10)}{n}$$

By splitting the summation, we get:

$$\bar{y} = \frac{\sum x_i}{n} + \frac{10n}{n}$$

This simplifies beautifully to $\bar{y} = \bar{x} + 10$. The center has shifted, just as we expected.

### The Invariance of Variance

Now, we arrive at the heart of the mystery: the variance. Variance, denoted as $\sigma^2$ or $Var(X)$, is defined as the average of the squared deviations from the mean:

$$Var(X) = \frac{\sum (x_i - \bar{x})^2}{n}$$

This formula captures the 'spread' of the data. It measures how far, on average, each student's score is from the class average.

Let's calculate the variance of our new, shifted marks, $Var(Y) = \frac{\sum (y_i - \bar{y})^2}{n}$. If we substitute our expressions for $y_i$ and $\bar{y}$, something magical happens.

The deviation term becomes:

$$(y_i - \bar{y}) = (x_i + 10) - (\bar{x} + 10)$$

When we open those brackets, the $+10$ and $-10$ cancel out perfectly. We are left with $(x_i - \bar{x})$.

### The Elegant Conclusion

Because the deviation of the new data is exactly the same as the deviation of the old data, the squared deviations are also identical. When we sum them up and divide by $n$, we find that:

$$Var(Y) = Var(X)$$

The spread of the curve did not change because the relative distances between the students' marks remained constant. Shifting the entire class by $10$ marks didn't make the class performance more or less consistent; it just made everyone look a little better.

And that, my friend, is why **variance is independent of the change of origin**. It is a fundamental property of dispersion that remains untouched, no matter how much you shift the starting point.

Animated Solution for Mathematics - Statistics: All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

Select Answer:

Visualized Solution

The Original Marks Distribution

Original Mean

Adding Grace Marks

Change of Origin

The Shifted Distribution

New Mean

Median and Mode

What is Variance?

Original Variance Formula

New Variance Setup

Deviations Remain Unchanged

Simplifying the Deviation

Variance is Independent of Origin

Final Conclusion

The Sigma Insight: Measures of Dispersion

Analyzing the Setup

The Change of Origin

Why the Mean Follows

The Invariance of Variance

The Elegant Conclusion

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