If each of the observation $x_1, x_2, \ldots, x_n$ is increased by '$a$', where $a$ is a negative or positive number, show that the variance remains unchanged.

Question

Accepted Answer

> [!TLDR]
> Adding a constant $a$ to every observation shifts the entire data set by $a$, so the mean also shifts by $a$. Since variance measures spread around the mean, and every point and the mean shift by the same amount, the deviations $x_i - \bar{x}$ stay exactly the same — hence variance is unchanged.

## Why this works — the intuition

Variance is a measure of *spread*, not location. If you add the same number to every data point, you slide the whole distribution left or right along the number line. The distances between points remain identical, and crucially, the distance from each point to the new mean is the same as before. So the "scatter" doesn't change — only the centre moves.

This is a fundamental property: variance is **translation invariant**. It doesn't care where the data sits, only how spread out it is.

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## Step-by-step proof

**1. Original variance formula**

For the original observations $x_1, x_2, \ldots, x_n$, the variance is defined as:

$$
\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2
$$

where $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$ is the original mean.

**2. New observations after adding $a$**

Let the new observations be $y_i = x_i + a$ for each $i = 1, 2, \ldots, n$.

**3. New mean**

The new mean $\bar{y}$ is:

$$
\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{1}{n} \sum_{i=1}^{n} (x_i + a) = \frac{1}{n} \left( \sum_{i=1}^{n} x_i + na \right) = \bar{x} + a
$$

So the mean also shifts by exactly $a$.

**4. New deviations from the new mean**

For each $i$:

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

The $a$'s cancel out perfectly. Every deviation is identical to the original deviation.

> [!WARNING]
> A common mistake is to think the deviations change because "the numbers got bigger". But the mean also got bigger by the same amount — the difference stays zero. Always check what happens to *both* the data point and the mean.

**5. New variance**

The new variance $\sigma_y^2$ is: …

Q.If each of the observation $x_1, x_2, \ldots, x_n$ is increased by ' $a$ ', where $a$ is a negative or positive number, show that the variance remains unchanged.

Concept understanding — Effect of Scaling Variance

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Intuition first

The precise statement

What about adding a constant?

A quick example

Common mistake to avoid

Why this works — the intuition

Step-by-step proof

Q.If each of the observation x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​ is increased by 'aaa', where aaa is a negative or positive number, show that the variance remains unchanged.

Why this works — the intuition

Step-by-step proof

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Q.If each of the observation $x_1, x_2, \ldots, x_n$ is increased by ' $a$ ', where $a$ is a negative or positive number, show that the variance remains unchanged.

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