Understanding Variance and Adding a Constant

The question asks about how the variance of a set of observations changes when a constant value is added to each observation. We are given that the variance of 25 observations is 4. We need to find the new variance after adding 2 to each of these 25 observations.

Variance Definition

Variance is a measure of how spread out a set of data is from its mean. It is calculated as the average of the squared differences from the mean.

Let $X_1, X_2, \dots, X_n$ be a set of $n$ observations with mean $\bar{X}$. The variance, denoted by $\sigma^2$ or $Var(X)$, is given by:

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

Effect of Adding a Constant on Variance

Consider a new set of observations $Y_1, Y_2, \dots, Y_n$, where each new observation $Y_i$ is obtained by adding a constant $c$ to the original observation $X_i$. So, $Y_i = X_i + c$.

Let's see how the mean changes. The new mean, $\bar{Y}$, is:

$\bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i = \frac{1}{n} \sum_{i=1}^{n} (X_i + c) = \frac{1}{n} \left( \sum_{i=1}^{n} X_i + \sum_{i=1}^{n} c \right) = \frac{1}{n} \left( \sum_{i=1}^{n} X_i + nc \right)$

$\bar{Y} = \frac{1}{n} \sum_{i=1}^{n} X_i + \frac{nc}{n} = \bar{X} + c$

So, adding a constant $c$ to each observation shifts the mean by the same constant $c$.

Now, let's look at the variance of the new observations $Y_i$. The variance of $Y$ is:

$Var(Y) = \frac{1}{n} \sum_{i=1}^{n} (Y_i - \bar{Y})^2$

Substitute $Y_i = X_i + c$ and $\bar{Y} = \bar{X} + c$ into the formula:

$Var(Y) = \frac{1}{n} \sum_{i=1}^{n} ((X_i + c) - (\bar{X} + c))^2$

$Var(Y) = \frac{1}{n} \sum_{i=1}^{n} (X_i + c - \bar{X} - c)^2$

$Var(Y) = \frac{1}{n} \sum_{i=1}^{n} (X_i - \bar{X})^2$

This is exactly the formula for the variance of the original observations, $Var(X)$.

Thus, we conclude that adding a constant to each observation does not change the variance of the observations.

Applying the Property to the Problem

In this question, we are given:

• Number of observations ($n$) = 25
• Original variance ($Var(X)$) = 4
• Constant added to each observation ($c$) = 2

Let the original observations be $X_i$ and the new observations be $Y_i$. Then $Y_i = X_i + 2$.

According to the property discussed, the variance of the new observations $Var(Y)$ is equal to the variance of the original observations $Var(X)$.

$Var(Y) = Var(X) = 4$

Therefore, the new variance of the resulting observations is 4.

Summarizing the Result

Adding a constant value to every observation in a dataset shifts the entire dataset along the number line. However, the spread of the data points relative to their new mean remains the same as their spread relative to the original mean. Variance measures this spread, so it is unaffected by the addition or subtraction of a constant.

Revision Table

Additional Information on Data Transformations and Variance

While adding a constant does not affect variance, other transformations do. For instance:

• Multiplying by a constant: If each observation $X_i$ is multiplied by a constant $k$ to get $Z_i = kX_i$, the new variance is $Var(Z) = k^2 Var(X)$. The standard deviation becomes $|k| \sigma_X$.
• Linear Transformation: If $W_i = aX_i + b$, where $a$ and $b$ are constants, then $Var(W) = a^2 Var(X)$. Adding the constant $b$ doesn't change variance, but multiplying by $a$ scales the variance by $a^2$.

Understanding these properties is crucial for analysing how data transformations impact measures of central tendency and dispersion.