Understanding Statistical Transformations: Mean and Variance

This question asks us to determine the new mean and variance of a dataset when every observation is multiplied by a constant. We are given the original mean and variance for 10 observations and the scaling factor.

Let's break down the problem:

• We have 10 observations.
• The original mean is 4.
• The original variance is 2.
• Every observation is multiplied by 2.

We need to find the mean and variance of this new series of observations.

Effect of Scaling on Mean and Variance

When each observation in a dataset is transformed by a linear transformation, the mean and variance also change in a predictable way. Consider a dataset with observations \(x_1, x_2, \ldots, x_n\). If each observation is transformed to \(y_i = ax_i + b\), where \(a\) and \(b\) are constants, then the new mean (\(\bar{y}\)) and new variance (\(\sigma_y^2\)) are related to the original mean (\(\bar{x}\)) and original variance (\(\sigma_x^2\)) as follows:

• New Mean: \(\bar{y} = a\bar{x} + b\)
• New Variance: \(\sigma_y^2 = a^2\sigma_x^2\)

In our specific problem, the transformation is simply multiplying each observation by 2. This means our transformation is \(y_i = 2x_i\). Comparing this to the general form \(y_i = ax_i + b\), we have \(a = 2\) and \(b = 0\). The original mean (\(\bar{x}\)) is 4 and the original variance (\(\sigma_x^2\)) is 2. The number of observations (n=10) is relevant for calculating the original statistics but not directly needed for the transformation rules themselves.

Calculating the New Mean and Variance

Using the rules for the effect of scaling on mean and variance, we can calculate the new statistics:

Calculating the New Mean:

The new mean is given by \(\bar{y} = a\bar{x} + b\).

Substituting the values \(a=2\), \(b=0\), and \(\bar{x}=4\):

\(\bar{y} = (2)(4) + 0\)

\(\bar{y} = 8\)

So, the new mean is 8.

Calculating the New Variance:

The new variance is given by \(\sigma_y^2 = a^2\sigma_x^2\).

Substituting the values \(a=2\) and \(\sigma_x^2=2\):

\(\sigma_y^2 = (2)^2 \times 2\)

\(\sigma_y^2 = 4 \times 2\)

\(\sigma_y^2 = 8\)

So, the new variance is 8.

Therefore, the mean and the variance of the new series of observations, after multiplying every original observation by 2, are 8 and 8 respectively.

Revision Table: Key Statistics Concepts

Additional Information on Statistical Transformations

Understanding how statistics change under transformations is crucial in data analysis. Linear transformations like \(ax_i + b\) are common. While the mean is affected by both the multiplicative factor \(a\) and the additive constant \(b\), the variance (and standard deviation) are only affected by the multiplicative factor \(a\). Adding a constant \(b\) shifts the entire dataset, changing the mean but not affecting the spread (variance or standard deviation). Multiplying by \(a\) scales the data, affecting both the mean and the spread.

These rules apply regardless of the number of observations, as long as the original mean and variance are known. They are fundamental concepts in statistics and probability.