Understanding Variance and the Effect of Scaling

This problem asks how the variance of a dataset changes when each observation is multiplied by a constant factor. Variance is a key measure of the spread or dispersion of a dataset. It tells us, on average, how much each observation differs from the mean.

We are given:

• Number of observations: 20
• Original variance ($\sigma^2$): 5
• Each observation is multiplied by a constant: 3

We need to find the new variance after this transformation.

How Variance Changes with Multiplication by a Constant

When each observation in a dataset is multiplied by a constant value, say \(k\), the variance of the new dataset is related to the original variance by a specific formula. If the original observations are \(x_1, x_2, \dots, x_n\) with variance \(\sigma_{old}^2\), and the new observations are \(y_i = k \times x_i\), then the new variance, \(\sigma_{new}^2\), is given by:

\[ \sigma_{new}^2 = k^2 \times \sigma_{old}^2 \]

This formula shows that multiplying each observation by \(k\) results in the variance being multiplied by the square of the constant, \(k^2\). This is because variance is calculated using squared differences from the mean, and scaling each observation by \(k\) scales these differences by \(k\), so the squared differences are scaled by \(k^2\).

Calculating the New Variance

Using the given information and the formula:

• Original Variance ($\sigma_{old}^2$) = 5
• Constant (k) = 3

The new variance ($\sigma_{new}^2$) is:

\[ \sigma_{new}^2 = 3^2 \times 5 \]

\[ \sigma_{new}^2 = 9 \times 5 \]

\[ \sigma_{new}^2 = 45 \]

Therefore, the new variance of the resulting observations is 45.

Step-by-Step Solution

1. Identify the original variance: \(\sigma_{old}^2 = 5\).
2. Identify the constant by which each observation is multiplied: \(k = 3\).
3. Recall or apply the formula for the new variance when observations are scaled by \(k\): \(\sigma_{new}^2 = k^2 \times \sigma_{old}^2\).
4. Substitute the values into the formula: \(\sigma_{new}^2 = 3^2 \times 5\).
5. Calculate the result: \(\sigma_{new}^2 = 9 \times 5 = 45\).

Summary of Result

The original variance was 5. When each observation is multiplied by 3, the new variance becomes 45.

Revision Table: Effect of Constants on Statistical Measures

Additional Information on Variance and Standard Deviation

Variance and Standard Deviation are fundamental concepts in statistics used to quantify the spread or dispersion of a set of data points around their mean. A low variance or standard deviation indicates that data points are generally close to the mean, while a high variance or standard deviation indicates that data points are spread out over a wider range of values.

• Variance (\(\sigma^2\)): It is the average of the squared differences from the Mean. It is calculated as: \[ \sigma^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n} \] (for population variance) or \[ s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1} \] (for sample variance). It is measured in squared units of the original data.
• Standard Deviation (\(\sigma\)): It is the square root of the variance. It brings the measure of spread back to the original units of the data. \[ \sigma = \sqrt{\sigma^2} \] or \[ s = \sqrt{s^2} \] Standard deviation is often easier to interpret than variance because it's in the same units as the data.

Understanding how these measures change under linear transformations (like adding a constant or multiplying by a constant) is crucial for data analysis and interpretation.