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When dealing with linear transformations in statistics, understanding how the mean changes is crucial. The process involves transforming the data linearly using an equation such as \( y_i = ax_i + b \). Here, \( a \) and \( b \) are constants. The original data has a mean \( \bar{x} \), which represents the central tendency.
The transformed mean \( \bar{y} \) is obtained using the same linear equation applied to individual data points. It results in the equation \( \bar{y} = a \bar{x} + b \). This means the new mean is a scaled version of the original mean, amplified by \( a \), and adjusted by adding \( b \). This change reflects how the shift and scale affect the central location of the data.
Key points to remember:

• Scaling the data by \( a \) changes the spread.
• Adding \( b \) affects the shift of the mean.

Variance is a measure of the spread of data values around the mean. When a linear transformation is applied to data, its variance also changes. The transformation \( y_i = ax_i + b \) affects the variance of the dataset, which is denoted \( s_y^2 \) for the transformed data.
The transformation impacts variance through scaling, where \( s_y^2 = a^2 s_x^2 \). Here, \( s_x^2 \) is the original variance before transformation. Notice that only the scaling factor \( a^2 \) impacts variance, while the addition of \( b \) does not.
This is because variance, being a measure of dispersion, is unaffected by uniform shifts:

• Larger values of \( a \) increase variance, making the data more spread out.
• Smaller values of \( a \) decrease variance, reducing the spread.
• Addition constant \( b \) has no effect on the variance.

When converting the standard deviation through a linear transformation, you modify how dispersed the data is around the mean. Standard deviation, denoted \( s \), is the square root of variance. When transforming data using the formula \( y_i = ax_i + b \), the conversion impacts standard deviation as \( s_y = |a| s_x \).
This transformation shows that the standard deviation scales linearly with \( |a| \), while the addition of \( b \) does not alter it. It's crucial to remember that although standard deviation is a measure of spread, transformations only affect it through scaling. Here's why that matters:

• Multiplying the data by \( a \) scales the deviation proportionally.
• Adding a constant \( b \) shifts the data but doesn't impact the deviation.

Converting temperatures between scales, such as Celsius and Fahrenheit, involves linear transformations that apply to both means and standard deviations. For Celsius to Fahrenheit conversions, the formula used is \( F = \frac{9}{5}C + 32 \). This represents a linear transformation.
To convert the mean temperature, apply the formula to the mean in Celsius. If the mean Celsius temperature is \( \bar{C} \), the Fahrenheit mean \( \bar{F} \) becomes \( \frac{9}{5} \times \bar{C} + 32 \). This transformation affects the central figure but keeps the relational dynamics of temperature consistent.
For standard deviation, the focus is only on the scaling factor of \( \frac{9}{5} \). This is because a shift (constant addition like \( +32 \)) does not affect deviation. Thus, the Fahrenheit standard deviation \( s_F \) is \( \frac{9}{5} \times s_C \), where \( s_C \) is the initial Celsius deviation.

• Transforming mean emphasizes shifts and scaling in temperature.
• Converting deviation highlights the effect of scaling without shifts.