AI Tutor
Imagine a weather station records a week of daily temperatures in Celsius with a mean of 20∘C20^\circ\text{C} and a standard deviation of 3∘C3^\circ\text{C}.
To report these values in Fahrenheit, you apply the formula F=1.8C+32F = 1.8C + 32 to every single data point.
Do you need to convert all individual readings first to find the new summary stats, or can you transform the original mean and spread directly?

What is Linear Coding?

Applying an equation of the form y=ax+by = ax + b to every value in a dataset is called a linear transformation or data coding.
Here, aa represents a scale factor (multiplying or dividing data), while bb represents a location shift (adding or subtracting a constant offset).
Historically, before computers existed, 19th-century statisticians like Karl Pearson relied on linear coding by hand to simplify calculations with enormous numbers.
How does shifting the entire dataset to the right or left affect where the middle lands?

Effects on Measures of Center

Every measure of central tendencyβ€”including the mean (ar{x}) and the medianβ€”is affected by both scaling and shifting.
If yi=axi+by_i = ax_i + b, the new mean is given by yˉ=axˉ+b\bar{y} = a\bar{x} + b.