Change units without changing the question accidentally

Try

A temperature conversion changes numbers without changing the physical temperatures. A logarithm changes how differences are compared: equal distances on a log scale represent equal ratios on the original scale. Predict which transformation could help when variation grows in proportion to the measurement’s magnitude.

Predict

If every value becomes y=3x+5, predict the new mean and standard deviation. The added five shifts everyone equally, so it should not change distances between observations. Then consider a negative multiplier: order reverses, but a standard deviation cannot become negative.

Experiment

Suppose a dataset has mean12 and SD4. Under y=3x+5, the mean becomes41 and SD12. Under y=−2x+7, the mean becomes−17 and SD8. Verify the spread rule directly with a small dataset of your own.

For the positive observations1 and100, base-10 logs are0 and2. Their average log is1; exponentiating gives10. The arithmetic mean of the original observations is50.5. Write both operations in order to see that transforming an average differs from averaging transformed values. This is a concrete warning against back-transforming a fitted average without considering its target.

Understand

For an affine transformation Y=a+bX, mean changes to a+b times the original mean, more precisely E(Y)=a+bE(X). Standard deviation becomes the absolute value of b times the original SD, and variance becomes b² times the original variance. Here a is the additive shift and b the scale multiplier. These formulas hold for sample summaries as well as population moments when defined. The case b=0 collapses all values to a constant.

A logarithm requires positive input. For base10, log10(y)=u means y=10^u. Equal differences in u correspond to equal multiplicative factors in y. A fitted relationship log10(y)=1+0.2x corresponds to y=10·10^(0.2x): increasing x by one multiplies the fitted value by10^0.2, approximately1.585. It does not add0.2 original y units.

The exponential of the average logarithm is the geometric mean. It can be a useful multiplicative center but is not the arithmetic mean in general. Similarly, exponentiating a predicted mean log response need not estimate the conditional arithmetic mean of the original response. Residual variability and its distribution affect that target. This unit teaches the distinction without supplying a universal correction factor.

Logs do not accept zeros or negative values in the real-number setting. Adding a constant before logging is a modeling decision that changes ratios, particularly near zero. It needs a scientific justification and a stated interpretation. A transformation should address the question or a diagnosed pattern, not just improve an attractive fit statistic.

After transforming, inspect the new graph and residuals, record the units or dimensionless ratios inside logarithms, and explain how predictions return to the original scale. If y has physical units, a logarithmic expression can be understood relative to a declared reference unit or scale.

Apply

Create a two-column table showing original and transformed values for1,10,100. Explain why equal log spacing corresponds to multiplication by ten. In the written practice, distinguish a fitted original-scale value from a guaranteed unbiased mean prediction. For a dashboard, state explicitly whether a displayed “average growth factor” is arithmetic or geometric.

Continue

Next develop probability models. Their assumptions will provide the bridge between descriptive data and uncertainty about repeated samples.