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Mean and unit conversion

About this app

365 morning temperatures from Reutlingen cross a bridge from °C to °F one by one, or only their mean x̄ does. Both land at 47.12 °F: the arithmetic mean is invariant under linear transformations, m·x̄ + c. Kelvin as a second example, squaring as a counter-check where the shortcut misses.

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

What it is about

The arithmetic mean is compatible with linear transformations: if every value is converted with y = m·x + c, the mean of the new values is exactly m·x̄ + c. Averaging divides every term by n, the factor m comes out of the sum, and n times c divided by n is c again. So to get the mean in Fahrenheit you do not need to convert every measurement; a single calculation is enough. This only holds for linear conversions. When squaring, the mean of the squares is larger than the square of the mean.

What the app shows

A climate researcher measures the temperature in Reutlingen every morning at 8 am for a year. At the top is the annual course with 365 values, next to it the effort as 365 tiles against one. Below are the °C scale and the °F scale, joined by a bridge of 365 lines. On the long way each value travels down its line, then the values are averaged: 47.12 °F. On the shortcut only x̄ = 8.40 °C crosses the bridge and lands at the same spot.

Try it yourself

Start the long way and watch a second mound build up below while the tiles fill. Then run the shortcut: the solid diamond lands exactly inside the hollow diamond of the long way. Switch to Kelvin: the lines now run parallel and both ways hit 281.55 K. Switch to squaring: the lines cross and the shortcut misses by 50.34. Tap a day in the annual course to see its own path across the bridge.

Frequently asked questions

How does the mean change under a linear transformation?
If all values are converted with y = m·x + c, the arithmetic mean satisfies ȳ = m·x̄ + c. So you can average first and convert once. Example: x̄ = 8.40 °C gives 1.8 · 8.40 + 32 = 47.12 °F.
Why does this not work for every conversion?
The shortcut needs the factor and the constant to come out of the sum, which means a linear conversion. When squaring, the mean of the squares is larger than the square of the mean; the difference is the mean squared deviation. Square root, logarithm and reciprocal do not allow the shortcut either.
How do you convert Celsius to Fahrenheit and Kelvin?
Fahrenheit: °F = 1.8 · °C + 32, so 0 °C = 32 °F and 100 °C = 212 °F. Kelvin: K = °C + 273.15. Both conversions are linear, so ȳ = m·x̄ + c holds for both.
What does the bridge in the app show?
Each line connects a °C value with its converted value. For a linear conversion the lines form an orderly ladder without crossings, and the middle stays the middle. When squaring, the lines cross because, for example, −5 and 5 both land on 25.

Subject: Statistics

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