Interval Scale
The interval scale is the third level in Stanley Smith Stevens's 1946 typology: values are ordered and the distances between them are equal and meaningful, but the zero point is arbitrary and does not mean "none of the quantity." Temperature in Celsius or Fahrenheit is the textbook case: the 10-degree gap between 20 and 30 is the same physical step as between 70 and 80, yet 0 degrees Celsius is a convention (the freezing point of water), not an absence of heat.
The permissible transformation on an interval scale is a positive linear one, y = a × x + b, which changes the unit and shifts the origin without disturbing anything that matters. Stevens called this the linear group. Converting Celsius to Fahrenheit (multiply by 9/5 and add 32) is exactly such a transformation, and it shows why the scale preserves differences and ratios of differences but not ratios of values. A change from 20 to 30 degrees Celsius is half as large as a change from 20 to 40, and that statement holds in Fahrenheit too, because ratios of intervals are invariant; but 40 degrees Celsius is not "twice as hot" as 20, since the same two temperatures in Fahrenheit are 104 and 68, a ratio of about 1.5.
Because differences are meaningful, the full toolkit of means, standard deviations, Pearson correlation, t-tests, regression, and analysis of variance is licensed — what is missing is only the right to form ratios of the raw values. The sample mean of daily temperatures and its standard deviation are interpretable, and z-scores work because they use only differences from the mean divided by a spread. What fails is multiplicative language: percent increase in Celsius temperature, the geometric mean, and the coefficient of variation (standard deviation divided by mean) all change when the arbitrary zero moves, so a coefficient of variation computed on Celsius readings differs from the one computed on Fahrenheit readings and so carries no real information.
Real interval-scale examples extend beyond temperature to calendar dates, clock times, and many standardized scores. Years on the Gregorian calendar are interval data: the gap between 1900 and 1950 equals that between 1950 and 2000, but year 0 is a historical choice, so 2000 is not "twice as late" as 1000. Longitude, pH (which is also logarithmic), and standardized test scores such as IQ, set to a mean of 100 and a standard deviation of 15 or so, are treated as interval measures with arbitrary origins, although whether psychological scores truly have equal intervals is an ongoing debate in psychometrics. In machine learning, standardizing a feature (subtracting the mean and dividing by the standard deviation) is an interval-scale operation that makes features comparable regardless of where their zeros sit.
The pitfalls come from reading the zero as real and from over-claiming equal intervals. Saying that a 40-degree day is "twice as warm" as a 20-degree day, or that a 120 IQ person is "20% smarter" than someone at 100, uses ratios the scale cannot support. Time-of-day is subtly cyclic, so averaging clock times near midnight needs circular statistics. And Stevens's categories are not always cleanly assigned: psychological and social measures are often assumed to be interval because the assumption makes powerful methods available, yet that assumption is a modeling choice that should be tested, with sensitivity analyses in rank-based or ordinal models, rather than stated as a fact.
Read the interval scale as measurement where the ruler's spacing is real but its starting point is a convention: differences, averages, and spreads are all legitimate, while any claim of the form "twice as much" quietly depends on a zero that was never an absence.