Measurement vs Count Data

Three things get run together and they are not the same. The variable is height. The observational unit is student 12. The measurement is the one recorded value, 168 centimeters, that ties them together. Every measurement arrives with a scale, a unit, and a precision, and all three were chosen by whoever collected the data.

Reliability and validity are separate properties and a measurement can have one without the other. Reliability is agreement between repeated measurements of the same thing. Validity is agreement with the quantity you meant to measure. A bathroom scale weighs a 150 pound object five times and reads 153.1, 153.0, 153.2, 153.1, 153.1. The mean is 153.1 and the standard deviation is 0.0707 pounds, so the scale is about as reliable as a bathroom scale gets. It is also wrong by 3.1 pounds every single time, so it is not valid.

"We measured it five times and got the same answer, so the measurement must be accurate." Repetition exposes random error only. A constant offset shifts every reading the same way, so it never shows up in the spread, and averaging does not touch it: at that same spread, 100 weighings put the standard error of the mean at 0.0071 pounds while the 3.1 pound gap sits exactly where it was. Precision improved. Accuracy did not. The survey version of the same problem is response bias, where answers miss the truth in one direction.

Changing units changes some summaries and leaves others alone. Recording that height as $168 / 2.54 = 66.14$ inches divides the mean and the standard deviation by 2.54, leaves the shape of the distribution unchanged, and leaves any correlation exactly where it was. Precision is a separate limit: recording to the nearest centimeter builds in rounding, which is one reason repeated measurements of one object rarely match to the last digit.

Topic 1.7 covers calculating summary statistics in different units of measurement, and the framework's claim there is the narrow one, that changing units changes the calculated statistics.

A count is a whole number, zero or more, saying how many. Counts are quantitative and discrete: you can add them, average them, and 3 really is one more than 2. What makes them feel categorical is where they come from, since most counts are produced by sorting units into categories and tallying.

Two different objects get called counts, and the observational unit tells them apart. Eight households report 0, 0, 1, 1, 1, 2, 2 and 3 pets. The unit is a household, the count is a variable measured on each one, and $\bar{x} = 10/8 = 1.25$ pets is a real average of eight measurements. Now tally 100 students by favorite sport and get 40, 25 and 35. The unit is still a student, but those three numbers summarize one categorical variable rather than measuring three things: they add to $n = 100$, and their average of about 33.3 is the mean category size, which describes no student at all.

The wrong sentence is "the counts 40, 25 and 35 are categorical data." Favorite sport is the categorical variable. The tallies are frequencies, and frequencies are numbers. The type belongs to the thing being counted, not to the tally, which is exactly why a chi-square procedure runs on counts while the variable it is testing is categorical.

A count depends on how many units you counted, so raw counts mislead across groups of different sizes. 45 walkers in a 300-student school is a larger share than 60 walkers in a 600-student school, since $45/300 = 0.15$ against $60/600 = 0.10$, and converting to a proportion or a rate is the fix. The exception runs the other way: chi-square needs the counts themselves, because the expected counts it compares against are counts, so feeding it percentages produces a statistic that answers nothing.

Tables and summary statistics for one categorical variable are topic 1.3. The chi-square tests for homogeneity and independence, topics 3.14 and 3.15, are where counts carry the inference.