Frequency table

A frequency table pairs each category, or each class interval, with the number of observations in it. The counts add to $n$. A relative frequency column divides each count by $n$ and adds to 1. The table is the numeric form of a bar graph or a histogram and holds exactly the same information, which is why the table is usually the more precise thing to report.

Thirty exam scores fall into the classes 60 to 70, 70 to 80, 80 to 90 and 90 to 100 with counts 4, 9, 12 and 5, under the usual rule that a class holds values at or above its lower bound and below its upper one. The relative frequency of the 80 to 90 class is $12 / 30 = 0.40$. Estimating the mean from the table means assuming every score sits at its class midpoint: $\frac{65(4) + 75(9) + 85(12) + 95(5)}{30} = \frac{2430}{30} = 81$.

The wrong sentence is "the mean is $(4 + 9 + 12 + 5)/4 = 7.5$." That averages the counts. The 7.5 is the average number of students per class interval, not the average score. One column says how many and the other says how much, and only the second one is the variable.

Grouping is where the information goes. That 81 is an estimate, not the mean. Had every score sat at the bottom of its class the mean would be 76, and at the top of its class it would be 86, so the table pins the true mean only to somewhere between 76 and 86. No further arithmetic on the table narrows that, because the individual values are gone. An ungrouped table, one row per distinct value, keeps them.

For a categorical variable there is no mean to estimate at all, so the table supports the mode and the proportions and stops there. Tables for one categorical variable are topic 1.3.