Read two-way frequency tables and compute relative frequencies.
Free, timed SHSAT practice sets on two-way tables— every question includes a full worked explanation, and there's no account required.
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A two-way table sorts the same people by two things at once, such as grade (rows) and pet preference (columns). Every row and every column adds up to its total.
Relative frequency: the words after "of" tell you the whole, so they give the denominator. "What fraction of 7th graders prefer cats?" means divide by the 7th grade row total. If 20 of 50 seventh graders prefer cats, the answer is \(\dfrac{20}{50} = \dfrac{2}{5}\). Simplify if needed.
Missing cell: use the row or column that has only one unknown. Total \(-\) known part = missing part.
Common traps: dividing by the grand total or by the column total instead of the row the question names, using the other column's number, and adding when you should subtract.
| Prefer cats | Prefer dogs | Total | |
|---|---|---|---|
| 7th grade | 23 | 25 | 48 |
| 8th grade | 27 | 17 | 44 |
| Total | 50 | 42 | 92 |
Relative frequency within 7th grade = (7th grade & cats) \(\div\) (7th grade total) = 23 \(\div\) 48 = \(\dfrac{23}{48}\). (Dividing by the grand total (\(23 \div 92\)) gives \(\dfrac{1}{4}\). Dividing by the "cats" column total (\(23 \div 50\)) gives \(\dfrac{23}{50}\). Comparing to the 7th grade students who prefer dogs (\(23 \div 25\)) gives \(\dfrac{23}{25}\).)
| Plays basketball | Plays soccer | Total | |
|---|---|---|---|
| 6th grade | 23 | ? | 41 |
| 7th grade | 10 | 13 | 23 |
| Total | 33 | ? | 64 |
The 6th grade row total is 41 and 23 of those play basketball, so 41 \(-\) 23 = 18 play soccer. (Subtracting from the grand total instead of the 6th grade total (\(64 - 23\)) gives 41. Adding instead of subtracting (\(41 + 23\)) gives 64. Doing the subtraction in the 7th grade row (\(23 - 10\)) gives 13.)