The Class Survey Graph
A Grade 5 class surveys 20 students about their favourite recess activity. Results: Soccer 8, Tag 5, Skipping 4, Other 3. A student draws a bar graph where the Soccer bar is 8 squares tall and the Tag bar is 5 squares tall, but the vertical axis starts at 4 instead of 0. A classmate looks at the graph and says, 'Soccer looks about three times bigger than Tag.' Another student makes a pictograph where one symbol stands for 2 students and draws 4 symbols for Soccer and 2.5 symbols for Tag.
1. Explain why the truncated-axis bar graph is misleading. What should be changed, and how would the corrected graph change the visual comparison?
Read the worked answer
Starting the axis at 4 cuts off the bottom of every bar. Soccer's visible height becomes 8 − 4 = 4 squares and Tag's becomes 5 − 4 = 1 square, so Soccer appears four times taller than Tag even though the real counts are 8 and 5, a ratio of 8:5 (about 1.6 times). The truncated axis exaggerates differences. To fix it, start the vertical axis at 0 and label equal intervals (0, 2, 4, 6, 8). Then Soccer's bar is 8 units and Tag's is 5 units, and the visual heights match the data. A reader can then correctly see Soccer is somewhat more popular, not three or four times more popular.
2. Evaluate the pictograph. Is it accurate, and what problem arises with the half symbol? Suggest a scale that avoids fractional symbols.
Read the worked answer
The pictograph is partly accurate: with one symbol = 2 students, Soccer needs 8 ÷ 2 = 4 symbols, which is correct. Tag needs 5 ÷ 2 = 2.5 symbols, which forces a half symbol. A half symbol is hard to read and can be misread as a whole symbol, overstating Tag as 6 students. Skipping (4 students) needs 2 symbols and Other (3 students) needs 1.5 symbols, so two categories need halves. A better scale is one symbol = 1 student, giving 8, 5, 4, and 3 symbols, all whole. Alternatively, keep 2 per symbol but add a key showing a half symbol clearly. The total should still check: 8 + 5 + 4 + 3 = 20 students.