An ogive is a cumulative frequency graph. You plot upper boundaries against cumulative frequency, start from zero at the first lower boundary, and read the median and quartiles from the curve.
This lesson is part of dispersion of grouped data. It uses the table from estimating mean from grouped data.
What do you plot?
Use the 40-student marks table. Each cumulative frequency is paired with the upper boundary of its class.
| Upper boundary | 9.5 | 19.5 | 29.5 | 39.5 | 49.5 | 59.5 |
|---|---|---|---|---|---|---|
| Cumulative frequency | 0 | 4 | 13 | 27 | 35 | 40 |
The first pair is the starting point: 9.5 is the lower boundary of the first class, and nothing has been counted yet.
The plotting checklist
- Mark the zero point (9.5, 0).
- Use upper boundaries, not midpoints or limits.
- Join the points with a smooth curve, not with ruled segments, unless your teacher asks for a polygon.
- Label both axes and use an even scale.
Worked example: reading the ogive
The total is 40, so the median is at cumulative frequency 20. The curve passes through (29.5, 13) and (39.5, 27). Reading across at 20 gives a median of about 34.5 marks.
The first quartile is at 10. It lies between (19.5, 4) and (29.5, 13), which gives about 26.2 marks. The third quartile is at 30, between (39.5, 27) and (49.5, 35), which gives about 43.3 marks.
The interquartile range is 43.3 − 26.2 = 17.1 marks.
Students scoring above 45. At 45 the curve reads about 31 students. So 40 − 31 = 9 students scored above 45.
Readings from a hand-drawn graph vary slightly, so draw clear guide lines on the paper and write the values you read.
The slips to avoid
The first slip is plotting against the midpoint, which shifts every point half a class to the left. The second is leaving out the zero point. The third is reading the vertical axis value at N instead of N ÷ 2 for the median.
Write “N ÷ 2 = 20” on your paper before reading the graph. That one line protects the mark.
Check yourself
Use the cumulative frequencies 0, 3, 10, 22, 28, 30 at the upper boundaries 0.5, 10.5, 20.5, 30.5, 40.5, 50.5. Find the median and the interquartile range by reading between the points.
Answer
N = 30, so the median is at 15. Between (20.5, 10) and (30.5, 22), the value is 20.5 + (5 ÷ 12) × 10 ≈ 24.7.
Q1 is at 7.5, between (10.5, 3) and (20.5, 10): 10.5 + (4.5 ÷ 7) × 10 ≈ 16.9. Q3 is at 22.5, between (30.5, 22) and (40.5, 28): 30.5 + (0.5 ÷ 6) × 10 ≈ 31.3.
The interquartile range is 31.3 − 16.9 = 14.4.
What to study next
Move to comparing grouped data using quartiles and spread, where two ogives are read together. Test the chapter with the practice set.
For a teacher to check your graph and your reading lines, see online one-to-one Mathematics tuition.