Measures of Dispersion for Grouped Data · 7.1.3
Ogive, quartiles and percentiles
Students plot cumulative frequency against the upper boundary of each class to draw a smooth rising ogive, building on the cumulative frequency table and histogram. They read the quartiles Q1 and Q3, and the median, at positions N/4, N/2 and 3N/4 up the vertical axis, and estimate any percentile the same way, at position kN/100.
The official learning standard (7.1.3)
“Construct an ogive for a set of grouped data and determine the quartiles. The construction of ogives needs to be related to the cumulative frequency histogram, and percentiles need to be involved.”
What it means
Students plot cumulative frequency against the upper boundary of each class to draw a smooth rising ogive, building on the cumulative frequency table and histogram. They read the quartiles Q1 and Q3, and the median, at positions N/4, N/2 and 3N/4 up the vertical axis, and estimate any percentile the same way, at position kN/100.
How it is examined
Tested almost entirely in Paper 2: students complete a cumulative frequency column from a grouped frequency table, plot the ogive accurately on graph paper through the upper class boundaries, then read off the median, Q1, Q3, and sometimes a stated percentile. Paper 1 may ask which axis or which position corresponds to a given quartile or percentile.
Worked example
The table shows the marks of 50 students in a test: 1–10 (4), 11–20 (9), 21–30 (14), 31–40 (15), 41–50 (8). Construct the cumulative frequency table, describe how the ogive is drawn, then determine the first quartile Q1, the third quartile Q3 and the 70th percentile.
- Cumulative frequencies (running totals): 4, 13, 27, 42, 50.
- Plot cumulative frequency against the upper boundary of each class, (0.5, 0), (10.5, 4), (20.5, 13), (30.5, 27), (40.5, 42), (50.5, 50), and join the points with a smooth rising curve to form the ogive.
- N = 50, so Q1 is at cumulative frequency N/4 = 12.5, which falls in the class 11–20 (boundaries 10.5–20.5, cumulative frequency running from 4 to 13).
- Reading (or interpolating) at this position: Q1 = 10.5 + [(12.5 − 4)/9] × 10 = 10.5 + 9.44 = 19.9 marks.
- Q3 is at cumulative frequency 3N/4 = 37.5, which falls in the class 31–40 (boundaries 30.5–40.5, cumulative frequency running from 27 to 42).
- Q3 = 30.5 + [(37.5 − 27)/15] × 10 = 30.5 + 7 = 37.5 marks.
- The 70th percentile is at cumulative frequency 70N/100 = 35, also in the class 31–40.
- P70 = 30.5 + [(35 − 27)/15] × 10 = 30.5 + 5.33 = 35.8 marks.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What is the difference between a cumulative frequency histogram and an ogive?
A cumulative frequency histogram is a set of bars showing the running total up to each class boundary; the ogive is the smooth curve traced by joining the top corner of each bar (or each plotted point), giving a continuous curve you can read at any position, not only at class boundaries.
Why is the percentile position kN/100 instead of just k?
Because a percentile marks the point below which k% of the data lies, so you need k/100 of the total frequency N, not k itself. For the 70th percentile with N = 50, the position is 70/100 × 50 = 35, not 70, always convert the percentage to a fraction of N first.
Can I find the quartiles by interpolation instead of drawing the ogive?
Yes, interpolation with the formula L + [(position − F)/f] × c gives the same value the ogive would show, using the cumulative frequency table directly, where position is N/4, 3N/4 or kN/100. Examiners accept either method, but if a question specifically says 'from the ogive,' you must draw and use the graph.