Ogives & Quartiles
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Master the construction of cumulative frequency curves (ogives) and determine quartiles, percentiles, and the interquartile range (IQR).
The Running Total Concept: Imagine a farmer in Kericho delivering tea leaves every morning over 30 days. If we record how many days yielded at most 20 kg, at most 40 kg, up to 100 kg, we are accumulating totals. A cumulative frequency curve (or ogive) visualises this running total to show the proportion of data falling below any chosen benchmark.
1. Cumulative Frequency & Upper Class Boundaries
An ogive is always plotted by placing the Upper Class Boundary (UCB) on the horizontal (\(x\)) axis and the Cumulative Frequency (\(CF\)) on the vertical (\(y\)) axis. Because cumulative frequency never decreases, an ogive is an 'S-shaped' monotonically increasing curve that starts at the lower boundary of the first class with \(CF = 0\).
2. Locating Quartiles on an Ogive
Quartiles divide an ordered dataset into four equal quarters (25% each):
- Lower Quartile (\(Q_1\)): The value at \(\frac{1}{4}N\) (or \(\frac{N+1}{4}\) for discrete data). 25% of observations fall below \(Q_1\).
- Median (\(Q_2\)): The middle value at \(\frac{1}{2}N\) (or \(\frac{N+1}{2}\)). 50% of observations fall below \(Q_2\).
- Upper Quartile (\(Q_3\)): The value at \(\frac{3}{4}N\) (or \(\frac{3(N+1)}{4}\)). 75% of observations fall below \(Q_3\).
Interquartile Range (IQR): \(\text{IQR} = Q_3 - Q_1\). It measures the spread of the central 50% of the distribution and is resistant to extreme outliers.
Interactive Ogive Explorer
Drag the slider to move along the cumulative frequency axis (\(CF\)) and observe how \(Q_1\), Median (\(Q_2\)), and \(Q_3\) are read horizontally and projected down to the \(x\)-axis.
Key Formulas
1. Cumulative Frequency Definition
\[CF_{i} = \sum_{j=1}^{i} f_{j}\]The cumulative frequency of class \(i\) is the sum of frequencies from the first class up to class \(i\).
2. Quartile Positions for Grouped Continuous Data
\[P_{Q1} = \frac{1}{4}N, \quad P_{Q2} = \frac{1}{2}N, \quad P_{Q3} = \frac{3}{4}N\]For ogives and continuous data grouped into classes, quartiles are determined using fractions of the total frequency \(N\).
3. Linear Interpolation Formula for Quartiles
\[Q_{k} = L + \left(\frac{\frac{k}{4}N - CF_{\text{prev}}}{f_{k}}\right) \times c\]Where:
- \(L\) = Exact lower class boundary of the quartile class
- \(N\) = Total cumulative frequency (\(\sum f\))
- \(CF_{\text{prev}}\) = Cumulative frequency of the class preceding the quartile class
- \(f_{k}\) = Frequency of the quartile class
- \(c\) = Class width (interval size: \(c = \text{Upper Boundary} - \text{Lower Boundary}\))
4. Interquartile Range & Semi-Interquartile Range
\[\text{IQR} = Q_3 - Q_1\]\[\text{Semi-IQR} = \frac{Q_3 - Q_1}{2}\]Worked Examples
Problem: An agricultural officer in Nakuru evaluates the weekly milk yields of dairy cows. From an ogive, she reads the lower quartile as \(Q_1 = 18.5\text{ litres}\) and the upper quartile as \(Q_3 = 34.5\text{ litres}\). Calculate:
- The Interquartile Range (IQR).
- The Semi-Interquartile Range (Quartile Deviation).
Step-by-step Solution:
- Calculate IQR:\[\text{IQR} = Q_3 - Q_1 = 34.5 - 18.5 = 16\text{ litres}\]
- Calculate Semi-IQR:\[\text{Semi-IQR} = \frac{\text{IQR}}{2} = \frac{16}{2} = 8\text{ litres}\]
Answer: \(\text{IQR} = 16\text{ litres}\), \(\text{Semi-IQR} = 8\text{ litres}\).
Problem: The table below shows the distribution of marks scored by 40 students in a mathematics test in Machakos High School.
| Marks | Class Boundaries | Frequency (\(f\)) | Cumulative Frequency (\(CF\)) |
|---|---|---|---|
| 10 - 19 | 9.5 - 19.5 | 6 | 6 |
| 20 - 29 | 19.5 - 29.5 | 10 | 16 |
| 30 - 39 | 29.5 - 39.5 | 14 | 30 |
| 40 - 49 | 39.5 - 49.5 | 8 | 38 |
| 50 - 59 | 49.5 - 59.5 | 2 | 40 |
Calculate the median mark using linear interpolation.
Step-by-step Solution:
- Identify median rank: \[\text{Rank} = \frac{1}{2}N = \frac{40}{2} = 20\]
- Identify median class: The 20th student falls into the class 30 - 39 (since \(CF\) before is 16 and ends at 30).
- Identify parameters: Lower boundary \(L = 29.5\), \(CF_{\text{prev}} = 16\), \(f = 14\), class width \(c = 39.5 - 29.5 = 10\).
- Apply interpolation:\[\text{Median} = L + \left(\frac{\frac{N}{2} - CF_{\text{prev}}}{f}\right) \times c = 29.5 + \left(\frac{20 - 16}{14}\right) \times 10\]\[\text{Median} = 29.5 + \left(\frac{4}{14}\right) \times 10 = 29.5 + 2.857 = 32.36\]
Answer: \(\text{Median} \approx 32.4\text{ marks}\).
Problem: A tea processing factory in Kericho weighs 50 bags of tea. The results are summarized below: Classes (kg): 20–29 (\(f=6\)), 30–39 (\(f=12\)), 40–49 (\(f=18\)), 50–59 (\(f=10\)), 60–69 (\(f=4\)). Total \(N=50\). Calculate the Interquartile Range (IQR) correct to 2 decimal places.
Step-by-step Solution:
- Compute Cumulative Frequencies:
- 20–29 (19.5–29.5): \(CF = 6\)
- 30–39 (29.5–39.5): \(CF = 6 + 12 = 18\)
- 40–49 (39.5–49.5): \(CF = 18 + 18 = 36\)
- 50–59 (49.5–59.5): \(CF = 36 + 10 = 46\)
- 60–69 (59.5–69.5): \(CF = 46 + 4 = 50\)
- Find Lower Quartile (\(Q_1\)):\[\text{Position} = \frac{1}{4}(50) = 12.5\]Falls in class 30–39: \(L = 29.5\), \(CF_{\text{prev}} = 6\), \(f = 12\), \(c = 10\).\[Q_1 = 29.5 + \left(\frac{12.5 - 6}{12}\right) \times 10 = 29.5 + \left(\frac{6.5}{12}\right) \times 10 = 29.5 + 5.42 = 34.92\text{ kg}\]
- Find Upper Quartile (\(Q_3\)):\[\text{Position} = \frac{3}{4}(50) = 37.5\]Falls in class 50–59: \(L = 49.5\), \(CF_{\text{prev}} = 36\), \(f = 10\), \(c = 10\).\[Q_3 = 49.5 + \left(\frac{37.5 - 36}{10}\right) \times 10 = 49.5 + \left(\frac{1.5}{10}\right) \times 10 = 49.5 + 1.50 = 51.00\text{ kg}\]
- Compute IQR:\[\text{IQR} = Q_3 - Q_1 = 51.00 - 34.92 = 16.08\text{ kg}\]
Answer: \(Q_1 = 34.92\text{ kg}\), \(Q_3 = 51.00\text{ kg}\), \(\text{IQR} = 16.08\text{ kg}\).
Common Mistakes
Why students do it: For frequency polygons, midpoints are plotted against frequency. Students confuse polygons with ogives.
Why students do it: Students are used to looking up \(x\) to find \(y = f(x)\).
Why students do it: In a class like \(20 - 29\), students take \(L = 20\) instead of \(19.5\).
Real World
Practice