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Learning Resources

Cumulative Frequency & Histograms

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

N/A Pathway: N/A

First Principles

Objective: Construct and interpret cumulative frequency curves (ogives) and histograms with unequal class widths using frequency density.

In a histogram, the area of each bar represents the frequency of that class, not its height. When class widths are unequal, we must plot Frequency Density on the vertical axis:

\[\text{Frequency Density } (fd) = \frac{\text{Frequency}}{\text{Class Width}} = \frac{f}{w}\]

(a) Cumulative Frequency Curves (Ogives)

Cumulative frequency (\( cf \)) represents the running total of frequencies. Points are plotted using \( (\text{Upper Class Boundary}, cf) \) and joined with a smooth curve.

(b) Estimating Quartiles from an Ogive

  • Median (\( Q_2 \)): Value at \( 50\% \) of total frequency (\( 0.5N \)).
  • Lower Quartile (\( Q_1 \)): Value at \( 25\% \) of total frequency (\( 0.25N \)).
  • Upper Quartile (\( Q_3 \)): Value at \( 75\% \) of total frequency (\( 0.75N \)).
  • Interquartile Range (IQR): \( IQR = Q_3 - Q_1 \).
fd = 2.0 fd = 3.0 Wider Class (w=20, fd=1.0)

Key Formulas

\[\text{Frequency Density } (fd) = \frac{\text{Frequency } (f)}{\text{Class Width } (w)}\] — Frequency Density: Vertical axis measure for histograms.
\[\text{Frequency } (f) = \text{Frequency Density } (fd) \times \text{Class Width } (w)\] — Bar Area Principle: Area of histogram rectangle equals class frequency.
\[\text{Ogive Plotting Coordinates} = (\text{Upper Class Boundary}, \text{Cumulative Frequency})\] — Ogive Points: Upper boundary used for cumulative plots.
\[\text{IQR} = Q_3 - Q_1\] — Interquartile Range: Difference between upper (\( 75\% \)) and lower (\( 25\% \)) quartiles.

Worked Examples

Example 1 (Easy - Total Frequency from Histogram): A histogram has 3 intervals: Interval A (width 5, height/fd 4), Interval B (width 5, height/fd 6), and Interval C (width 10, height/fd 3). Calculate total frequency \( N \).
  1. Calculate frequency for each interval using \( f = fd \times w \).
  2. \( f_A = 4 \times 5 = 20 \).
  3. \( f_B = 6 \times 5 = 30 \).
  4. \( f_C = 3 \times 10 = 30 \).
  5. Total \( N = 20 + 30 + 30 = 80 \).

Answer: 80

Example 2 (Medium - Cumulative Frequency Interpolation): An ogive passes through \( (30, 18) \) and \( (50, 30) \) in a straight line segment. Estimate the cumulative frequency at value \( x = 35 \).
  1. Calculate gradient of line segment: \( m = (30 - 18) / (50 - 30) = 12 / 20 = 0.6 \).
  2. Use linear interpolation formula: \( cf(35) = 18 + m(35 - 30) \).
  3. \( cf(35) = 18 + 0.6(5) = 18 + 3 = 21 \).

Answer: 21

Example 3 (Hard - Grouped Data Median Estimation): KCSE Mathematics mock trial scores out of 50 are grouped as follows: 0–10 (\( f=5 \)), 10–20 (\( f=8 \)), 20–30 (\( f=12 \)), 30–40 (\( f=9 \)), 40–50 (\( f=6 \)). Estimate the median score to 1 decimal place.
  1. Total frequency \( N = 5 + 8 + 12 + 9 + 6 = 40 \).
  2. Median position \( = N / 2 = 20^{\text{th}} \) observation.
  3. Cumulative frequencies: 0–10 (\( cf=5 \)), 10–20 (\( cf=13 \)), 20–30 (\( cf=25 \)). The \( 20^{\text{th}} \) score lies in interval 20–30.
  4. Linear interpolation: \( \text{Median} = L + \left(\frac{0.5N - cf_{\text{prev}}}{f_{\text{median}}}\right) \times w = 20 + \left(\frac{20 - 13}{12}\right) \times 10 \).
  5. \( \text{Median} = 20 + \left(\frac{7}{12}\right) \times 10 = 20 + 5.833 = 25.83 \approx 25.8 \).

Answer: 25.8

Common Mistakes

Mistake Setting histogram bar height equal to raw frequency when class widths are unequal.
Correction Bar height must strictly equal Frequency Density (\( fd = f / w \)) so that bar area represents frequency.
Why it feels right Standard bar charts use height for frequency, leading students to apply the same convention to histograms.
Mistake Plotting cumulative frequency points at midpoints or lower class boundaries.
Correction Cumulative frequency accumulates all data up to the upper boundary, so points must be plotted at \( (\text{Upper Boundary}, cf) \).
Why it feels right Midpoints are used when estimating means from grouped tables, creating confusion with ogive plots.

Real World

KNEC National Exam Score Distributions: Examination boards use cumulative frequency curves to determine cut-off scores for KCSE grade boundaries and percentile rankings.
KNBS Income Analytics: Economists at the Kenya National Bureau of Statistics plot income histograms with unequal class widths (since high income brackets span wide ranges) using frequency density to prevent visual distortion.

Practice

An ogive passes through points (30, 18) and (50, 30) along a straight line segment. Estimate the cumulative frequency for a value of 35. (Type only the number, e.g., 21)
Review the concepts above.
A class interval 20 - 50 has a frequency of 15. What is the frequency density of this class interval? (Type only the number, e.g., 0.5)
Review the concepts above.
KCSE Mathematics mock trial scores grouped into intervals: 0-10 (f=5), 10-20 (f=8), 20-30 (f=12), 30-40 (f=9), 40-50 (f=6). Estimate the median score to 1 decimal place. (Type only the number, e.g., 25.8)
Review the concepts above.
A histogram bar has a frequency density of 1.2 and a class width of 25. What is the frequency represented by this bar? (Type only the number, e.g., 30)
Review the concepts above.
A histogram has three intervals: A (width 5, height 4), B (width 5, height 6), C (width 10, height 3), where height represents frequency density. What is total frequency? (Type only the number, e.g., 80)
Review the concepts above.
A histogram has four intervals: 0-10, 10-20, 20-30, 30-40 with frequencies 5, 15, 10, 20. What is the total number of observations N? (Type only the number, e.g., 50)
Review the concepts above.