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Learning Resources

Histograms & Cumulative Freq

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

Form 2 Pathway: N/A

First Principles

Objective: Master the construction, interpretation, and analysis of grouped continuous data using Histograms (via Frequency Density) and Cumulative Frequency Curves (Ogives).

Context: The Kericho Tea Harvest

Imagine a tea cooperative in Kericho weighing daily harvests from hundreds of smallholder farmers. Because weights vary continuously (e.g., \(12.4\text{ kg}, 18.7\text{ kg}, 35.2\text{ kg}\)), we group them into class intervals. But when classes have different widths, drawing simple bar heights is misleading! We must draw Frequency Density so that the area of each rectangular bar represents the true number of farmers.

Interactive Ogive & Histogram Explorer

Adjust class frequencies to see how the Cumulative Frequency Curve (Ogive) develops and locate the median position (\(n/2\)).

Class Frequencies (Marks)
Total (\(n\)): 30
Median Position (\(n/2\)): 15

Target CF: 15 | Estimated Mark: —

Core Concepts & Principles

  • Continuous Data & Histograms: In a histogram, intervals touch without gaps. The area of each rectangle is directly proportional to the class frequency: \[\text{Area} = \text{Class Width} \times \text{Frequency Density} = \text{Frequency}\]
  • The Cumulative Frequency Curve (Ogive): An ogive represents the running total of frequencies. Points are strictly plotted at the upper class boundaries against cumulative frequency, starting from \((\text{lower boundary of first class}, 0)\).
  • Finding Statistical Measures:
    • Median (\(Q_2\)): Read at cumulative frequency \(\frac{N}{2}\).
    • Lower Quartile (\(Q_1\)): Read at cumulative frequency \(\frac{N}{4}\).
    • Upper Quartile (\(Q_3\)): Read at cumulative frequency \(\frac{3N}{4}\).
    • Interquartile Range (IQR): \(\text{IQR} = Q_3 - Q_1\).

Key Formulas

1. Frequency Density

\[\text{Frequency Density} = \frac{\text{Class Frequency}}{\text{Class Width}}\]

Used as the vertical axis of a histogram to ensure areas remain strictly proportional to frequencies even when intervals are unequal.

2. Cumulative Frequency (\(CF\))

\[CF_k = \sum_{i=1}^{k} f_i = f_1 + f_2 + \dots + f_k\]

The total number of observations scoring less than or equal to the upper boundary of class \(k\).

3. Ogive Quartile Positions

\[\text{Lower Quartile Position } (Q_1) = \frac{N}{4}\] \[\text{Median Position } (Q_2) = \frac{N}{2}\] \[\text{Upper Quartile Position } (Q_3) = \frac{3N}{4}\] \[\text{Interquartile Range (IQR)} = Q_3 - Q_1\] \[\text{Semi-Interquartile Range} = \frac{Q_3 - Q_1}{2}\]

4. Relative Frequency & Proportion

\[\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Total Observations } (N)}\]

Worked Examples

Example 1 (Easy): Calculating Frequency Density

Problem: In a test distribution, the class interval \(20 - 30\) contains \(16\) students. Find the frequency density (height of the histogram bar) for this interval.

Solution:

  1. Identify class limits: Lower boundary = \(20\), Upper boundary = \(30\).
  2. Calculate class width \(w\): \[w = 30 - 20 = 10\]
  3. Compute Frequency Density: \[\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} = \frac{16}{10} = 1.6\]

Answer: The bar is drawn to a height of \(1.6\) units.

Example 2 (Medium): Cumulative Frequency Table

Problem: The table below shows the distribution of weights (in kg) of 40 bags of maize harvested in Nakuru:

Weight (kg)\(10-19\)\(20-29\)\(30-39\)\(40-49\)
Frequency614128

Determine the upper class boundaries and the cumulative frequency for each interval.

Solution:

  1. Because data is continuous, upper class boundaries are: \(19.5, 29.5, 39.5, 49.5\).
  2. Calculate running totals:
    • Class 1 (\(\le 19.5\)): \(CF = 6\)
    • Class 2 (\(\le 29.5\)): \(CF = 6 + 14 = 20\)
    • Class 3 (\(\le 39.5\)): \(CF = 20 + 12 = 32\)
    • Class 4 (\(\le 49.5\)): \(CF = 32 + 8 = 40\)
  3. The ogive is plotted through points: \((9.5, 0), (19.5, 6), (29.5, 20), (39.5, 32), (49.5, 40)\).

Example 3 (Hard): Interpreting an Unequal Interval Histogram

Problem: A histogram consists of three bars with the following dimensions:

  • Class A: Interval \(0 - 10\), Frequency Density = \(1.8\)
  • Class B: Interval \(10 - 30\), Frequency Density = \(1.2\)
  • Class C: Interval \(30 - 40\), Frequency Density = \(0.8\)

Find the total number of items \(N\) and the percentage of items scoring greater than 10.

Solution:

  1. Calculate frequency (area) of each interval: \[f_A = \text{Width} \times \text{Density} = (10 - 0) \times 1.8 = 10 \times 1.8 = 18\] \[f_B = (30 - 10) \times 1.2 = 20 \times 1.2 = 24\] \[f_C = (40 - 30) \times 0.8 = 10 \times 0.8 = 8\]
  2. Total observations \(N\): \[N = 18 + 24 + 8 = 50\]
  3. Items scoring greater than 10: \[f_B + f_C = 24 + 8 = 32\]
  4. Calculate percentage: \[\text{Percentage} = \frac{32}{50} \times 100\% = 64\%\]

Common Mistakes

Misconception 1: Plotting cumulative frequency at class midpoints

Why it feels right Frequency polygons are plotted at midpoints, so students often do the same for ogives.

Correction Cumulative frequency represents observations up to the boundary. Therefore, cumulative frequency points must always be plotted at upper class boundaries, starting from 0 at the lower boundary of the very first class.

Misconception 2: Using bar height as frequency in unequal class widths

Why it feels right In standard discrete bar charts, bar height directly equals frequency.

Correction In histograms, Area = Frequency. When class widths vary, the vertical axis is Frequency Density: \[\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\].

Misconception 3: Reading the median at \(50\) instead of \(N/2\)

Why it feels right Many students confuse "percentiles (50%)" with cumulative frequency and look for the mark 50 on the vertical axis.

Correction Always locate \(\frac{N}{2}\) where \(N\) is the total frequency (the top of the ogive curve), not 50 (unless \(N = 100\)).

Real World

Setting the Median (\(Q_2\)): Halfway through the candidate population (\(N/2 = 400{,}000\)), examiners determine the national median score.
University Direct Entry (\(C+\) and above): By determining the mark corresponding to the top 15–20% of candidates on the ogive curve, the government establishes fair cut-off criteria for university sponsorship under KUCCPS.
Standard Deviation & Quality Control: Plotting ogives across consecutive years allows KNEC to monitor if examination difficulty remains consistent year-on-year.

Practice

In a histogram, the class interval 20–25 has a class width of 5 units. If the frequency density (height of the bar) is 8 units, calculate the frequency for this class interval. (Type only the number, e.g., 42)
Review the concepts above.
A continuous distribution has two classes: 0–10 with frequency 15, and 10–30 with frequency 30. Calculate the height (frequency density) of the second class (10–30). (Type only the number, e.g., 2.3)
Review the concepts above.
A teacher recorded the test scores of 15 students as follows: 12, 15, 18, 22, 25, 27, 30, 33, 35, 38, 40, 42, 45, 48, 50. The data is grouped into classes: 10–19, 20–29, 30–39, 40–49, 50–59. What is the cumulative frequency up to the upper limit 39? (Type only the number, e.g., 42)
Review the concepts above.
A set of 20 test scores are: 12, 15, 17, 19, 22, 24, 24, 26, 28, 30, 31, 33, 35, 37, 38, 40, 42, 45, 48, 50. When grouped into intervals 10–19, 20–29, 30–39, 40–49, 50–59, what is the cumulative frequency up to the end of the 30–39 interval? (Type only the number, e.g., 42)
Review the concepts above.
A frequency distribution has classes and frequencies: 10–20 (5), 20–30 (8), 30–40 (12), 40–50 (7). What is the cumulative frequency up to and including the class 30–40? (Type only the number, e.g., 42)
Review the concepts above.
A histogram represents grouped data with intervals 0–5 (frequency 10), 5–10 (frequency 15), and 10–15 (frequency 25). What proportion of the total observations is represented by the area of the bar for the interval 10–15? (Type only the number, e.g., 0.75)
Review the concepts above.