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Learning Resources

Data Collection & Graphs

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

Form 1 Pathway: N/A

First Principles

Objective

Understand how to systematically collect raw data, organise it into frequency distribution tables, and accurately represent it using bar charts, pie charts, and line graphs.

Interactive Data Collector & Bar Chart

Click + or − to adjust attendance across weekdays at a school in Nakuru. Observe how the bar heights and relative frequency scale in real time.

Total Absences: 0

(a) Concrete Scenario: The Market Tally

Imagine Mama Sarah selling fruit at Kongowea Market in Mombasa. As customers purchase pineapples, mangoes, and watermelons, recording each fruit name repeatedly becomes messy. Instead, she draws a tally stroke \(|\) for each item sold, bundling groups of five with a diagonal line (\(\bcancel{||||}\)). Each tally represents a single raw datum, and the bundle sum is the frequency.

Data refers to distinct pieces of factual information. Frequency (\(f\)) simply counts how many times a particular value occurs.

(b) Visual Representation: Choosing the Right Graph

  • Bar Chart: Uses vertical or horizontal bars separated by uniform spaces. Best for comparing categorical data (e.g., types of vehicles passing a checkpoint).
  • Line Graph: Connects consecutive data points with line segments. Ideal for showing continuous trends over time (e.g., daily temperature or stock price).
  • Pie Chart: A circle subdivided into sectors where each angle is proportional to the category's share of the total \(360^\circ\).

Key Formulas

\[ N = \sum f_i \] — Total Frequency: The sum of all individual category frequencies across the dataset.
\[ \text{Relative Frequency} = \frac{f_i}{N} \] — Relative Frequency: The proportion of the whole represented by a single group or category.
\[ \theta_i = \frac{f_i}{N} \times 360^\circ \] — Pie Chart Sector Angle: The central angle allocated to category \(i\) within a full circle of \(360^\circ\).
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] — Arithmetic Mean: The sum of the products of each value (or class midpoint) and its corresponding frequency, divided by total observations.
\[ \text{Range} = \text{Maximum Value} - \text{Minimum Value} \] — Range: A basic measure of dispersion reflecting the total span of values.

Worked Examples

Example 1 (Easy): Basic Frequency & Relative Frequency
Problem: A roadside count in Machakos recorded the following vehicles: Cars: 18, Matatus: 24, Boda-bodas: 30, Trucks: 8. Find the total number of vehicles observed and calculate the relative frequency of Matatus as a simplified fraction.
  1. Sum total frequencies: \[ N = 18 + 24 + 30 + 8 = 80 \]
  2. Calculate relative frequency: \[ \text{Relative Frequency of Matatus} = \frac{f_{\text{Matatu}}}{N} = \frac{24}{80} = \frac{3}{10} \]
  3. Answer: Total = \(80\), Relative Frequency = \(\frac{3}{10}\) (or \(0.30\)).
Example 2 (Medium): Pie Chart Sector Angles
Problem: In a Form 1 class of 72 students, 20 walk to school, 32 board matatus, 12 use bicycles, and 8 are driven by personal car. Calculate the sector angle for the 'Matatu' category on a pie chart.
  1. Identify total frequency: \[ N = 20 + 32 + 12 + 8 = 72 \]
  2. Apply the sector angle formula: \[ \theta = \frac{f_{\text{Matatu}}}{N} \times 360^\circ = \frac{32}{72} \times 360^\circ \]
  3. Simplify: \[ \frac{32}{72} = \frac{4}{9} \implies \frac{4}{9} \times 360^\circ = 4 \times 40^\circ = 160^\circ \]
  4. Answer: The sector angle is \(160^\circ\).
Example 3 (Hard): Mean from a Grouped Frequency Table
Problem: The masses (in kg) of 40 bags of maize delivered to a collection depot are grouped as follows:
Mass (kg): 40–44, 45–49, 50–54, 55–59
Frequency: 6, 14, 12, 8
Calculate the estimated mean mass of the bags.
  1. Determine the class midpoints (\(x_i\)):
    • 40–44: \(\frac{40+44}{2} = 42\)
    • 45–49: \(\frac{45+49}{2} = 47\)
    • 50–54: \(\frac{50+54}{2} = 52\)
    • 55–59: \(\frac{55+59}{2} = 57\)
  2. Compute \(f_i \times x_i\):
    • \(6 \times 42 = 252\)
    • \(14 \times 47 = 658\)
    • \(12 \times 52 = 624\)
    • \(8 \times 57 = 456\)
  3. Sum the products and total frequency: \[ \sum f_i = 40 \] \[ \sum f_i x_i = 252 + 658 + 624 + 456 = 1990 \]
  4. Calculate the mean: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{1990}{40} = 49.75\text{ kg} \]
  5. Answer: \(49.75\text{ kg}\).

Common Mistakes

Mistake Confusing the category value \(x\) with its frequency \(f\) when identifying the mode.
Why it happens: Learners often see the highest number in the table (the highest frequency) and mistakenly write that frequency number as the mode.
Correction The mode is the category or score itself (\(x\)) that has the highest frequency, NOT the frequency value itself.
Mistake Leaving spaces between bars when drawing a histogram, or omitting spaces in a bar chart.
Why it happens: Bar charts and histograms look similar at first glance.
Correction In a bar chart (discrete/categorical data), separate bars with equal gaps. In a histogram (continuous numerical intervals), bars must touch without gaps.
Mistake Assuming pie chart sectors are proportional to percentages out of \(100^\circ\) rather than \(360^\circ\).
Why it happens: Percentages sum to 100%, leading students to multiply fractions by 100 instead of 360 when finding angles.
Correction Always multiply the fraction \(\frac{f}{N}\) by \(360^\circ\) to calculate degrees for a circle.

Real World

Agribusiness & Crop Yields: Tea factories in Kericho record daily kilogram deliveries from hundreds of smallholder farmers. Grouped frequency distributions help managers determine average yield per hectare and identify production peaks.
Transport & Logistics: Matatu Saccos track passenger volumes across different routes and times using line graphs to adjust fleet distribution during morning and evening rush hours.
County Budget Allocations: County governments present annual financial budgets using pie charts to clearly display allocations across Health, Education, Infrastructure, and Agriculture to citizens.
Public Health Monitoring: Local dispensaries use daily attendance line charts to spot sudden surges in malaria or flu cases, triggering timely medical supply restocking.

Practice

A frequency table records the ages of students in a school club: 10–14 years: 5 students 15–19 years: 12 students 20–24 years: 8 students 25–29 years: 5 students How many students are in the club in total? (Type only the number, e.g., 42)
Review the concepts above.
A class of 30 learners was asked how many siblings they have. The responses are shown in the frequency table below: Siblings: 0, 1, 2, 3, 4 Frequency: 5, 10, 8, 4, 3 What is the modal number of siblings in this class? (Type only the number, e.g., 2)
Review the concepts above.
A weather station in Eldoret recorded daily maximum temperatures over five days as follows: 22°C, 24°C, 23°C, 25°C, and 26°C. What is the mean temperature in °C for the five days? (Type only the number, e.g., 25)
Review the concepts above.
A survey of 120 secondary school students asked about their preferred mode of transport to school. The results were: Matatu: 45 Boda-boda: 30 Walking: 25 Bus: 20 What percentage of the students prefer Matatu? (Type only the number, e.g., 35.5)
Review the concepts above.
A school surveyed 200 students about their favourite sport. The results showed that 25% chose Basketball. What is the central angle (in degrees) of the sector representing Basketball on a pie chart? (Type only the number, e.g., 90)
Review the concepts above.
The number of books read last term by 30 students are listed in ascending order: 0, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16. What is the median number of books read? (Type only the number, e.g., 6.5)
Review the concepts above.