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Learning Resources

Data Presentation and Interpretation

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

Grade 08 Pathway: N/A

First Principles

Objective: Organize raw data into frequency tables, construct and interpret charts, and calculate the mean, median, mode, and range.

Interactive Data & Averages Explorer

Add values or pick a sample dataset to watch the summary statistics & chart update!

Dataset: []
Mean (\(\bar{x}\))
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Median
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Mode
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Range
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(a) Concrete Scenario: A dairy farmer in Kiambu tracks daily milk production in liters over one week: \(12, 15, 12, 18, 10, 14, 12\). Raw numbers can look confusing in a list. Organizing them into a frequency table or bar chart makes patterns instantly visible.

(b) Central Tendency & Spread:

  • Mean (\(\bar{x}\)): The equal balance point. Add all values and divide by the total number of entries \(n\).
  • Median: The exact middle value when data is sorted in ascending order.
  • Mode: The most frequently occurring score. Useful for inventory (e.g., most popular shoe size).
  • Range: The difference between the highest and lowest values (\(\text{Max} - \text{Min}\)), indicating spread.

(c) Sector Angle in Pie Charts: When representing frequency \(f\) out of total sample \(n\) on a pie chart: \[\text{Sector Angle } (\theta) = \frac{f}{n} \times 360^\circ\]

Crucial Rule: When finding the median, ALWAYS sort your numbers from smallest to largest first!

Key Formulas

\(\bar{x} = \frac{\sum x}{n}\) — Mean: Sum of all observations divided by total count \(n\).
\(\bar{x} = \frac{\sum (f \cdot x)}{\sum f}\) — Mean from a Frequency Table: Sum of (frequency \(\times\) value) divided by total frequency.
\(\text{Median Position} = \frac{n + 1}{2}\) — The position of the middle item in a sorted list of \(n\) numbers.
\(\text{Mode} = \text{Value with highest frequency } (f_{\max})\)
\(\text{Range} = x_{\max} - x_{\min}\) — Measure of dispersion (highest minus lowest value).
\(\theta = \frac{f}{n} \times 360^\circ\) — Sector angle for a category in a pie chart.

Worked Examples

Example 1 (Easy - Calculating Mean & Range): A student scored the following marks in 5 weekly math quizzes: \(14, 18, 12, 16, 20\). Calculate: (a) Mean score, and (b) Range.
  1. Sum of scores: \(\sum x = 14 + 18 + 12 + 16 + 20 = 80\).
  2. Count: \(n = 5\).
  3. Mean: \(\bar{x} = \frac{80}{5} = 16\).
  4. Range: \(\text{Max} - \text{Min} = 20 - 12 = 8\).
  5. Answer: Mean = 16 ; Range = 8.
Example 2 (Medium - Mean from Frequency Table): A shopkeeper recorded shoe sizes sold in a day as shown in the table below. Find the mean shoe size sold.
\(\begin{array}{|c|c|c|} \hline \text{Shoe Size } (x) & \text{Frequency } (f) & f \cdot x \\ \hline 38 & 3 & 114 \\ 39 & 5 & 195 \\ 40 & 8 & 320 \\ 41 & 4 & 164 \\ \hline \text{Total} & \sum f = 20 & \sum f \cdot x = 793 \\ \hline \end{array}\)
  1. Calculate \(\sum f\): \(3 + 5 + 8 + 4 = 20\).
  2. Calculate \(\sum (f \cdot x)\): \(114 + 195 + 320 + 164 = 793\).
  3. Calculate Mean: \(\bar{x} = \frac{793}{20} = 39.65\).
  4. Answer: Mean shoe size = 39.65.
Example 3 (Hard - Pie Chart Angle & Frequency Deduction): In a school election of \(180\) students, the vote distribution is shown in a pie chart. If the sector angle for candidate Mutua is \(108^\circ\), how many students voted for Mutua?
  1. Set up proportion formula: \(\theta = \frac{f}{n} \times 360^\circ\).
  2. Substitute known values: \(108^\circ = \frac{f}{180} \times 360^\circ\).
  3. Simplify: Since \(\frac{360}{180} = 2\), we have \(108 = 2f\).
  4. Solve for \(f\): \(f = \frac{108}{2} = 54\).
  5. Answer: 54 students voted for Mutua.

Common Mistakes

Mistake Dividing by the number of unique categories instead of total frequency \(\sum f\) when calculating the mean from a frequency table.
Correction The denominator must always be the total number of individual data points (\(n = \sum f\)), not the number of rows/categories in the table.
Why it feels right Students see 4 rows in a table and naturally want to divide the sum by 4, ignoring that each row represents multiple items.
Mistake Finding the median without first arranging numbers in numerical order.
Correction Always sort the list from smallest to largest before picking the middle value!

Real World

Retail Inventory Planning: A supermarket manager in Nakuru uses the mode of clothing sizes sold to decide which sizes to restock most heavily.
Agricultural Yield Analysis: A coffee cooperative calculates the mean kilograms of coffee berries delivered per farmer to distribute annual dividend bonuses fairly.
Media & Demographic Surveys: TV networks analyze pie charts of audience shares. A sector angle of \(90^\circ\) indicates exactly \(25\%\) of total viewership.

Practice

A boda boda rider recorded his daily fuel expenditure (in KSh) for 5 consecutive days: 400, 450, 380, 520, and 450. What is the mode of his daily fuel expenditure? (Type only the number, e.g., 450)
Review the concepts above.
The following frequency distribution shows how many students obtained marks within each category on a mathematics quiz. Score | Frequency 5 | 3 6 | 7 7 | 12 8 | 9 9 | 5 What is the total number of students who sat for the quiz? (Type only the number, e.g., 36)
Review the concepts above.
A farmer in Nyandarua recorded the mass of potato bags harvested over 6 days (in kg): 120, 140, 160, 150, 190, 180. Calculate the median harvest mass in kg. (Type only the number, e.g., 155)
Review the concepts above.
A fruit vendor in Kawangware market earned the following daily profits (in KES) over a week: 800, 1200, 950, 1400, 1100, 1500, 750. What is the range of her daily profits in KES? (Type only the number, e.g., 750)
Review the concepts above.
In a survey of 120 agricultural produce buyers in Eldoret, 40 buyers selected maize as their primary purchase. What angle (in degrees) will represent maize on a pie chart? (Type only the number, e.g., 120)
Review the concepts above.
The table below shows the number of goals scored by a soccer team across 10 matches: Goals scored: 0, 1, 2, 3 Frequency: 2, 4, 3, 1 Calculate the mean number of goals scored per match. (Give your answer to 1 decimal place, e.g., 1.3) (Type only the number, e.g., 1.3)
Review the concepts above.