Subject: Mathematics | Level: JHS (Basic 7 – 9) | Topic: Statistics and Data Handling | Difficulty: Intermediate
Introduction
Statistics is the branch of Mathematics that deals with collecting, organising, presenting, analysing, and interpreting data. In everyday life, statistics is everywhere – in sports scores, market prices, school examination results, hospital records, and weather reports. Understanding statistics helps you make sense of information and make better decisions. This lesson covers the key statistical concepts examined at BECE.
1. Collecting and Organising Data
Types of Data
- Quantitative data: data expressed in numbers (e.g. test scores, heights, temperatures)
- Qualitative data: data described in words/categories (e.g. favourite colour, type of transport)
- Discrete data: can only take specific values (e.g. number of students: 1, 2, 3…)
- Continuous data: can take any value within a range (e.g. height: 1.63 m, 1.64 m…)
Frequency Tables
A frequency table organises raw data into groups (or individual values) and shows how many times (the frequency) each value or group occurs.
Example: The following scores were recorded for 10 students in a quiz: 6, 8, 7, 9, 6, 8, 8, 7, 6, 9
| Score | Tally | Frequency |
|---|---|---|
| 6 | ||| | 3 |
| 7 | || | 2 |
| 8 | ||| | 3 |
| 9 | || | 2 |
| Total | 10 |
2. Measures of Central Tendency
Mean (Arithmetic Mean)
The mean is the average value. It is the most commonly used measure of central tendency.
Formula: Mean = Sum of all values ÷ Number of values
Example: Find the mean of: 12, 15, 18, 20, 25
Sum = 12 + 15 + 18 + 20 + 25 = 90
Number of values = 5
Mean = 90 ÷ 5 = 18
Mean from a Frequency Table:
Mean = Σ(fx) ÷ Σf, where f = frequency and x = value
Example: Using the quiz score table above:
Σ(fx) = (6×3) + (7×2) + (8×3) + (9×2) = 18 + 14 + 24 + 18 = 74
Σf = 3 + 2 + 3 + 2 = 10
Mean = 74 ÷ 10 = 7.4
Median
The median is the middle value when data is arranged in order from smallest to largest. If there is an even number of values, the median is the average of the two middle values.
Example 1 (odd number of values): Find the median of: 4, 7, 9, 12, 15
Already in order. Middle value (3rd of 5) = 9. Median = 9.
Example 2 (even number of values): Find the median of: 3, 5, 8, 12, 14, 20
Already in order. Two middle values are 8 and 12.
Median = (8 + 12) ÷ 2 = 20 ÷ 2 = 10
Mode
The mode is the value that appears most often in a data set. There can be one mode, more than one mode, or no mode.
Example: Find the mode of: 3, 5, 5, 7, 8, 5, 9, 3
The value 5 appears 3 times (more than any other). Mode = 5.
Example (bimodal – two modes): 2, 4, 4, 6, 6, 8
Both 4 and 6 appear twice. The data is bimodal: modes are 4 and 6.
Range
The range measures the spread of data.
Formula: Range = Highest value – Lowest value
Example: Data: 8, 12, 5, 19, 3, 15. Range = 19 – 3 = 16
3. Which Average to Use?
- Mean: best for data without extreme values; gives a precise average
- Median: best when there are extreme values (outliers) that would distort the mean
- Mode: best for non-numerical data, or when the most common value matters (e.g. most popular shoe size)
Example: Monthly salaries in GHS: 800, 850, 900, 900, 12,000
Mean = (800+850+900+900+12,000) ÷ 5 = 15,450 ÷ 5 = 3,090 (distorted by 12,000)
Median = 900 (middle value)
The median gives a better picture of a typical salary here.
4. Presenting Data Graphically
Bar Charts
A bar chart uses rectangular bars to show frequencies. Each bar represents a category, and the height shows the frequency. Bars should be equal width with equal spaces between them. Always include a title, labelled axes, and a scale on the vertical axis.
Pie Charts
A pie chart shows data as slices of a circle. The whole circle = 360°. Each slice represents a proportion of the total.
Formula for angle of each sector: Angle = (Frequency ÷ Total) × 360°
Example: 40 students were asked their favourite subject: Maths (12), English (10), Science (8), Social Studies (10).
- Maths: (12/40) × 360 = 108°
- English: (10/40) × 360 = 90°
- Science: (8/40) × 360 = 72°
- Social Studies: (10/40) × 360 = 90°
- Total: 108 + 90 + 72 + 90 = 360° ✓
Line Graphs
Line graphs are used to show changes over time. Points are plotted and connected with straight lines. They are particularly useful for showing trends (e.g. changes in temperature, population, or prices over months or years).
Summary
- Mean = sum of values ÷ number of values (or Σfx ÷ Σf from a frequency table)
- Median = middle value when arranged in order (average of two middle values for even count)
- Mode = most frequently occurring value
- Range = highest value – lowest value
- Pie chart angle = (Frequency ÷ Total) × 360°
Practice Questions
- The ages of 7 children are: 8, 10, 9, 12, 8, 11, 9. Find the mean, median, mode, and range.
- The scores of 5 students in a test are: 55, 70, 48, 70, 82. Find the mean and median.
- In a class survey, 30 students chose their favourite Ghanaian food: fufu (12), jollof rice (9), banku (6), waakye (3). Calculate the angle for each sector of a pie chart.
- A data set has a mean of 15 and 8 values. What is the sum of all the values?
- Which measure of central tendency would be most appropriate to describe the modal shoe size sold in a shop? Give a reason.
Answers: (1) Mean=9.57, Median=9, Mode=8 and 9 (bimodal), Range=4 (2) Mean=65, Median=70 (3) Fufu=144°, Jollof rice=108°, Banku=72°, Waakye=36° (4) Sum=15×8=120 (5) Mode, because we want the most common size, not an average
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