3.4 Data Handling, Graphs & Probability

Data Handling, Graphs & Probability

Quick Summary: Statistics and data handling involve collecting, organising, presenting, and interpreting quantitative information. Probability allows us to calculate the mathematical likelihood of specific outcomes. This section covers frequency tables, graphical displays (bar charts, line graphs, pictographs, pie charts), statistical measures (mean, median, mode, range), and basic probability calculations.

Data handling empowers students to make sense of information presented in tables, charts, and news reports across the Caribbean.

Data Collection & Frequency Tables

Raw data consists of unorganised numbers or observations. To analyse raw data effectively, statisticians group and organise it using tally charts and frequency tables.

The Tallying System

• Single items are marked as vertical strokes ($|$).
• The fifth item is drawn diagonally across four vertical strokes ($|!|!|!|!!/!!$) to complete a bundle of 5, making counting efficient.

Sample Frequency Table: Favourite Sports in Grade 6

Visual Data Representation: Bar Graphs, Line Graphs & Pictographs

Graphical representations allow data to be visually analysed and compared quickly.

1. Bar Graphs

• Used for discrete categorical data (e.g., favourite foods, eye colours).
• Bars must have equal width and uniform spacing between them.
• The vertical axis ($y$-axis) represents frequency, while the horizontal axis ($x$-axis) lists categories.

2. Line Graphs

• Used to track continuous changes over time (e.g., monthly rainfall, daily temperatures, stock prices).
• Points are plotted and connected with straight line segments.

3. Pictographs

• Use visual symbols or icons to represent data quantities.
• Must always include a Key / Legend defining what each symbol represents (e.g., ⚽ = 4 students).

Constructing & Interpreting Pie Charts

A pie chart is a circular graph divided into proportional sectors, where each sector represents a fraction of the whole dataset ($360^\circ$).

Formula for Sector Angle

$\text{Sector Angle} = \frac{\text{Category Frequency}}{\text{Total Frequency}} \times 360^\circ$

Worked Example: Pie Chart Sector Angles

Problem: In a class of 40 CPEA students, 15 choose Agriculture, 10 choose Computer Science, 10 choose Art, and 5 choose Music as their elective. Calculate the sector angle for each subject on a pie chart.

• Total Frequency: $40 \text{ students}$.
• Agriculture: $\text{Angle} = \frac{15}{40} \times 360^\circ = \frac{3}{8} \times 360^\circ = 135^\circ$
• Computer Science: $\text{Angle} = \frac{10}{40} \times 360^\circ = \frac{1}{4} \times 360^\circ = 90^\circ$
• Art: $\text{Angle} = \frac{10}{40} \times 360^\circ = 90^\circ$
• Music: $\text{Angle} = \frac{5}{40} \times 360^\circ = \frac{1}{8} \times 360^\circ = 45^\circ$
• Verification Check: $135^\circ + 90^\circ + 90^\circ + 45^\circ = 360^\circ$.

Measures of Central Tendency & Range

Statistical measures describe the central summary point and the spread of values in a dataset.

Key Definitions & Formulas

1. Mean (Average): The arithmetic sum of all data values divided by the total number of values. $\text{Mean} = \frac{\sum x}{n} = \frac{\text{Sum of all scores}}{\text{Total number of scores}}$
2. Median: The middle value when data points are arranged in ascending or descending order.
   • For an odd count of data points, it is the exact middle number.
   • For an even count of data points, it is the average of the two middle numbers.
3. Mode: The value that occurs most frequently in a dataset. A dataset may have one mode (unimodal), multiple modes (bimodal), or no mode.
4. Range: The difference between the highest (maximum) and lowest (minimum) values in a dataset, measuring data dispersion. $\text{Range} = \text{Maximum Value} - \text{Minimum Value}$

Worked Example: Test Scores Analysis

Problem: Calculate the Mean, Median, Mode, and Range for the following set of student test scores: $85, 92, 78, 85, 90, 65, 85$.

• Step 1 (Order data ascending): $65, 78, 85, 85, 85, 90, 92$ ($n = 7$ scores).
• Step 2 (Calculate Mean): $\text{Sum} = 65 + 78 + 85 + 85 + 85 + 90 + 92 = 580$ $\text{Mean} = \frac{580}{7} \approx 82.86$
• Step 3 (Determine Median): With 7 items, the 4th item is the middle value. The 4th number is 85.
• Step 4 (Determine Mode): The score 85 appears 3 times (more than any other score).
• Step 5 (Calculate Range): $\text{Range} = 92 - 65 = 27$
• Summary: Mean $\approx 82.86$, Median $= 85$, Mode $= 85$, Range $= 27$.

Introduction to Simple Probability

Probability measures the numerical likelihood of a specific event occurring. It is expressed as a fraction, decimal, or percentage ranging from 0 (impossible event) to 1 (certain event).

Probability Scale

• 0 (0%): Impossible (e.g., rolling a 7 on a standard 6-sided die).
• 0.5 (50%): Equally likely (e.g., flipping a fair coin and landing on Heads).
• 1 (100%): Certain (e.g., selecting a green marble from a bag containing only green marbles).

Simple Probability Formula

$P(\text{Event}) = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}}$

Complementary Probability

The probability of an event NOT happening is given by: $P(\text{NOT Event}) = 1 - P(\text{Event})$

Worked Example: Marble Bag Probability

Problem: A bag contains 4 red marbles, 5 blue marbles, and 3 yellow marbles. A student draws one marble at random without looking.

1. What is the probability of drawing a blue marble?
2. What is the probability of NOT drawing a red marble?

• Total Marbles: $4 + 5 + 3 = 12 \text{ marbles}$.
• Part 1 (Blue Marble): $P(\text{Blue}) = \frac{5}{12}$
• Part 2 (Not Red Marble):
  • Favourable non-red outcomes $= 5 \text{ blue} + 3 \text{ yellow} = 8 \text{ marbles}$. $P(\text{Not Red}) = \frac{8}{12} = \frac{2}{3}$
• Answer: Probability of blue is $\frac{5}{12}$; probability of not red is $\frac{2}{3}$.