# IGCSE Higher: Statistical Measures and Representation

## Scheme of work: IGCSE Higher: Year 10: Term 2: Statistical Measures

#### Prerequisite Knowledge

1. Understanding and Constructing Frequency Tables
• Concept: Knowing how to create and interpret frequency tables to organize data.
• Example: A frequency table showing the number of students scoring different ranges of marks in a test. $\begin{array}{|c|c|} \hline \text{Marks Range} & \text{Frequency} \\ \hline 0-10 & 5 \\ 10-20 & 12 \\ 20-30 & 9 \\ \hline \end{array}$
2. Calculating Mean, Median, and Mode from Listed Data
• Concept: Knowing how to calculate the mean, median, and mode from a list of data points.
• Example: Given the data set $$\{3, 7, 7, 2, 9\}$$: $\text{Mean} = \frac{3+7+7+2+9}{5} = \frac{28}{5} = 5.6 \\ \text{Median} = 7 \quad (\text{middle value when ordered}) \\ \text{Mode} = 7 \quad (\text{most frequent value})$
3. Calculating Mean, Median, and Mode from Data in a Frequency Table
• Concept: Knowing how to calculate the mean, median, and mode from data presented in a frequency table.
• Example: Given the frequency table: $\begin{array}{|c|c|} \hline \text{Value} & \text{Frequency} \\ \hline 1 & 2 \\ 2 & 3 \\ 3 & 5 \\ \hline \end{array}$ $\text{Mean} = \frac{1 \times 2 + 2 \times 3 + 3 \times 5}{2+3+5} = \frac{23}{10} = 2.3 \\ \text{Median} = 2 \quad (\text{value at the middle of cumulative frequency}) \\ \text{Mode} = 3 \quad (\text{highest frequency})$
4. Drawing and Interpreting Pie Charts
• Concept: Understanding how to draw and interpret pie charts to represent data.
• Example: Drawing a pie chart for the distribution of different colors of cars in a parking lot: $\text{Red: } 30\% \\ \text{Blue: } 20\% \\ \text{Green: } 25\% \\ \text{Other: } 25\%$

#### Success Criteria

1. Construct and Interpret Histograms
• Objective: Students should be able to construct and interpret histograms for continuous variables with unequal class intervals.
• Example: Construct a histogram for the following data: $\begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-10 & 5 \\ 10-20 & 12 \\ 20-30 & 9 \\ \hline \end{array}$ \text{Interpret the histogram to find the modal class interval.}
2. Construct Cumulative Frequency Diagrams from Tabulated Data
• Objective: Students should be able to construct cumulative frequency diagrams from tabulated data.
• Example: Given the following data, construct a cumulative frequency diagram: $\begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-10 & 5 \\ 10-20 & 12 \\ 20-30 & 9 \\ 30-40 & 14 \\ 40-50 & 10 \\ \hline \end{array}$ \text{Use the diagram to estimate the median.}
3. Use Cumulative Frequency Diagrams
• Objective: Students should be able to use cumulative frequency diagrams to find quartiles and percentiles.
• Example: Given a cumulative frequency diagram, find the lower quartile ($$Q_1$$), median ($$Q_2$$), and upper quartile ($$Q_3$$). $\text{Estimate } Q_1 \text{ at the 25th percentile}, \\ \text{Estimate the median } Q_2 \text{ at the 50th percentile}, \\ \text{Estimate } Q_3 \text{ at the 75th percentile}.$
4. Estimate the Median from a Cumulative Frequency Diagram
• Objective: Students should be able to estimate the median value from a cumulative frequency diagram.
• Example: Given a cumulative frequency diagram with 50 data points, find the median: $\text{Median position} = \frac{n+1}{2} = \frac{50+1}{2} = 25.5 \\ \text{Locate the 25.5th value on the cumulative frequency diagram to estimate the median.}$

#### Key Concepts

1. Construct and Interpret Histograms
• Key Concept: Understanding that histograms represent the distribution of continuous data, where the area of each bar represents the frequency of data within each interval. Unlike bar charts, histograms do not have gaps between the bars.
2. Construct Cumulative Frequency Diagrams from Tabulated Data
• Key Concept: Recognizing that cumulative frequency diagrams plot the cumulative total of frequencies up to each class interval, providing a visual representation of how frequencies accumulate over the range of data.
3. Use Cumulative Frequency Diagrams
• Key Concept: Knowing that cumulative frequency diagrams can be used to find quartiles and percentiles, which provide measures of position within the data set. The diagram allows for quick estimation of these values.
4. Estimate the Median from a Cumulative Frequency Diagram
• Key Concept: Understanding that the median is the middle value of an ordered data set and can be found by locating the position $$\frac{n+1}{2}$$ on the cumulative frequency diagram.

#### Common Misconceptions

1. Construct and Interpret Histograms
• Common Misconception: Students might incorrectly treat histograms as bar charts, leaving gaps between the bars and not adjusting the bar widths for unequal class intervals.
• Example: When constructing a histogram, a student might leave gaps between the bars or not adjust the width for unequal intervals, leading to incorrect representation of the data. $\text{Incorrect: Treating histogram bars as separate entities like in bar charts.}$
2. Construct Cumulative Frequency Diagrams from Tabulated Data
• Common Misconception: Students might misunderstand the cumulative frequency concept and incorrectly plot frequencies instead of cumulative frequencies.
• Example: Given the following data, a student might plot the frequency values directly on the cumulative frequency diagram instead of adding them cumulatively: $\begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-10 & 5 \\ 10-20 & 12 \\ 20-30 & 9 \\ 30-40 & 14 \\ 40-50 & 10 \\ \hline \end{array}$ \text{Correct: Add frequencies cumulatively (5, 17, 26, 40, 50).}
3. Use Cumulative Frequency Diagrams
• Common Misconception: Students might incorrectly interpret the quartiles and percentiles by not accurately reading the cumulative frequency values on the graph.
• Example: A student might estimate the lower quartile ($$Q_1$$) at the 20th percentile instead of the 25th percentile on the cumulative frequency diagram. $\text{Incorrect: Reading } Q_1 \text{ at the 20th percentile.} \\ \text{Correct: Reading } Q_1 \text{ at the 25th percentile.}$
4. Estimate the Median from a Cumulative Frequency Diagram
• Common Misconception: Students might incorrectly identify the median by misinterpreting the cumulative frequency diagram or using the wrong data point.
• Example: Given a cumulative frequency diagram with 50 data points, a student might incorrectly identify the median by taking the 25th data point instead of the 25.5th: $\text{Median position} = \frac{n+1}{2} = \frac{50+1}{2} = 25.5 \\ \text{Correct approach: Locate the 25.5th value on the cumulative frequency diagram.}$

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