Statistical Reasoning and its Application in Real Life

In today’s information age, data collection and analysis are crucial. Researchers use surveys, interviews and observations to gather numerical, categorical and ordinal data. Effective organisation and visualisation through graphs, charts and tables all help to facilitate understanding.

Mastering data analysis and interpretation enable you to develop informed decision-making and problem-solving skills. You can apply data concepts to real- world projects, building a foundation for statistics, economics and data science studies.

KEY IDEAS

• A data collection instrument is a tool or method used to gather, record and measure data from respondents, participants or sources. Its purpose is to collect accurate, reliable and relevant data for research, analysis, or decision–making.

• Data Analysis is the process of extracting insights, patterns and meaning from data to inform business decisions, solve problems or answer questions.

• Data is information that can be used for analysis, decision–making, or communication

• Data organisation refers to the process of structuring, categorising and arranging data in a logical and systematic way to facilitate efficient storage, retrieval, analysis and visualisation.

• Data representation refers to the process of presenting data in a clear, concise and meaningful way to facilitate understanding, analysis and decision-making.

• Measures of Dispersion, also known as variability or spread, describes how spread out or scattered data points are from the central tendency (mean, median, mode).

Data collection instruments play a crucial role in gathering accurate and relevant information from diverse sources. These tools are vital for successful research projects, experiments and surveys. In year one we discovered that data collection instruments fall into two primary categories: quantitative and qualitative.

Quantitative instruments, such as surveys and questionnaires, gather numerical data for statistical analysis. Conversely, qualitative instruments, including interviews, focus groups and observations, provide rich insights into individuals’ thoughts, behaviours and experiences. This session will focus on developing effective data collection tools to gather high -quality data from participants.

Some Data collection instruments are as follows;

1. A questionnaire

2. An interview guide

3. Observation guide

4. A Survey

Activity 4.1: Designing and validating a questionnaire Working individually or in pairs, read through the steps to design a good questionnaire. Consider what topics you would want to investigate and how you would create a questionnaire to support this. Look at the sample questionnaire. Could you improve it? How would you go about this? Are there any areas which could cause upset to any communities? Could there be any bias creeping into the question style which may influence responses?

Such things must always be considered when constructing data collection instruments.

Steps to develop a Simple Questionnaire and validate it to collect data

Step 1: Define the research objective and scope a Clearly say what the main purpose of your research is.

b. Mention how long the study will take and point out any challenges you might face.

Step 2: Identify the target population and sample size

a. Clearly explain the group of people you are studying.

b. Try to choose a sample that truly reflects the whole group you are interested in.

Step 3: Determine the questionnaire format a Online, b Paper-based, c Mixed

Step 4: Decide on the question types a Closed-ended questions – require specific options for answers, for

example, yes/no, multiple choice, rating scale.

b Open-ended questions – allow respondents to answer in their own words and give more detailed responses.

Step 5: Develop clear, concise and logical questions. Use simple language and avoid jargon

Step 6: Organise the questionnaire a Give a brief introduction explaining the purpose of the questionnaire, remembering to assure respondents that their responses will be confidential.

b Think about question order. Begin with the simple questions, keeping the more challenging ones for later.

c Keep the questionnaire short and to the point. It should not take too long to be completed.

Step 7: Pilot-test the questionnaire (small group of about 5-10 respondents), requesting feedback on it.

Step 8: Revise and refine the questionnaire based on the feedback.

Step 9: Distribute the questionnaire.

a Decide how you will distribute. For example, in person, via email or through social media.

b Provide clear instructions on how to complete and return the questionnaire.

A Sample Questionnaire

Climate Resilience Questionnaire for Senior High Students

Introduction Thank you for participating in this survey. Your responses will help us understand your perceptions and experiences of climate change and resilience. All answers will be kept confidential.

Section 1: Demographics

1. Age: _____________________________

2. Gender:

a. Male

b. Female

3. Grade level: _____________________________

4. School location (urban/rural): _____________________ Section 2: Climate Change Awareness

1. How often do you hear about climate change in the news?

a. Daily

b. Weekly

c. Rarely

d. Never

2. What do you think is the main cause of climate change? (Select one)

a. Human activities

b. Natural processes

c. Both

d. Unsure

3. Have you learned about climate change in school?

a Yes b No Section 3: Climate Resilience Knowledge

1. What does climate resilience mean to you? (Open-ended)

2. Which of the following climate-related hazards have you experienced or learned about? (Select all that apply)

a. Floods

b. Droughts

c. Heatwaves

d. Storms

e. Other (please specify) _____________________

3. How do you think individuals can contribute to climate resilience? (Select all that apply)

a. Reducing energy consumption

b. Conserving water

c. Planting trees

d. Reducing waste

e. Other (please specify) _____________________ Section 4: Climate-Related Experiences

1. Have you or your family experienced any climate-related impacts (e.g., flooding, drought)?

a. Yes

b. No

2. How concerned are you about climate change affecting your community?

a. Very concerned

b. Somewhat concerned

c. Not very concerned

d. Not at all concerned

3. Have you participated in any climate-related activities (e.g., tree planting, clean-ups)?

a. Yes

b. No Section 5: Attitudes and Behaviours

1. How important is addressing climate change to you? (Scale: 1-5, where 1 is “not important at all” and 5 is “very important”)

2. Do you think your actions can make a difference in addressing climate change?

a. Yes

b. No

c. Unsure

3. Which climate-friendly behaviours do you practice regularly? (Select all that apply)

a. Recycling

b. Using public transport

c. Reducing meat consumption

d. Using energy-efficient appliances

e. Other (please specify) _____________________ Section 6: Conclusion Thank you for taking the time to complete this survey! Your input is valuable.

Activity 4.2: Designing an interview guide Working individually or in pairs, read through the steps to design an interview guide. Once again, consider what topics you would want to investigate in this way. Look at the sample guide. Think about what will work well and what you think could be done better.

Steps to develop an interview guide:

Step 1: Define the research objective Identify the specific goal and outcome of the interview

Step 2: Determine the interview type a Structured interview (Standard questions) b Semi–structured interview (Flexible questions) c Unstructured interview (Open–ended conversation) d Focus group interview (Group discussion)

Step 3: Identify the target population a Define the demographic characteristics b Determine the sample size

Step 4: Develop the interview protocol a Create an outline of topics and questions b Ensure questions are clear, concise, relevant, non-leading and open- ended for qualitative data

Step 5: Select the interview method a Face to face interview b Telephone interview c Video conference interview d Online interview (E.g. Email, chat)

Step 6: Prepare the interview Questions a Background questions (Demographics) b Introductory questions (Context) c Main questions (research focus) d Probing questions (follow–up) e Closing questions (Final thoughts)

Step 7: Pilot – Test the Interview a Conduct a trial interview b Refine questions and protocol

Step 8: Ensure validity and reliability a Validate questions through expert review b Ensure consistency in questioning

Step 9: Plan for data analysis a Determine data analysis methods b Develop a coding scheme Step 10: Obtain informed consent a Explain the research purpose b Ensure confidentiality c Obtain participant consent A Sample Interview Guide Here is a sample interview guide on mathematics as a core subject:

Introduction a Introduce yourself b Explain the purpose of the interview c Ensure confidentiality Section 1: Mathematics Background a What sparked your interest in mathematics?

b How long have you been studying/teaching mathematics?

c What area of mathematics do you specialise in (e.g., algebra, geometry)?

Section 2: Teaching/Learning Mathematics

a What makes mathematics challenging?

b How do you make mathematics engaging and fun?

c What teaching methods do you use to explain complex concepts?

d How do you assess understanding in mathematics?

e What role does technology play in mathematics education?

Section 3: Importance and Applications

a Why is mathematics a core subject?

b How does mathematics impact everyday life?

c Can you share examples of real-world applications of mathematics?

d How does mathematics relate to other subjects (e.g., science, engineering)?

e What career opportunities require strong mathematics skills?

Section 4: Challenges and Future

a What challenges do you face teaching/learning mathematics?

b How can mathematics education be improved?

c What are your thoughts on mathematics curriculum development?

d How can we increase student interest in mathematics?

e What’s the future of mathematics education?

Conclusion a Thank you for your time b Any final thoughts/questions

Activity 4.3: Designing an observation guide Working individually or in pairs, read through the steps to design an observation guide. Consider what topics you would want to investigate in this way. Look at the sample guide. How could it be improved? Which areas do you like?

Steps to develop an observation guide:

Step 1: Define the Research Objective

a Clearly articulate the research question or hypothesis b Identify the specific behaviours or phenomena to observe

Step 2: Choose the Observation Method

a Participant observation (active participation) b Non-participant observation (passive observation) c Structured observation (pre-defined checklist) d Unstructured observation (open-ended notes)

Step 3: Select the Observation Site

a Natural setting (e.g., classroom, workplace) b Controlled setting (e.g., laboratory) c Public or private space

Step 4: Develop an Observation Guide

a Identify key behaviours or events to observe b Create a checklist or coding scheme c Establish clear definitions and criteria

Step 5: Train Observers

a Ensure inter-rater reliability b Provide observer training and guidelines c Establish consistency in observation

Step 6: Conduct the Observation

a Record observations using, for example, field notes, audio or video recordings, photographs b Maintain observer neutrality c Ensure confidentiality

Step 7: Record and Transcribe Data

a Transcribe field notes or recordings b Code and categorise data c Use data management software

Step 8: Analyse Data

a Content analysis b Thematic analysis c Coding d Data visualisation

Step 9: Validate Findings

a Member checking (validate with participants) b Peer debriefing (discuss with classmates) c Triangulation (use multiple methods) Step 10: Report Findings a Write a detailed report b Include methodology, results and conclusions c Use visual aids (e.g., tables, figures) A Sample Observation Guide Here is a sample observation guide for data collection for an inspection of a lesson in a classroom:

Observer’s Name: ______________________________________ Date: ______________________________________________ Time: ______________________________________________ Location: ___________________________________________ Subject/Group: ______________________________________

1. Classroom Environment (20 points)

• Organisation and layout (5)

• Is the room well-organised?

• Are materials easily accessible?

• Technology and resources (5)

• Are technology tools available?

• Are educational resources adequate?

• Safety and cleanliness (5)

• Are hazardous materials handled properly?

• Is the room clean and well-maintained?

• Is the lighting sufficient?

• Is the temperature comfortable?

2. Teacher Behaviour (30 points) a Instructional methods (10)

• Are methods engaging and effective?

• Are different learning styles accommodated?

b Communication (10)

• Is communication clear and concise?

• Does the teacher listen actively?

c Classroom management (5)

• Are transitions smooth?

• Are behavioural expectations clear?

d Feedback and assessment (5)

• Is feedback constructive?

• Are assessments aligned with objectives?

3. Student Behaviour (30 points) a Engagement and participation (10)

• Are students actively engaged?

• Do students participate willingly?

b Motivation and interest (10)

• Are students motivated?

• Is the lesson relevant?

c Collaboration and teamwork (5)

• Do students work together effectively?

• Do students respect each other’s ideas?

d Self-directed learning (5)

• Are students encouraged to take ownership?

• Do students set goals?

4. Lesson Planning (20 points) a Clear objectives (5)

• Are objectives clearly stated?

• Are objectives aligned with curriculum?

b Relevant materials (5)

• Are materials relevant and engaging?

• Are materials adequate?

c Effective lesson pacing (5)

• Is pacing appropriate?

• Are transitions smooth?

d Assessment and evaluation (5)

• Is assessment aligned with objectives?

• Is feedback constructive?

Additional Notes:

…………………………………………………………………………………… Observer’s Comments:

……………………………………………………………………………………

Activity 4.4: A Survey

Working individually or in pairs, read through the steps to design a survey.

Note the similarities with a questionnaire. Consider what topics you think would suit this type of data collection. Look at the sample survey. Does this seem appropriate? How could it be improved?

Steps to develop a survey:

Step 1: Define the Goal

Clearly state what you want to achieve with the survey.

Step 2: Identify the Audience

Determine who will participate in the survey.

Step 3: Choose a Method

Decide how you will collect data (e.g., online, phone, in-person).

Step 4: Develop Questions

Create clear, concise questions that will get you the information you need.

Step 5: Test the Survey

Pilot-test with a small group to ensure the questions are understood and work as you expected.

Step 6: Collect Data

Gather responses from the identified audience.

Step 7: Analyse Data

Examine and summarise the results.

Step 8: Interpret Results

Draw conclusions based on the data analysis.

Step 9: Report Findings

Share the results with stakeholders.

Step 10: Evaluate and Improve

Assess the survey process and make adjustments for future surveys.

A Sample Survey

Here is a sample survey to investigate student subject choice:

Student Subject Choice Survey

Introduction:

Thank you for participating in this survey. Your responses will help us understand your interests and preferences in subject choices. All answers will be kept confidential.

Section 1: Demographics

1. Age: _____________________________

2. Grade Level: _____________________________

3. Current subjects: _____________________________ Section 2: Subject Interests

1. Which subjects do you enjoy learning the most? (Select up to 3)

a. Mathematics

b. Science

c. English

d. History

e. Foreign Language

f. Arts

g. Other (please specify) _____________________

2. What factors influence your subject choices? (Select all that apply)

a. Interest in the subject

b. Career prospects

c. Teacher influence

d. Parental advice

e. Peer recommendations

f. Other (please specify) _____________________ Section 3: Subject Difficulty

1. How challenging do you find each subject? (Scale: 1-5, where 1 is “easy” and 5 is “difficult”)

a. Mathematics: _______

b. Science: _______

c. English: _______

d. History: _______

e. Foreign Language: _______

f. Arts: _______ Section 4: Career Aspirations

1. What career do you aspire to? _____________________________

2. How relevant are your current subjects to your career goals? (Scale: 1-5, where 1 is “not relevant” and 5 is “very relevant”)_______________________ Section 5: Open-Ended Questions

1. What motivates you to learn a particular subject?

2. Are there any subjects you wish were offered in your school? If yes, please specify.

Conclusion:

Thank you for taking the time to complete this survey. Your input is valuable.

Application of Digital Technology in Data Collection

In designing these tools, computer application software such as Microsoft Word, Excel, Notepad, etc. can help to make the creating of these data collections easier and faster.

Figure 4.1: Sample for data collection Some ideas for topic areas which you might like to consider for your own data collection instruments:

1. Science and Technology

For example, climate change, renewable energy, artificial intelligence or plastic pollution

2. Health and Medicine

For example, mental health, nutrition, vaccines, addiction or pandemics

3. Social Sciences

For example, education systems, bullying, gender equality or cultural diversity

4. Environmental Studies

For example, sustainable agriculture, deforestation or water conservation

5. Economics and Business

For example, globalisation, consumer behaviour or labour markets

6. History and Politics

For example, democracy, elections, colonialism or human rights

7. Arts and Literature

For example, art movements, film studies, literary analysis or music therapy

8. Technology and Society

For example, social media, cybersecurity or the digital divide.

Data organisation and representation are crucial for making information easily understandable, analysable and interpretable. Effective organisation involves structuring data logically and coherently. Visualisation methods, such as charts, graphs, tables and diagrams, facilitate clear presentation. Various representation methods suit different data types and purposes.

Each method has unique advantages. For example, bar graphs highlight categorical differences, while line graphs track temporal changes. By selecting the appropriate representation, data can be communicated effectively, enabling informed decision-making and insightful analysis. Proper data organisation and representation transform complex information into actionable knowledge.

Pictorial Representation of Data

Pictorial representation of data, or data visualisation, graphically displays information to facilitate understanding and insight. Its purpose is to simplify complex data, enhance comprehension and communicate information effectively.

Pictorial representation includes bar charts, pie charts, line graphs, scatter plots and histogram.

Real–world applications span business, healthcare, education, finance, etc.

Consideration for Effective data visualisation includes clarity, accuracy, simplicity, color scheme, labelling etc.

Cumulative Frequency Curve (Ogive)

An ogive, or cumulative frequency curve, visually displays a dataset’s cumulative frequency distribution. This graphical representation helps analyse the distribution and identify key statistics like median, quartiles and percentiles.

The ogive provides a clear picture of how cumulative totals accumulate, facilitating understanding of the dataset’s distribution and statistical measures.

Activity 4.5: Constructing a cumulative frequency curve (or Ogive) Working in small groups follow the steps below to create a cumulative frequency curve:

Step 1: Gather data: collect the raw data you want to represent with a cumulative frequency curve

Step 2: Put your data into a frequency distribution table showing class intervals and frequencies

Step 3: Calculate cumulative frequencies by adding each class’s frequency to previous totals.

Step 4: Plot cumulative frequencies against each class interval’s upper boundary.

Step 5: Connect plotted points with a smooth curve to form the ogive.

Step 6: Give your graph a title and label your axes: Make sure to properly label the x-axis and the y-axis (cumulative frequency).

Example 4.1:

Draw an ogive for the distribution of test scores for twenty students in music class

Table 4.1: Frequency Distribution Table

Test Score in class intervals Frequency 0 < x ≤ 20 4 20 < x ≤ 40 6 40 < x ≤ 60 5 60 < x ≤ 80 3 80 < x ≤ 100 2

Solution

Table 4.2: Cumulative Frequency Distribution Table

Test Score Frequency Upper boundary Cumulative Frequency 0 < x ≤ 20 4 20 4 20 < x ≤ 40 6 40 10 (4+6) 40 < x ≤ 60 5 60 15(4+6+5) 60 < x ≤ 80 3 80 18(4+6+5+3) 80 < x ≤ 100 2 90 20(4+6+5+3+2) Plot the following points from the cumulative frequency table above on the graph sheet; (20, 4), (40, 10), (60, 15), (80, 18) and (100, 20).

Scale: On the x-axis, 2 cm : 20 units and 1 cm : 2 units on y-axis

Figure 4.2: A cumulative frequency curve to show the results from a music test

Example 4.2

Construct a cumulative frequency distribution graph, also known as an ogive, using the data on the heights of 50 students, displaying the cumulative frequency against the corresponding height values.

Table 4.3: Frequency Table

Height (cm) Frequency

140 < x ≤ 145 8 145 < x ≤ 150 12 150 < x ≤ 155 18 155 < x ≤ 160 10 160 < x ≤ 165 2

Solution

Calculate the cumulative frequency for each class interval:

Table 4.4: Cumulative Frequency Table

Height (cm) Frequency Upper boundary Cumulative freq.

140 < x ≤ 145 8 145 8 145 < x ≤ 150 12 150 20 150 < x ≤ 155 18 155 38 155 < x ≤ 160 10 160 48 160 < x ≤ 165 2 165 50 Plot your cumulative frequency curve, remembering to use the upper boundary.

Figure 4.3: A cumulative frequency curve to show the heights of students Types of Ogives The graphs above demonstrate the more common ‘less than’ ogive curves, which shows cumulative frequency. However, there is also another type – the ‘more than ogive’, which shows the cumulative relative frequency.

Less than Ogive (or cumulative frequency Ogive) displays the cumulative frequency of values less than or equal to a specific value.

For example, a school wants to know how many students are shorter than 165 cm.

We would plot the cumulative frequency against the upper boundary of each class interval as we have done above.

More than Ogive (or Cumulative Relative frequency Ogive) displays the cumulative frequency of values greater than or equal to a specific value.

For example, a company wants to know how many employees earn more than GHC 200,000.00 per year. We would plot the cumulative frequency against the lower boundary of each interval. Therefore, if the class intervals are 0 – 10, 11 – 20, 31- 40 etc, the ‘more than’ ogive will plot the cumulative frequency against 0, 11, 31 etc

Activity 4.6: Constructing a More than Ogive Working in small groups follow the steps below to create a more than ogive:

Step 1: Calculate Cumulative Frequency:

• Determine the cumulative frequency for each class interval by successively adding the frequencies, starting from the highest-class interval.

Step 2: Identify Lower Class boundaries

• Use the lower boundary of each class interval for plotting.

Step 3: Plot Points:

• Plot the cumulative frequency values against the lower-class boundaries on a graph.

Step 4: Draw the Curve:

• Connect the plotted points with a smooth curve.

Example 4.3:

Graph the two ogives for the following frequency distribution of the weekly wages of the given number of workers at Serene Hotel. Use your curves to find the median wage.

Table 4.5: Frequency Distribution Table

Weekly wages Number of workers 0< x ≤ 20 4 20< x ≤ 40 5 40< x ≤ 60 6 60< x ≤ 80 3

Solution

Table 4.6: Cumulative Frequency Table

Weekly wages Number of workers C.F. (Less than) C.F. (More than) 0 < x ≤ 20 4 4 18 (total) 20 < x ≤ 40 5 9 (4+5) 14 (18-4) 40 < x ≤ 60 6 15 (9+6) 9 (14-5) 60 < x ≤ 80 3 18 (15+3) 3 (9-6) For plotting the less than curve, use the points (20,4), (40,9), (60,15) and (80,18).

These are joined, freehand, to obtain the less than ogive.

For plotting the more than curve, use the points (0,18), (20,14), (40,9) and (60,3).

These are joined, freehand, to obtain the more than ogive.

These are shown in the graph below.

Figure 4.4: A cumulative frequency curve showing the weekly wages of hotel workers The median: A perpendicular line on the x-axis is drawn from the point of intersection of these curves. This perpendicular line meets the x-axis at a certain point. This determines the median. In this case, the median is 40.

Waffle Charts

Waffle charts are a visually appealing and effective way to represent data proportions and percentages. They consist of a grid of small squares or “waffles”, each representing a specific portion of the whole, typically 1% or another fixed unit of measure. This makes waffle charts particularly useful for comparing parts of a whole in a clear and concise manner.

Waffle charts are an excellent alternative to pie charts and bar graphs when you want to emphasise the composition of different categories within a dataset. They are easy to read and understand, making them a popular choice for presentations and reports where visual clarity is crucial.

Activity 4.7: Constructing a Waffle Chart

Working in pairs follow the steps below to create a waffle chart.

Step 1: Gather the data you want to represent.

Step 2: Convert the data into percentages if it is not already in this form.

Step 3: Design a 10x10 grid for 100 squares, or adjust the grid size based on your data.

Step 4: Fill the squares according to the data proportions using distinct colours for each category.

Step 5: Add labels and a legend (or key) to help interpret the chart.

What are the benefits of waffle charts?

• Waffle charts facilitate easy comparison of categorical data

• They provide a clear, visual representation of data making it easy to understand.

• A waffle chart is ideal for displaying large datasets in a compact space.

• They break down complex data into simple visual chunks.

Example 4.4

The following data illustrate the three-year enrollment growth in a Senior High School in Keta. Represent it using a waffle chart.

Table 4.7: Enrollment Data

Year Enrollment Increase (%)

2018 25 2019 45 2020 78

Solution

Figure 4.5: Waffle chart showing enrollment growth Now think about data that you consider would suit being represented in a waffle chart.

Box Plots

A box and whisker plot (or box plot) is a graph that displays the data distribution by using five numbers. Those five numbers are the minimum, first (lower) quartile, median, third (upper) quartile and maximum.

Figure 4.6: A box and whisker plot In a box and whisker plot: The left and right sides of the ‘box’ are the lower and upper quartiles. The box covers the interquartile interval, where 50% of the data is found.

• The vertical line that splits the box in two is the median. Sometimes, the mean is also indicated by a dot or a cross on the box plot.

• The whiskers are the two lines outside the box, which go from the minimum data value to the lower quartile (the start of the box) and then from the upper quartile (the end of the box) to the maximum data value.

Activity 4.8: Creating a box and whisker plot Working individually create a box-and-whisker plot to visualise the distribution of Dziifa’s 20 dice rolls, which yielded the following results: 6, 3, 3, 6, 3, 5, 6, 1, 4, 6, 3, 5, 5, 2, 2, 2, 2, 3, 2, 3.

Follow these steps to do this:

Step 1: The first thing is to order the data from smallest to largest: 1 2 2 2 2 2 3 3 3 3 3 3 4 5 5 5 6 6 6 6

Step 2: Calculate the median value.

Since the number of data points is even, we have:

Me = x₁₀ + x₁₁______ 2 = 3 + 3/2 = 3

Step 3: Next calculate the lower quartile (Q1). Remember that this is the ‘median’ of the lower half of the data. Q1 = 2

Step 4: Calculate the upper quartile (Q3). Remember that this is the ‘median’ of the upper half of the data. Q3 = 5

Step 6: Find the minimum and maximum data values. Here, minimum = 1 and maximum = 6

Step 7: Choose an appropriate scale for your data.

Step 8: Draw a box from the lower quartile value 2 to the value 3, which is the median, and put a vertical line through the median.

Step 9: Then, draw the box from the median to the upper quartile.

Step 10: Then draw your “whiskers”. Those are the lines that extend parallel with the scale from the box. The whisker goes from the lower quartile to the minimum value, 1, and from the upper quartile to the maximum value, 6.

And this is what your box plot should look like:

Figure 4.7: A box plot of Dziifa’s 20 dice rolls Now let us do an example where we need to interpret a box plot.

Example 4.5

Find the range, the interquartile range and the median of the data from the box plot below.

Solution

The minimum value of the given data is 5 and the maximum is 50 ∴ The range is R = 50 – 5 = 45.

The lower quartile is 15 and the upper quartile is 35.

∴ The interquartile range is IQR = 35 – 15 = 20 The median is line in the centre of the box which is 25.

Now think about data that you consider would suit being represented in a box plot and why it would work well.

Activity 4.9: Model and solve real-life problems involving data Embracing our immediate environment, we can uncover valuable data on pressing community concerns. By harnessing this data and leveraging the data presentation techniques we have acquired, we can uncover insights and drive meaningful discussions on these issues. Consider the following mini- project, designed to bridge the gap between theory and practice, enabling you to apply your data collection, analysis and visualisation skills to authentic, community-focused initiatives.

Can you undertake a similar project addressing concerns within your community?

Survey Title: Student Preferences and Habits

Objective: To collect, organise and present data on student preferences and habits regarding extracurricular activities, study habits and social media usage Introduction for Respondents:

1. Please answer honestly.

2. Responses are confidential.

3. Ask questions if unsure.

Section 1: Demographic Information

1. Age:

o Under 15 o 15-17 o 18-20 o Over 20

2. Gender:

o Male o Female

3. Grade Level:

o SHS 1 o SHS 2 o SHS 3 Section 2: Extracurricular Activities

1. What extracurricular activities do you participate in? (Check all that apply) o Sports o Music o Drama o Clubs o Other (please specify)

2. How many hours per week do you dedicate to extracurricular activities?

Section 3: Study Habits

1. How many hours per day do you spend studying?

2. What study methods do you prefer? (Check all that apply) o Online resources o Physical textbooks o Notes o Study groups o Other (please specify) Section 4: Social Media Usage

1. Which social media platforms do you use? (Check all that apply) o Facebook o Instagram o Twitter o TikTok o Other (please specify)

2. How many hours per day do you spend on social media?

Process

1. Data Collection:

o Distribute surveys to 50 students o Collect responses through online forms or paper surveys o Record data in a spreadsheet (e.g., Google Sheets, Excel)

2. Data Organisation:

o Categorise data by section (extracurricular activities, study habits, social media usage) o Calculate frequencies, means and percentages o Create tables and charts to visualise data

3. Data Presentation:

o Create a bar chart to display popular extracurricular activities o Use a pie chart to show study method preferences o Create a histogram to illustrate social media usage hours o Write a brief report summarising key findings

Activity 4.10: Investigating the Disadvantages of the Mean

Six students each from 3 Arts 1, 3 Arts 2 and 3 Arts 3 in ANNASS took a Core Mathematics examination. The results, in percentages, were as follows:

3 Arts 1: 100, 80, 70, 30, 20 and 0 3 Arts 2: 60, 55, 55, 45, 45 and 40 3 Arts 3: 53, 52, 51, 49, 48 and 47 Working in a group, find the mean mark for each of the three classes. Discuss the results within your group. Can you draw any conclusions from your results?

This activity shows that the mean does not tell us anything about the spread of the data. In each class, the mean mark is 50% yet the performances are very different. In 3 Arts 1, the mean hides the fact someone got 100% and another person got 0%.

Mathematicians have found ways of measuring the spread, or dispersion, of the data, not merely the central tendency. With measures of dispersion, mathematicians can determine how close together or how far apart the values under discussion are from each other. In this way, we can conclude whether the data is closely clustered around the mean, or widely scattered.

The simplest measure of dispersion is the range.

Activity 4.11: Finding the range Still within your groups, find the range for each of the three classes in activity 10 by finding the difference between the highest and lowest marks in each class. Discuss the outcome of this activity with your group. Can you interpret the results?

For 3 Arts 1, 3 Arts 2 and 3 Arts 3, the ranges are 100%, 20% and 6% respectively. Thus, you can appreciate the fact that the wider the range, the more scattered the data.

Let us know look at another measure of dispersion, the mean absolute deviation.

Activity 4.12: Finding the mean deviation If _ x ₁, _ x ₂ and _ x ₃ are the means for 3 Arts 1, 3 Arts 2 and 3 Arts 3 respectively, copy and complete the table below.

Table 4.8: Means and mean deviations for 3 Arts 1, 3 Arts 2 and 3 Arts 3 Marks in percentages Deviations from the mean 3 Arts 1 3 Arts 2 3 Arts 3 3 Arts 1 x− _ x ₁ 3 Arts 2 x− _ x ₂ 3 Arts 3 x− _ x ₃ 100 60 53 100 − 50 = 50 60 − 50 = 10 53 − 50 = 3 80 55 52 80 − 50 = 55 − 50 = 52 − 50 = 70 55 51 70 − 50 = 55 − 50 = 51 − 50 = 30 45 49 30 − 50 = 45 − 50 = 49 − 50 = 20 45 48 20 − 50 = 45 − 50 = 48 − 50 = 0 40 47 0 − 50 = 40 − 50 = 47 − 50 = ∑ ( x− ¯x₁) = ∑ (x− ¯x₂) = ∑ (x− ¯x₃)= From this you should have discovered that in each class the sum of deviations from the mean, _ x, is always zero. For this reason, this activity looks like an exercise in futility. However, activity 13 will build upon this activity to derive another measure of dispersion.

Activity 4.13: Finding the mean absolute deviation The absolute value of 10, denoted by |10| = 10. Also, |− 10| = 10 .

Therefore, when finding the absolute value, choose the number and neglect the sign. With this knowledge, copy and complete the table below. You can do this alone or with some friends.

Table 4.9: Mean absolute deviation for 3 Arts 1, 3 Arts 2 and 3 Arts 3 3 Arts 1 x− _ x ₁ 3 Arts 2 x− _ x ₂ 3 Arts 3 x− _ x ₃ 3 Arts 1 | x− ¯x ₁| 3 Arts 2 |x− ¯x ₂| 3 Arts 3 |x− ¯x ₃| 50 10 3 50 30 5 2 30 20 5 1 20 − 20 − 5 − 1 20 − 30 − 5 − 2 30 − 50 − 10 − 3 50 ∑ |x− x₁| = 200 ∑ |x− x₂| = ∑ |x− x₃| = After completing the table, you can appreciate the fact that there is a relationship between the sum of the absolute deviations from the mean and how scattered the data is. That is, the smaller the value of ∑ |x− ¯x|, the more clustered the data is about the mean. When you divide the sum of the absolute values of the deviations from the mean by the number of values, you will get the measure of dispersion known as the mean absolute deviation. The mean absolute deviations for 3 Arts 1, 3 Arts 2 and 3 Arts 3 are 200/6 = 33.3% , 40/6 = 6.7% and 12/6 = 2.0% respectively. Thus, the smaller the mean absolute deviation, the more clustered the data about the mean.

Let us now look at another measure of dispersion known as the variance. Activity 14 will assist you to find the variance of a given set of data.

Activity 4.14: Finding the variance of a given set of data Copy and complete the table below, working alone or with a small group. Here, you must find the square of the absolute deviations from the mean. When you find the sum of the squares of the deviations from the mean and divide this result by the number of values, you will get the measure of dispersion called the variance.

Table 4.10: Finding variance 3 Art 1 x− _ x ₁ 3 Art 2 x− _ x ₂ 3 Art 3 x− _ x ₃ 3 Art 1 (x− ¯x ₁)² 3 Art 2 (x− ¯x ₁)² 3 Art 3 (x− ¯x ₁)²50 10 3 30 5 2 20 5 1 − 20 − 5 − 1 − 30 − 5 − 2 − 50 − 10 − 3 ∑ (x− ¯x₁)²∑ (x− ¯x₂)²∑ (x− ¯x₃)² The variance is given by v = ∑(x− _ x )²_________ n . Like all the measures of dispersion, the smaller the variance the more clustered that data about the mean.

The square root of the variance is called the standard deviation. The standard deviation is an important measure in banking, the social and physical sciences as well as engineering.

Example 4.6

The ages, in years, of six boys playing football on a park are 12, 14, 16, 18 and 10.

1. Calculate the:

a. range;

b. mean absolute deviation;

c. variance;

d. standard deviation.

2. What conclusion can you draw from your answers?

Solution

Table 4.11: Calculating variance and standard deviation x x− _ x |x − ¯x | (x − x)²x²12 − 2 2 4 144 14 0 0 0 196 16 2 2 4 256 18 4 4 16 324 10 − 4 4 16 100 ∑ x = 70 ∑ |x − ¯x| =12 ∑ (x− ¯x)²= 40 ∑x²= 1020

1. a Range = highest value – lowest value = 18 − 10 = 8 years b The mean absolute deviation = ∑|x − _ x |________ n = 12/5 = 2.4 years c The variance = ∑(x− _ x )²_________ n = 40/5 = 8 Alternatively, we can use another formula to find the variance.

The variance = ∑x²____ n − (∑ x/n ) 2 = 1020/5 − (70/5 ) 2 = 204− 14²= 204 − 196 = 8 d The standard deviation is the square root of the variance. Hence, the standard deviation is √8 = 2.828 ≈ 3 years.

2. One standard deviation of approximately 3 years suggests that most of the values should be about ± 3 years away from the mean. Therefore, with a mean of 14 years and a standard deviation of 3 years shows that many of the data points will be from about 11 to 17 years.

Note, with large datasets and the data being symmetrical about the mean about 68% of data points will fall within 1 standard deviation either side of the mean and 95% of data points within 2 standard deviations. When a data set is small or skewed this will not be as definitive.

Example 4.7

The ages, in years, of the students in 2 Agric. are as follows: 19, 15, 18, 16, 15, 17 and 16.

Find the:

a. median

b. interquartile range

c. quartile deviation.

Solution

a. To find the median, pick the central value after arranging the values of the data in ascending or descending order of magnitude: 15, 15, 16, 16, 17, 18,

19. Thus, the median is 16 years.

b. The interquartile range = upper quartile − lower quartile. Note that the upper and lower quartiles are the values of the data that are associated with the 3∑ f/4 th and the ∑ f/4 th positions. Because the quartiles depend on the position of the values of the data, it is expedient to get natural numbers.

For this reason, if there is an odd number of values, mathematicians add one to the total frequency before dividing it by 2 or 4. As you can see, the given data has seven values. Thus, we add one to the total frequency before we find the lower and upper quartiles. Thus, the values associated with the second and sixth positions are the lower and upper quartiles respectively.

Thus, 15 years and 18 years are the lower and upper quartiles respectively.

Since the difference between the upper and lower quartiles is 3, 3 years is the interquartile range.

c. The quartile deviation, also known as the semi-interquartile range, is obtained when the interquartile range is divided by 2. Thus, the quartile deviation is 1.5 years.

Review Questions Section 4A

1. What is the primary purpose of a data collection instrument in mathematics?

2. What are the two main types of data collection instruments?

3. What is the difference between a survey and a questionnaire?

4. What are the key characteristics of a well-designed data collection instrument?

5. What is pilot-testing and why is it essential in data collection instrument design?

6. How do you ensure validity in a data collection instrument?

7. What is the difference between open-ended and closed-ended questions?

8. What are some common data collection instruments used in mathematics education research?

9. How do you ensure reliability in a data collection instrument?

10. What are some potential sources of error in data collection instruments?

Review Questions Section 4B

1. The followings are the marks scored by 80 candidates in an examination

Table 4.12: Frequency Table

Marks (%) Frequency

40 < x ≤ 50 2 50 < x ≤ 60 6 60 < x ≤ 70 8 70 < x ≤ 80 18 80 < x ≤ 90 28 90 < x ≤ 100 18

a) Construct a cumulative frequency table and use it to draw cumulative frequency curve (less than) Ogive

b) Constructing a Greater Than Ogive:

2. A high school is analysing the distribution of students’ participation in various extracurricular activities. The data for a total of 100 students is as follows:

a. Sports: 35 students

b. Music: 25 students

c. Science: 18 students

d. Art: 13 students

e. Drama: 9 students Create a waffle chart to visually represent the distribution of students’ participation in these extracurricular activities.

3. A statistics teacher collects the final exam scores of 15 students in a class.

The scores are as follows: 72, 85, 90, 65, 78, 80, 92, 88, 76, 84, 91, 87, 69, 95, 82. Calculate the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum of the exam scores. Using this information draw a box plot to represent the distribution of the exam scores.

Review Questions Section 4C

1. Which of the following measures is not a measure of dispersion?

a. mean absolute deviation

b. median

c. range

d. standard deviation

2. The mean and the standard deviation are closely related, true or false?

3. The table below illustrates the marks obtained by 40 students in St. Mary’s Boys’ SHS in a class test marked out of 10.

Table 4.13: Frequency Table

Marks 1 2 3 4 5 Frequency 6 14 12 5 3 Find the quartile deviation for the distribution below and interpret your answer.

4. An African-American millionaire wants to set up a factory in West Africa to process gold into bullions and jewellery. His agents provided him with the information in the table below.

Table 4.14: Frequency Table

Country Mean interest rates per annum Standard deviation of interest rates Burkina Faso 18% 5% Ghana 20% 10% Ivory Coast 24% 20% You are a member of the mathematical club in your school. You have been tasked to advise the millionaire based on the data given in the table as to which country is best suited for the investment. Your report should not be more than 250 words.