Statistics with Ungrouped Data
Senior 1 · Mathematics
Statistics with Ungrouped Data
Lesson 1 of 5
22 min
In August 2022 enumerators visited every household in Rwanda for the Fifth Population and Housing Census, and counted about 13.2 million people. The National Institute of Statistics of Rwanda (NISR) then organised the answers into tables and graphs, and the government uses them to decide where to build schools and health centres. Collecting, organising and explaining numbers like these is the work of statistics, and in this unit you will do it with data from your own school and village.
Definition
Statistics: the branch of mathematics that collects, organises, presents, analyses and interprets data.
Data: facts or measurements collected about people or things, such as ages, marks or favourite subjects. Raw data are data as they were collected, before they are sorted.
Definition
Variable: a property that can take different values from one person or thing to another, for example "height" or "district of birth".
Population: the whole group you want to find out about. Sample: the part of the population from which data are actually collected.
Example: to study the heights of all 240 S1 learners in a school (the population), you might measure 40 of them (a sample).
Definition
Qualitative (categorical) data: data that are words or categories, not amounts. Example: favourite sport, eye colour, the district where you were born.
Quantitative (numerical) data: data that are numbers obtained by counting or measuring. Example: number of siblings, mass in kg.
Definition
Discrete data: quantitative data that can take only separate values, usually whole numbers, because they are counted. Example: the number of goals in a match can be 0, 1, 2, …, but never 2.5.
Continuous data: quantitative data that can take any value in a range, because they are measured. Example: a height can be , , , …; the answer is only as exact as the measuring tool.
Worked example 1: classifying variables
Classify each variable as qualitative, discrete or continuous: (a) favourite subject (b) number of people in a household (c) the time a learner takes to walk to school (d) the district a learner was born in (e) shoe size (f) the mass of a bag of beans.
Solution
| Variable | Number? | Counted or measured? | Type |
|---|---|---|---|
| (a) favourite subject | no | – | qualitative |
| (b) people in a household | yes | counted | discrete |
| (c) time to walk to school | yes | measured | continuous |
| (d) district of birth | no | – | qualitative |
| (e) shoe size | yes | chosen from fixed sizes (38, 39, 40, …) | discrete |
| (f) mass of a bag of beans | yes | measured | continuous |
Shoe size is discrete even though feet are measured, because shoes come only in fixed sizes: nobody buys size 38.7.
Data you collect yourself are primary data; data that someone else has already collected, such as NISR census tables or Meteo Rwanda rainfall records, are secondary data. There are four main ways to collect primary data:
| Method | How it works | Good for |
|---|---|---|
| questionnaire or survey | the same written questions are given to many people | opinions, habits, facts about people |
| interview | questions are asked face to face and the answers recorded | people who cannot read, or detailed answers |
| observation | you watch and record what happens, often with tally marks | traffic, behaviour, events |
| measurement or experiment | you measure with an instrument | heights, masses, times, rainfall |
When you observe, record each event as it happens with a tally mark, then count the marks at the end. S1 learners at a school in Nyamirambo counted the vehicles passing their gate for ten minutes:
Definition
Tally mark: a stroke drawn for each item as it is counted. Every fifth stroke is drawn across the previous four, making a "gate" of five.
Frequency: the number of times a value or category occurs. In the sheet above, the frequency of "car" is 14.
Worked example 2: population and sample
A head teacher wants to know how long the 240 S1 learners in her school spend on homework each evening. She asks the 40 learners of S1 A, her own class.
(a) What are the population and the sample? (b) What type of data will she collect? (c) Is her sample a fair one? How could she improve it?
Solution
Worked example 3: judging a survey question
Two learners wrote questions for a survey on Umuganda, the community work held on the last Saturday of each month:
Q1: "Don't you agree that Umuganda is the best way to keep our village clean?"
Q2: "How many of the last three Umuganda sessions did you attend? ☐ 0 ☐ 1 ☐ 2 ☐ 3"
Which question will give more useful data? Explain.
Solution Q2 is better. Q1 is a leading question: its wording pushes people to answer "yes", so the answers tell you more about the question than about the people. It also asks for an opinion with no clear scale. Q2 asks for a fact over a fixed period, and its tick boxes cover every possible answer exactly once, so the answers can be counted in a frequency table and compared.
Warning
Numbers that are not quantitative data
Tip
Count → discrete; measure → continuous. If you can ask "how many?", the data are discrete; if you ask "how long?", "how heavy?" or "how much time?", they are continuous.
FuseSchool: Types of Data (4 min). Watch the difference between qualitative and quantitative data, and between discrete and continuous data, with an example of each.
Practice
Answer in full sentences where you are asked to explain.
Sources and credits
The Rwandan examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.
