Statistical terms, types of data and frequency tables – Descriptive Statistics (Senior 4 Basic Mathematics) | Ganzaa

Descriptive Statistics

Lessons

Senior 4 · Basic Mathematics

Descriptive Statistics

5 lessons

Lesson 1 of 5

Statistical terms, types of data and frequency tables

25 min

Thirty numbers from the 2022 census

From 16 to 30 August 2022 the National Institute of Statistics of Rwanda (NISR) carried out the Fifth Population and Housing Census, which counted 13 246 394 residents. Its main report lists the population of each of the 30 districts, from 318 126 in Nyaruguru to 879 505 in Gasabo (NISR, RPHC5 Main Indicators Report, Table 1). Thirty numbers are already too many to take in at a glance. How many districts have fewer than 400 000 people? Is a district of half a million unusual? To answer, statisticians first organise data, and this lesson shows how.

Definition

Statistics: the science of collecting, organising, presenting, analysing and interpreting data. Descriptive statistics summarises the data you have with tables, graphs and numbers such as averages.

Population: the whole group being studied. Sample: a part of the population chosen to represent it. A census collects data from every member of the population; a sample survey collects it from a sample.

Variable: a characteristic that can take different values, such as age or district. Observation (datum): one recorded value of a variable.

Definition

Primary data: collected first-hand for the purpose at hand (your own questionnaire). Secondary data: collected by someone else and reused (the NISR census tables).

Frequency $f$ : the number of times a value, or a value in a class, occurs. Relative frequency: $\frac{f}{n}$ , where $n = \sum f$ . Cumulative frequency: the running total of the frequencies up to and including a value or class.

Types of data

Frequency tables

A frequency distribution lists each value (or class of values) with its frequency. For data with few different values, list every value: an ungrouped table. For data with many different values, such as district populations, group the values into classes: a grouped table.

Term	Meaning	For the class 40–49 of whole-number marks
Class limits	The smallest and largest values that can be written in the class	40 and 49
Class boundaries	Halfway between one class's upper limit and the next class's lower limit, so classes touch without gaps	39.5 and 49.5
Class width $c$	Upper boundary minus lower boundary	$49.5 - 39.5 = 10$
Midpoint (class mark)	$\frac{\text{lower limit} + \text{upper limit}}{2}$ , the value that represents the class	$\frac{40 + 49}{2} = 44.5$

Choose 5 to 10 classes of equal width where possible, with no gaps or overlaps, so that every observation belongs to exactly one class.

Tip

Ask "can it be counted or measured?" If the answer is a word or a label, the data are qualitative. If it is a count (you can say "how many"), it is discrete. If it is a measurement (you can say "how much" or "how long", and a finer instrument would give more decimals), it is continuous.

✦

Worked example 1: an ungrouped table

A survey records the number of people in 20 households in a village: 4, 3, 5, 2, 4, 1, 3, 4, 6, 3, 2, 4, 5, 7, 3, 4, 2, 6, 1, 5. (Illustrative data.) Make a frequency table with relative and cumulative frequencies.

Household size $x$	1	2	3	4	5	6	7	Total
Frequency $f$	2	3	4	5	3	2	1	20
Relative frequency $\frac{f}{20}$	0.10	0.15	0.20	0.25	0.15	0.10	0.05	1.00
Cumulative frequency	2	5	9	14	17	19	20

Reading the table: the most common size is 4 (a quarter of households); 9 households, that is $\frac{9}{20} = 45\%$ , have 3 people or fewer. The relative frequencies add up to 1, and the last cumulative frequency equals $n$ : two quick checks.

Worked example 2: a grouped table from the census

Group the populations of Rwanda's 30 districts (NISR, 2022) into classes of width 100 000, starting at 300 000.

Population (thousands)	Districts	$f$	Relative frequency	Cumulative $f$	Midpoint
$300 \le x \lt 400$	Nyaruguru, Nyabihu, Muhanga, Ruhango, Rulindo, Gakenke, Nyanza, Ngororero, Rutsiro, Nyamagabe, Karongi, Nyarugenge, Huye, Burera, Gisagara	15	0.500	15	350
$400 \le x \lt 500$	Ngoma, Nyamasheke, Gicumbi, Kamonyi, Kayonza, Kirehe, Musanze, Rwamagana, Rusizi, Kicukiro	10	0.333	25	450
$500 \le x \lt 600$	Rubavu, Bugesera, Gatsibo	3	0.100	28	550
$600 \le x \lt 700$	Nyagatare	1	0.033	29	650
$700 \le x \lt 800$	none	0	0.000	29	750
$800 \le x \lt 900$	Gasabo	1	0.033	30	850
Total		30	1.000 (rounded)

Interpretation: half the districts have between 300 000 and 400 000 residents, and 25 of the 30 (83%) have fewer than 500 000. The distribution has a long tail to the right: a few large districts, led by Gasabo in Kigali. Because a district's population is a count, the data are discrete; with values this large we write classes as $300\,000 \le x \lt 400\,000$ , treating them like continuous data.

Worked example 3: limits, boundaries, width and midpoints

The marks of a class in a test (whole numbers) are grouped as 30–39, 40–49, 50–59, 60–69, 70–79. Give the boundaries, width and midpoint of the class 60–69, and explain why boundaries are needed.

Boundaries: $\frac{59 + 60}{2} = 59.5$ and $\frac{69 + 70}{2} = 69.5$ . Width: $c = 69.5 - 59.5 = 10$ . Midpoint: $\frac{60 + 69}{2} = 64.5$ .

The limits 59 and 60 leave a gap; a mark of 59.6 (if halves were allowed) would belong nowhere. Boundaries close the gaps, and they are what you plot on the horizontal axis of a histogram in the next lesson.

Warning

Common mistakes. Overlapping classes such as 300–400 and 400–500 with no rule for 400: say $300 \le x \lt 400$ . Using the class limits instead of the boundaries to find the width (49 − 40 = 9 is wrong; the width is 10). Calling data "discrete" because they are rounded: a time rounded to the nearest minute is still continuous.

Corbettmaths, "Discrete and Continuous Data". After watching, classify every variable in this lesson's examples, including a district's population and its land area.

✦

Practice

Try each one before opening the answers. They get harder as you go.

Classify each variable as nominal, ordinal, discrete or continuous: (a) the number of cows on a farm; (b) the mass of a bag of coffee beans; (c) a pupil's home district; (d) a hotel's star rating; (e) the time a bus takes from Huye to Kigali.
Explain the difference between a census and a sample survey, and give one advantage of each.
A shop counts its customers on 24 days: 12, 15, 9, 14, 12, 18, 11, 15, 12, 20, 13, 9, 16, 12, 14, 17, 10, 12, 15, 11, 13, 19, 14, 12. Make a grouped frequency table with classes 9–11, 12–14, 15–17, 18–20, including cumulative frequencies.
For the classes in question 3, write the boundaries, width and midpoint of the class 15–17.
Using the census table in worked example 2, what percentage of districts have at least 500 000 residents?
A student groups the district populations (in thousands) as 300–400, 400–500, 500–600 and so on, and cannot decide where a district of exactly 400 000 people would go. What is wrong with the classes, and how should they be written?
Why might a researcher choose classes of width 50 000 instead of 100 000 for the district data? What would be lost and gained?

Show answersHide answers

(a) discrete; (b) continuous; (c) nominal; (d) ordinal; (e) continuous.
A census covers every member of the population; a survey covers a sample. Census: complete and exact for small areas. Survey: cheaper and quicker, and can ask more detailed questions.
9–11: 9, 9, 11, 10, 11 gives $f = 5$ ; 12–14: 12, 14, 12, 12, 13, 12, 14, 12, 13, 14, 12 gives $f = 11$ ; 15–17: 15, 15, 16, 17, 15 gives $f = 5$ ; 18–20: 18, 20, 19 gives $f = 3$ . Total 24; cumulative 5, 16, 21, 24.
Boundaries 14.5 and 17.5; width 3; midpoint $\frac{15 + 17}{2} = 16$ .
$3 + 1 + 0 + 1 = 5$ districts; $\frac{5}{30} \times 100\% \approx 16.7\%$ .
The classes overlap: 400 000 belongs to both "300–400" and "400–500". Write $300\,000 \le x \lt 400\,000$ , $400\,000 \le x \lt 500\,000$ , and so on, so that 400 000 goes only in the second class.
Narrower classes show more detail (for example, how many districts are between 350 000 and 400 000) but give more classes with small or zero frequencies, which hides the overall shape. Wider classes show the shape but lose detail.

Quick check

Which of these variables is continuous?

Key points

A census studies the whole population; a sample survey studies part of it. Primary data are collected first-hand; secondary data are reused.
Data are qualitative (nominal or ordinal) or quantitative (discrete or continuous).
A frequency table lists values or classes with their frequencies; relative frequency $= \frac{f}{n}$ ; the last cumulative frequency equals $n$ .
Classes must not overlap or leave gaps; width = upper boundary − lower boundary; midpoint = mean of the class limits.
Rwanda's 30 district populations (2022) are skewed to the right: half lie between 300 000 and 400 000.

Sources and credits

Data: Fifth Rwanda Population and Housing Census, 2022: Main Indicators Report, National Institute of Statistics of Rwanda (Table 1, resident population by district).
Photo: Digital Census Bangladesh 2022 Enumerator Self Portrait, Vugon kumar, Wikimedia Commons, CC BY-SA 4.0.
Video: Discrete and Continuous Data, Corbettmaths, embedded from YouTube.
Written by Ganzaa: the explanations, the diagram, the tables, the worked examples, the practice items and the questions. The household and shop data are illustrative. The unit follows the topic list of REB's S4 Basic Mathematics (2026–2027); no REB text is reproduced. This lesson uses CC BY-SA material and is shared under CC BY-SA 4.0.

1Thirty numbers from the 2022 census

2Types of data

3Frequency tables

Thirty numbers from the 2022 census

Types of data

Frequency tables