Individuals, variables and units
Someone hands you a page of numbers. Before you can learn anything from it, what do you need to know about each number?
Where you are 8 in Data and variables
Why this exists
Here are three numbers somebody wrote on a scrap of paper:
Nothing can be learned from them. Is 410 a length, a mass, a count? Of what? Measured how, in which unit? A number gets its meaning from three things that travel with it: which thing it describes, which feature of that thing it records, and the unit and method of the recording. A data table keeps all three attached, and every later piece of statistics starts by reading one.
Definition
Data are facts recorded about a group of things.
An individual is one of the things the data describe: a person, an animal, an object, a place, a day, an event, whatever was studied. Statistics books also say case or observational unit.
A variable is one feature recorded about every individual, whose value can differ from one individual to the next. Its values may be words (a turtle’s species) or numbers (its mass in grams); sorting variables by that difference is the job of Types of data: categorical, numerical, discrete, continuous.
A data table gives each individual one row and each variable one column. The cell where a row meets a column holds one value: that variable for that individual. Many tables begin with a column that only names each row, such as a tag; in this lesson it is not counted as a variable, since it only says which individual the row is. Some books do count it.
An observation is everything recorded about one individual, so one row. The number of observations, written n, is the number of individuals, which is the number of rows under the heading.
The units and context of a variable say what feature was recorded, how it was measured or counted, and in what unit: centimeters, grams, minutes. A count has a unit too, the thing counted: visits, frames, days. Context also covers who collected the data, where and when.
Intuition
Margit volunteers at a turtle rescue pond. She tags each turtle brought in and keeps this table.
| Tag | Shell length (cm) | Mass (g) | Days in care |
|---|---|---|---|
| T1 | 14.5 | 410 | 12 |
| T2 | 12 | 380 | 30 |
| T3 | 11.5 | 290 | 5 |
| T4 | 9 | 160 | 21 |
| T5 | 13.5 | 450 | 8 |
| T6 | 15 | 520 | 16 |
Read across a row and you get one turtle’s story: T3 has an 11.5 cm shell, it weighs 290 g, and it has been in care for 5 days. Read down a column and you see one feature changing from turtle to turtle: the masses run from 160 g to 520 g. That change is variability, and it is what a Statistical questions and variability asks about. The scrap of paper was the first row with its headings torn off.
Which things are the individuals is a choice, made by what you want to know. A bird feeder can be logged with one row per landing or one row per day; Examples 2 and 3 build one of each.
Results
Counting values. A table with n rows of individuals and v variable columns, every cell filled, holds
values.
One column, one unit, one method. Values in a column can be compared, ordered or added only when they share one unit and one way of measuring. Changing the unit changes the number, not the amount: 1.2 kg and 1{,}200 g are the same mass.
A value needs its context. A value is complete only when you can say which individual and which variable it belongs to, how it was measured, and in what unit.
Why it is true
Each of the n rows has one cell for each of the v variables. Adding v once per row gives v + v + \dots + v with n terms, which is n \times v. Counting by columns gives v \times n, the same number by the commutative property of multiplication.
A measurement says how many units fit into an amount. A gram is a thousandth of a kilogram, so 1{,}000 grams fit into each kilogram, and 1.2 kilograms holds 1.2 \times 1{,}000 = 1{,}200 grams. The amount did not change; only the size of the counting piece did. Numbers in different units count pieces of different sizes, so they cannot be compared until one is converted (Converting metric units by powers of ten).
Worked examples
Reading the turtle table
Use Margit’s table above. (a) What are the individuals, and what is n? (b) Name the variables and the unit of each. (c) The value 12 appears twice. What does it mean each time? (d) How many values does the table hold, not counting the tags?
Solution(a) Each row describes one turtle, so the individuals are the turtles brought to the pond, and with six rows n = 6.
(b) Shell length (centimeters), mass (grams) and days in care (days). The tag only names each turtle.
(c) The column says which variable, the row says which turtle. In row T1, 12 is days in care: T1 has been in care 12 days. In row T2, 12 is shell length: T2’s shell is 12 cm long. One number, two unrelated meanings.
(d) Six rows and three variable columns: 6 \times 3 = 18 values.
A log where each row is a visit
Noomi watches her bird feeder one Saturday and writes a line each time a bird lands.
| Visit | Minutes after 7 o’clock | Bird | Seconds on feeder |
|---|---|---|---|
| 1 | 5 | sparrow | 40 |
| 2 | 12 | finch | 25 |
| 3 | 12 | finch | 35 |
| 4 | 30 | blue tit | 15 |
| 5 | 41 | sparrow | 50 |
| 6 | 58 | robin | 20 |
| 7 | 63 | finch | 30 |
| 8 | 80 | sparrow | 45 |
(a) Are the individuals birds? (b) What is n, and what are the variables? (c) Why are rows 2 and 3 two rows? (d) How many visits were made by finches?
Solution(a) No. Each row is one landing, so the individuals are visits. Noomi cannot tell sparrows apart, and the sparrow of visit 1 may be the sparrow of visit 5. The table counts visits, not different birds.
(b) Eight rows, so n = 8. The variables are the start time (minutes after 7 o’clock), the kind of bird (a word) and the time on the feeder (seconds). The start time’s context includes its starting point: 63 means 8{:}03.
(c) Two finches landed together at 7{:}12. That is two visits, and each individual gets its own row.
(d) Finches appear in rows 2, 3 and 7, so 3 visits were by finches.
The same kind of feeder, one row per day
Jutta logs her own feeder with one row per day for a week.
| Day | Visits | Low (^\circ\text{C}) | Seed eaten (g) |
|---|---|---|---|
| Mon | 22 | 6 | 120 |
| Tue | 18 | 4 | 95 |
| Wed | 30 | 3 | 140 |
| Thu | 25 | 5 | 110 |
| Fri | 16 | 8 | 85 |
| Sat | 35 | 9 | 150 |
| Sun | 20 | 7 | 100 |
The “Low” column is the day’s lowest temperature. (a) What are the individuals, and what is n? (b) What is the unit of the visits column, and how does it connect to a log like Noomi’s? (c) How much seed was eaten in the week, in kilograms?
Solution(a) Each row is one day, so the individuals are days and n = 7.
(b) The unit is visits. Each value is the number of rows a visit log would have for that day, so a visit log for the whole week would need 22 + 18 + 30 + 25 + 16 + 35 + 20 = 166 rows. One row here stands for many rows there; which table to keep depends on whether the question is about days or single visits.
(c) The seed values are all in grams, so they can be added:
A kilogram is 1{,}000 grams, so this is 800 \div 1{,}000 = 0.8 kg.
A column with no unit
Gorm keeps four beehives. His notebook lists the honey taken this summer: hive A 12, hive B 9.5, hive C 14, hive D 11, with no unit. (a) Why can you not yet say how much honey that is? (b) The next page says it was weighed in kilograms, after being spun out of the comb. What is the total? (c) A friend’s best hive gave 13{,}500 g. Whose best hive gave more?
Solution(a) 12 could be kilograms, pounds or jars of any size, and it matters whether the comb was weighed too. Without unit and method the numbers can be ordered against each other but not against anything else.
(b) All four are kilograms, measured one way, so they add: 12 + 9.5 + 14 + 11 = 46.5 kg.
(c) Use one unit first: 13{,}500 \div 1{,}000 = 13.5 kg. Gorm’s hive C gave 14 kg, which is 14 - 13.5 = 0.5 kg more. The bare numbers 14 and 13{,}500 point the opposite way.
From notes to a table
Lucjan notes the turtles brought in one week, measuring each shell with a ruler held straight from front to back. “Tuesday: musk turtle, shell 8.5 cm, mass 140 g. Wednesday: painted turtle, shell 13 cm, mass 360 g. Thursday: box turtle, shell 12.5 cm, mass 400 g.” (a) Put the notes into a data table and count its values. (b) Noomi measures the Wednesday turtle with a tape laid along the curve of its shell and gets 14 cm. Who is wrong?
Solution(a) Three turtles, so three rows. Each has an arrival day, species, shell length and mass, so four variables; the arrival day is a recorded fact about each turtle, so it counts.
| Arrived | Species | Shell, straight (cm) | Mass (g) |
|---|---|---|---|
| Tue | musk | 8.5 | 140 |
| Wed | painted | 13 | 360 |
| Thu | box | 12.5 | 400 |
It holds 3 \times 4 = 12 values.
(b) Neither. They measured different features: the tape follows the dome, so it is longer than the straight line, here by 14 - 13 = 1 cm. Same unit, different method, and a column may hold only one method, which is why the heading says “straight”.
Choosing a different individual
Gorm’s notebook also splits the honey by month.
| Hive | June (kg) | July (kg) | August (kg) |
|---|---|---|---|
| A | 3 | 5 | 4 |
| B | 2 | 4.5 | 3 |
| C | 4 | 6 | 4 |
| D | 3 | 5 | 3 |
(a) What are the individuals, n and the variables? (b) Rewrite the data with one row for each hive in each month. What are the individuals and n now? (c) Find the total honey and compare it with Example 4.
Solution(a) Each row is a hive, so n = 4, with three variables: the honey in June, in July and in August, each in kilograms.
(b) Each row is now one hive in one month, with columns hive, month and honey (kg): A, June, 3; A, July, 5; and so on. Four hives in three months gives 4 \times 3 = 12 rows, so n = 12. The hive letter no longer names a row, since three rows share each letter; it is now a fact recorded about each hive-month, so hive and month are both variables.
(c) Both tables hold the same twelve honey amounts. Hive A gave 3 + 5 + 4 = 12 kg, B 2 + 4.5 + 3 = 9.5 kg, C 4 + 6 + 4 = 14 kg and D 3 + 5 + 3 = 11 kg: the totals of Example 4, adding to 46.5 kg. Only what counts as one individual changed. “Which hive gave most?” suits a row per hive; “Did the honey change month to month?” suits a row per hive-month.
Boundaries
One kind of individual per table. A table whose rows mix turtles and days has no single answer to “what is n?”.
Blank cells. A blank means the value was not recorded. The table still has n \times v cells but fewer values, and that variable has fewer than n observations.
A fact shared by every row is context, not a variable. If every turtle came to the same pond, the pond’s name does not vary across these individuals; it belongs in the title.
Words have no unit. A species has no unit, though its context still matters: who identified it, and by what guide.
Converting fixes the unit, not the method. Grams turned into kilograms are comparable only if the same feature was measured the same way.
Where this goes next
Types of data: categorical, numerical, discrete, continuous sorts variables by whether their values name a group or count or measure an amount. Data tables, tallies and frequency tables count how often each value appears down a column, and Collecting data with a survey is about gathering the rows. Reporting results in the units of the setting runs through Describing a distribution and Reading a graph: titles, labels and units. Later, two numbers recorded on one individual become one point in Plotting paired data: explanatory and response variables.
Exercises
Exercises 1 to 6 use this table from an inspection of five hives at Gorm’s beekeeping club. A brood frame is a wooden frame of comb holding young bees.
| Hive | Brood frames | Mass (kg) | Bees out per minute |
|---|---|---|---|
| P | 6 | 38.5 | 42 |
| Q | 4 | 31 | 27 |
| R | 7 | 41 | 55 |
| S | 5 | 35.5 | 38 |
| T | 3 | 29 | 19 |
Reading a table
- What are the individuals, and what is n?
- Name the variables, and give the unit of each one that has a unit.
- Say in a full sentence what the value 55 means.
- How many values does the table hold, not counting the hive letters?
- Which hive is heaviest, and how many kilograms heavier is it than the lightest?
- How many brood frames are there in all five hives together?
Choosing the individual
- Jutta writes a line each time a bird lands on her feeder, with the time and the kind of bird. In one hour she writes 14 lines. What are the individuals, and what is n?
- Noomi keeps a feeder record with one row for each day of April, recording visits, rain (mm) and seed eaten (g). What is n, and how many values does the table hold, not counting the dates?
- A feeder table has 4 variable columns, no blank cells, and holds 84 values. How many individuals does it describe?
- Name the individuals in each record. (a) For each nest box on a farm, whether it was used this spring and how many eggs were laid in it. (b) For each hour of one day, the number of birds seen at a feeder. (c) For each turtle found in a pond survey, its species and shell length.
- Margit weighs each of 7 turtles every Monday for 5 weeks. (a) If each row is one weighing, what is n? (b) If each row is one turtle, with a column for each week’s mass, what is n, and how many variables are there? (c) How many masses are recorded either way?
Units and context
- Three turtle masses are written as 0.43 kg, 370 g and 530 g. Write all three in grams and put them in order from lightest to heaviest.
- A hive gave 2.75 kg of honey in June and 3{,}100 g in July. How much did it give over the two months, in kilograms?
- A column headed “Time on feeder (min)” holds 0.5, 0.75, 1.25 and 2. What is the total time, in seconds?
- Lucjan’s straight shell lengths are 142 mm, 96 mm and 131 mm, and an older log gives a fourth turtle’s straight length as 11.8 cm. Put all four in millimeters, in order from shortest. Which is longest?
- A bag of seed holds 2.5 kg, and Jutta puts 125 g in her feeder each morning. How many mornings does one bag last?
- A note on the club’s wall says only “Q: 31.” Using the table above, say what 31 most likely means, and rewrite the note so nobody has to guess.
Explaining
- Example 2’s table records visits. Explain why it cannot tell you how many different birds came, and describe a record that could.
- Explain why a table with n rows and v variable columns has n \times v cells, and why it may hold fewer values than that.
- Gorm’s inspection table could have a column “Apiary” reading “Hillside” in every row. Explain why that fact belongs in the title rather than in a column.
- Two volunteers report the “size” of the same turtle as 16 and 470. Explain how both can be right, and what the headings should say so the question never comes up.
Answers
- The five hives inspected, one per row, so n = 5.
- Brood frames (frames); mass (kilograms); bees out per minute (bees per minute).
- At the inspection, 55 bees left the entrance of hive R in one minute.
- Five rows and three variable columns: 5 \times 3 = 15 values.
- Hive R, at 41 kg. The lightest is T at 29 kg, and 41 - 29 = 12 kg.
- 6 + 4 + 7 + 5 + 3 = 25 frames.
- Each line is one landing, so the individuals are visits, not birds, and n = 14.
- April has 30 days, so n = 30. With three variables the table holds 30 \times 3 = 90 values.
- Each individual fills a row of 4 values, so 84 \div 4 = 21 individuals.
- (a) Nest boxes. (b) Hours. (c) Turtles.
- (a) 7 \times 5 = 35 weighings, so n = 35. (b) n = 7, with 5 variables: the mass in week 1, week 2 and so on to week 5. (c) 35 either way; only the arrangement changes.
- 0.43 \times 1{,}000 = 430 g. In order: 370 g, 430 g, 530 g.
- 3{,}100 g is 3.1 kg, so the total is 2.75 + 3.1 = 5.85 kg.
- 0.5 + 0.75 + 1.25 + 2 = 4.5 minutes, and 4.5 \times 60 = 270 seconds.
- 11.8 \times 10 = 118 mm. In order: 96, 118, 131, 142 mm, so 142 mm is longest.
- 2.5 kg is 2{,}500 g, and 2{,}500 \div 125 = 20 mornings.
- The only 31 in row Q is the mass, so hive Q most likely weighed 31 kg. A complete note: “Hive Q, May inspection: whole hive weighed 31 kg.”
- One bird can land many times, so the rows count landings. Counting birds needs a way to tell them apart, such as a colored leg band on each, with one row per bird.
- Every row has one cell for each of the v variables, so n rows give v added n times, n \times v. A blank cell is a value not recorded, so the values can number fewer than the cells.
- A variable must be able to differ between individuals. A fact the same for every hive says nothing about how they differ; it is context, as in the title “Hillside apiary, May inspection”.
- They recorded different variables: 16 is a shell length in centimeters and 470 a mass in grams. Headings such as “Shell length, straight (cm)” and “Mass (g)” name the feature, method and unit, so a bare “size” never appears.