Statistical questions, and the five-point summary
What this lesson covers
- Multivariate data is data in a set that has more than two variables.
- Data can be collected from observational studies in which the observers do not alter or control the behaviour of the subjects.
- Statistical questions clearly identify the variable, group of interest, and the intent of an investigation. A summary investigation is about a group. A comparison investigation compares a variable across two clearly identified groups. A relationship investigation looks for a connection between paired numerical or paired categorical variables. A time-series investigation looks at a variable over time.
- Primary data is data that is collected first-hand.
- Secondary data is data collected by someone else.
Statistics starts before any data exists
- Everything here follows from the question you asked.
- Ask it badly and no amount of calculation rescues you.
- So the question comes first.
Which of these is a statistical question?
- How tall is Sam? That has one answer. Measure him.
- How tall are Year 9 students here? The answers will vary.
- That variation is the difference, and it is the whole subject.
| Question | Answers | Verdict |
|---|---|---|
| How tall is Sam? | one answer | not statistical |
| How tall are Year 9s here? | answers will vary | statistical |
| What did I score? | one answer | not statistical |
| How do our scores vary? | variation is the subject | statistical |
Three things a good question names
- The variable: what are you actually measuring?
- The group: who are you measuring it on?
- The intent: what do you want to find out about it?
| Part | Meaning | Example |
|---|---|---|
| The variable | what is being measured | height |
| The group | who it is measured on | Year 9 at our school |
| The intent | what you want to find out | the typical value |
Run that on your own question before you start
- If you cannot point at all three, the question is not ready.
- "What about sport?" names none of them.
- Thirty seconds here saves a whole afternoon later.
Four kinds of investigation
- A summary looks at one group.
- A comparison puts two named groups side by side.
- A relationship pairs two variables. A time series adds dates.
| Type | What it does | What you must collect |
|---|---|---|
| Summary | one group, one variable | just the measurement |
| Comparison | one variable, two groups | also record the group |
| Relationship | two variables, paired | both, from the same subject |
| Time series | one variable over time | also record the date |
And this is why that matters
- Look at the right-hand column. The type decides what you collect.
- Running a comparison but forgot to record the group?
- You have one pile of numbers. You cannot go back and add it.
So these are not four words to memorise
- They are a decision you make before collecting anything.
- The wrong choice is not a naming error.
- It is data you did not gather and now cannot.
Where the data came from
- Primary data is data you gathered yourself.
- Secondary data is somebody else’s, like Stats NZ.
- Observational means you watched without interfering.
| Kind | Meaning | Example |
|---|---|---|
| Primary | you collected it yourself | your own survey |
| Secondary | somebody else collected it | Stats NZ, a website |
| Observational | watched, did not interfere | counting cars |
Usually you record several things at once
- One student: height, year level, sport, travel time.
- More than two variables makes it multivariate data.
- Mix numerical and categorical, and you can ask much more.
| Idea | Detail | |
|---|---|---|
| One student | height, year, sport, travel | several facts |
| More than two variables | that is multivariate data | |
| At least one numerical | height, travel time | |
| At least one categorical | year level, sport |
Now you have numbers. Sort them first
- Eleven values, in the order they were collected.
- Nothing works until they are in order. Sort them.
- Every single thing that follows assumes this step happened.
| Stage | The data |
|---|---|
| Raw | 9, 4, 15, 7, 2, 20, 11, 5, 14, 8, 12 |
| Sorted | 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 20 |
Find the middle. Then the middle of each half
- Eleven values, so the median is the sixth one. Nine.
- The five below it: their middle is five. That is Q1.
- The five above: their middle is fourteen. That is Q3.
| Step | Working | Value |
|---|---|---|
| Median, Q2 | the middle value of all 11 | 9 |
| Lower half | 2, 4, 5, 7, 8 | its middle is 5 |
| So Q1 | 5 | |
| Upper half | 11, 12, 14, 15, 20 | its middle is 14 |
| So Q3 | 14 |
No formula, and that is deliberate
- You may have seen a position formula for quartiles.
- Different books give different ones, and they disagree.
- Halving, then halving again, always works. Use that.
Five numbers for the whole set
- Minimum, Q1, median, Q3, maximum. Two, five, nine, fourteen, twenty.
- Each quartile is a quarter of the way through the data.
- Five numbers, and you can describe a set of any size.
| Point | Value | What it means |
|---|---|---|
| Minimum | 2 | |
| Q1 | 5 | a quarter are below this |
| Median | 9 | half are below this |
| Q3 | 14 | three quarters are below this |
| Maximum | 20 |
The IQR, and why it beats the range
- Range is max minus min: eighteen. IQR is Q3 minus Q1: nine.
- Now change one value. Make the twenty a sixty.
- The range jumps to fifty-eight. The IQR does not move at all.
| Measure | Working | Value |
|---|---|---|
| Range | 20 − 2 | 18 |
| IQR | 14 − 5 | 9 |
| Now change the 20 to 60 | range becomes 58 | IQR stays 9 |
| Why | it uses the middle half | only |
What to take away
- Seven things, and the fourth is the one that ruins projects.
| A statistical question | expects its answers to vary |
| It must name | the variable, the group and the intent |
| The four types | summary, comparison, relationship, time series |
| The type | decides what you must collect, before you collect it |
| Primary or secondary | is about who gathered it |
| Five-point summary | min, Q1, median, Q3, max |
| IQR | is Q3 minus Q1, and one odd value cannot move it |
Your turn
- Pause here and try all four.
- Answers are in the description.
| Question | |
|---|---|
| 1 | Is "how long is the trip to school?" statistical? |
| 2 | Comparing two groups. What else must you record? |
| 3 | Find the five-point summary of 8, 3, 11, 5, 14, 6, 18, 9. |
| 4 | Find the IQR of that set. |
Show answer
1. Yes. Different students will take different times, so the answer varies, and it names the variable, the group and the intent. 2. Which group each person is in. Without that column you have one pile of numbers and no way to split it, and you cannot go back and add it later. 3. Sorted: 3, 5, 6, 8, 9, 11, 14, 18. Eight values, so the median is the average of the 4th and 5th, which is 8.5. The lower half is 3, 5, 6, 8, so Q1 is 5.5. The upper half is 9, 11, 14, 18, so Q3 is 12.5. The summary is 3, 5.5, 8.5, 12.5, 18. 4. 7, from 12.5 minus 5.5.
The answers
- Eight values, so the median is an average.
- Q1 is 5.5 and Q3 is 12.5.
- And you cannot add the group column later.
| Answer | |
|---|---|
| 1 | Yes, the answers vary |
| 2 | Which group each person is in |
| 3 | 3, 5.5, 8.5, 12.5, 18 |
| 4 | 7 |
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