Introduction to Statistics · Chapter 1 · Sampling and Data
A lab, not a reading: pull your own systematic sample, organize it two different ways, and argue about what each grouping shows.
bookSHelf · Introduction to Statistics · §1.5 · a self-paced section
Learning objectives — by the end of this section you will be able to
§1.5.1 — Stats Lab: Data Collection Experiment
This section is a lab, not a reading. Everything you have met so far in this chapter — population and sample, sampling method, frequency, relative frequency — you are now going to do with your own hands: collect real numbers from real people, organize them two different ways, and argue about what those two organizations show.
Keep a record of what you actually did — not what you meant to do. Those small departures from the plan are exactly what a statistician has to report honestly.
Context Pause — how you pick matters
The method is part of the answer
How you pick who to ask is not a detail you get to skip past. A sample gathered by a clear, repeatable rule can stand behind a claim about the whole class; a sample of whoever happened to be nearby cannot. That is why this lab makes you follow a specific selection rule instead of just grabbing names.
A repeatable rule is what lets a sample speak for the whole class — a convenience sample never can.
§1.5.2 — the task
Ask five classmates from a different class how many movies they saw at the theater last month — no rented movies.
The Rule — draw 12 names systematically
The instruction to move down four names is what makes this systematic rather than convenience — the randomness lives entirely in the one starting name you pick.
§1.5.2 — how systematic sampling compares
In a simple random sample, every possible group of 12 classmates is equally likely to be chosen. Once you have committed to a starting point and a step of four, eleven of your twelve names are already decided — there is exactly one genuinely random decision in the entire procedure.
Watch for structure in the list. If the class list happens to alternate between two lab groups and your step size lines up with that alternation, you would sample one group over and over and never touch the other. Glance at the list before you start, and say so if you can see a pattern in the ordering.
Insight Note — the census-worker analogy
Every fourth house on the street
Systematic sampling means you start at one random spot and then step through the list at a fixed interval. It is the same idea as a census worker knocking on every tenth house instead of choosing houses by feel — easy to carry out, hard to fudge, and it spreads your picks evenly across the whole list.
§1.5.2 — recording what you collect
| Marked name | Value 1 | Value 2 | Value 3 | Value 4 | Value 5 |
|---|---|---|---|---|---|
| Name 1 | |||||
| Name 2 | |||||
| ⋮ | |||||
| Name 12 |
Table 1.5.1: Movie counts collected by each of your 12 marked classmates (5 values each, 60 values total).
Fill this by hand in class — the rāSHio notes ahead show how to get the finished 60 values into a spreadsheet fast.
Try it in rāSHio — getting your 60 values in fast
Once the grid is full, open rāSHio and choose File → Delimited List… to get your class's 60 movie counts into the spreadsheet. Commas, semicolons, tabs, and one-per-line all parse, so you can type the values straight off your paper grid row by row without reformatting anything first.
Figure 1.5.1
File → Delimited List…
Paste or type a list of values with any separator, and rāSHio lands them in a spreadsheet column ready for analysis — no manual reformatting.
Figure 1.5.1: Getting your 60 collected values into rāSHio: File → Delimited List…
Your turn — walk the rule by hand
Try It Now 1.5.1 — twenty-five names, one starting point
A class list has 25 names, numbered 1 through 25. You randomly pick name number 7 as your starting point, then move down four names each time until you have marked 12 names, wrapping back to the top of the list when you run off the bottom. Which 12 positions do you mark?
Step 1 — set up the rule. Mark position 7, then add 4 each time. The only random decision you ever make is that first 7; everything after it is arithmetic.
Step 2 — count forward, wrapping at 25. 7, 11, 15, 19, 23. The next one would be 27, which runs off a 25-name list, so wrap back to the top: 27−25=2. Keep going from 2 the same way until you have 12 marks.
7, 11, 15, 19, 23, 2, 6, 10, 14, 18, 22, 1
Answer: positions 7, 11, 15, 19, 23, 2, 6, 10, 14, 18, 22, 1. It worked out cleanly because 4 and 25 share no common factor, so stepping by 4 walks through the whole list before it ever returns to the start. In a class of 24, stepping by 4 would loop back to the start after only 6 marks — the same 6 people, twice.
Try it in rāSHio — drawing the starting name
You do not have to close your eyes and point at the class list to get that starting name. Open rāSHio, choose File → Random Numbers…, and set Min 1, Max 25 (or however many names your list holds) and How many 1 — the number it returns is your starting position, and counting on by four from there is arithmetic you can do on the printed list.
Figure 1.5.2
File → Random Numbers…
Set the minimum, maximum, and how many values to draw, and read off the generated whole number — that is your starting position on the class list.
Figure 1.5.2: Drawing the random starting position in rāSHio: File → Random Numbers…
§1.5.3 — order the data
Complete two relative frequency tables from your class's 60 movie counts: one row per exact count, and one with the counts grouped into intervals.
Same 60 numbers, two different lenses — and the discussion questions ahead ask whether they always agree.
§1.5.3 — what each column asks for
The bottom row should come out to 1 — or very close, once you round. If your last cumulative entry is not 1, a value has been miscounted or dropped somewhere; find it before you answer the questions.
Table 1.5.2
| Number of movies | Frequency | Relative frequency | Cumulative relative frequency |
|---|---|---|---|
| 0 | |||
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 | |||
| 7+ |
Table 1.5.2: Frequency of number of movies viewed, one row per movie count.
Table 1.5.3
| Number of movies | Frequency | Relative frequency | Cumulative relative frequency |
|---|---|---|---|
| 0–1 | |||
| 2–3 | |||
| 4–5 | |||
| 6–7+ |
Table 1.5.3: Frequency of number of movies viewed, counts grouped into intervals.
Try it in rāSHio — building both tables at once
With your 60 counts loaded, rāSHio will build both of these tables for you: choose Graph → Frequency Table with Discrete values checked to fill Table 1.5.2's frequency, relative frequency, and cumulative relative frequency columns, then clear that checkbox and set Bins Start 0 and Width 2 to get Table 1.5.3's grouped version of the very same data. Fill the tables in by hand first — then use this to check yourself.
Figure 1.5.3
Graph → Frequency Table
Check Discrete values and Calculate for Table 1.5.2; clear it and set bins for Table 1.5.3 — same 60 values, two tables.
Figure 1.5.3: Building both frequency tables in rāSHio: Graph → Frequency Table.
Your turn — practise on a small set first
Try It Now 1.5.2 — Kenji's ten movie counts
Kenji ran the survey early and brought back ten movie counts: 0, 1, 1, 2, 2, 2, 3, 4, 4, 6. Build the frequency, relative frequency, and cumulative relative frequency columns from his data, then find the percent of the data that is at most two.
Step 1. 0 appears once, 1 appears twice, 2 appears three times, 3 once, 4 twice, 5 never, 6 once — the frequencies, adding to 10.
Step 2. Divide each frequency by 10 for relative frequency, then add down the column for cumulative relative frequency.
Step 3. "At most two" is 0, 1, or 2 movies — read the cumulative value on the row for 2: 0.10+0.20+0.30=0.60.
| Movies | Freq. | Rel. | Cum. |
|---|---|---|---|
| 0 | 1 | 0.10 | 0.10 |
| 1 | 2 | 0.20 | 0.30 |
| 2 | 3 | 0.30 | 0.60 |
| 3 | 1 | 0.10 | 0.70 |
| 4 | 2 | 0.20 | 0.90 |
| 5 | 0 | 0.00 | 0.90 |
| 6 | 1 | 0.10 | 1.00 |
Table 1.5.4: Worked practice table for the ten reported movie counts.
Answer: 60% of the data is at most two — the cumulative column had already done the adding.
Your turn — does grouping ever change the answer?
Try It Now 1.5.3 — Camila and Rosa disagree
Camila is sure that grouping the counts cannot change any answer; her wife Rosa, who teaches the other section, thinks it can. Take Kenji's ten counts — 0, 1, 1, 2, 2, 2, 3, 4, 4, 6 — grouped 0–1, 2–3, 4–5, 6–7+, and answer "what percent of the data is more than three?" from the grouped table. Does it agree with the ungrouped table?
Step 1. 0–1 holds 3 values, 2–3 holds 4, 4–5 holds 2, 6–7+ holds 1 — the four counts add to 10.
Step 2. Dividing by 10: 0.30, 0.40, 0.20, 0.10, with cumulative relative frequencies 0.30, 0.70, 0.90, 1.00.
Step 3. "More than three" is what's left after the first two rows (0.70): 1−0.70=0.30.
Answer: 30% either way — the grouped and ungrouped tables agree here.
The headline result
Grouping never changes the data — but it can hide the question you wanted to ask.
Ask "more than three" and both tables agree. Ask "more than two" instead, and the grouped table cannot answer it at all.
In Try It Now 1.5.3, the cut point landed exactly on a group boundary — so grouped and ungrouped agreed. That was the luck of the cut, not a property of grouping.
† 2 and 3 are locked together in one row of the grouped table; you cannot separate them without going back to the raw data. Rosa is right about the general case — grouping is a convenience, and it can cost you precision.
§1.5.4 — discuss with your group
Answer these with your group. There is no single right answer to any of them — what matters is that you can defend the answer you give with something you can point to in your own tables.
§1.5 — conclusions
A systematic sample turns "who do I ask" into a repeatable rule: one random start, then a fixed step, wrapping when you reach the bottom of the list. The same 60 values can then be organized two ways — and the choice of grouping can hide as much as it reveals.
A step size that lines up with a hidden pattern in the list resamples the same people. And a grouped table can only answer a question that respects its own boundaries — when the cut you need falls inside a group, only the raw data can answer it.
Next: Chapter 2 — Descriptive Statistics, where the data you just collected becomes a graph you can read and a handful of numbers that summarise it.