1.5 Data Collection Experiment

Aligned outcomes:

Learning Objectives

By the end of this section, you will be able to:

In this section, you will learn to:

demonstrate the systematic sampling technique on a real class list;
construct relative frequency and cumulative relative frequency tables from data you collected yourself;
interpret your results and explain how two different groupings of the same data can tell different-looking stories.

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. You will go out and collect real numbers from real people, organize them two different ways, and then argue about what those two organizations show.

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.

Work through the lab with your class, and keep a record of what you actually did — not what you meant to do. If you had to loop back to the top of the class list, or a marked classmate was absent, write that down. Those small departures from the plan are exactly the kind of thing a statistician has to report honestly.

1.5.2 Movie Survey

Ask five classmates from a different class how many movies they saw at the theater last month. Do not include rented movies.

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.

The instruction to "move down four names" is what makes this a systematic sample rather than a convenience sample. The randomness lives entirely in the one starting name you pick; after that the rule is mechanical, so nobody's preferences can sneak in — not yours, and not your classmates'. When you reach the bottom of the class list you wrap around to the top and keep counting, which is why a class of any size still yields exactly the 12 marked names the lab asks for.

It is worth being clear about what systematic sampling is not. It is not the same as a simple random sample, in which 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. Most of the time that is close enough, and it is far easier to carry out in a classroom than drawing twelve names from a hat. The one thing to watch for is a list that has a repeating pattern built into it. If the class list happened to alternate between two lab groups, and your step size lined up with that alternation, you would sample from one lab group over and over and never touch the other. Glance at the list before you start, and if you can see structure in the ordering, say so when you report your results.

Record the data.
In class, randomly pick one person. On the class list, mark that person's name. Move down four names on the class list. Mark that person's name. Continue doing this until you have marked 12 names. You may need to go back to the start of the list. For each marked name record the five data values. You now have a total of 60 data values.
For each name marked, record the data.

**Table 1.5.1** — Movie counts collected by each of your 12 marked classmates (5 values each, 60 values total).
Marked name	Value 1	Value 2	Value 3	Value 4	Value 5
Name 1
Name 2
Name 3
Name 4
Name 5
Name 6
Name 7
Name 8
Name 9
Name 10
Name 11
Name 12

Try it in rāSHio

Once the grid above 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 — Getting your 60 collected values into rāSHio: File → Delimited List…

Try It Now 1.5.1

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?

Solution

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 the end of 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

Step 3 — check that nobody got marked twice. All 12 positions are different, which is what you want — 12 different classmates giving 5 values each is the 60 data values the lab asks for.

Answer: positions 7, 11, 15, 19, 23, 2, 6, 10, 14, 18, 22, 1. It is worth noticing why this worked out cleanly: 4 and 25 share no common factor, so stepping by 4 walks through the whole list before it ever returns to the start. Had the class held 24 names, stepping by 4 would loop back to your starting name after only 6 marks and you would sample the same 6 people twice. When that happens, say so and change the step size rather than pretending you have 12 independent picks.

Try it in rāSHio

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 — Drawing the random starting position in rāSHio: File → Random Numbers…

1.5.3 Order the Data

Complete the two relative frequency tables below using your class data.

**Table 1.5.2** — Frequency of number of movies viewed, one row per movie count.
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.
Number of movies	Frequency	Relative frequency	Cumulative relative frequency
0–1
2–3
4–5
6–7+

Try it in rāSHio

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 — Building both frequency tables in rāSHio: Graph → Frequency Table.

Remember what each column asks for. Frequency is the plain count of how many of your 60 values landed in that row. Relative frequency is that count divided by 60, which turns it into a share of the whole. Cumulative relative frequency adds up the relative frequencies from the top row down, so 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 — go back and find it before you answer the questions.

Now use your completed tables to answer these:

Using the tables, find the percent of data that is at most two. Which table did you use and why?
Using the tables, find the percent of data that is at most three. Which table did you use and why?
Using the tables, find the percent of data that is more than two. Which table did you use and why?
Using the tables, find the percent of data that is more than three. Which table did you use and why?

Try It Now 1.5.2

Before you fill in your own tables, practise on a small set. Kenji ran the survey early and brought back the ten movie counts he collected: 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.

Solution

Step 1 — count each value. 0 appears once, 1 appears twice, 2 appears three times, 3 once, 4 twice, 5 never, 6 once. Those counts are the frequencies, and they must add to 10.

Step 2 — divide each frequency by 10 to get relative frequency, then add them down the column to get cumulative relative frequency.

**Table 1.5.4** — Worked practice table for the ten reported movie counts.
Number of movies	Frequency	Relative frequency	Cumulative relative frequency
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

Step 3 — read the answer off the cumulative column. "At most two" means 0, 1, or 2 movies, so we want the cumulative relative frequency on the row for 2.

$$ 0.10 + 0.20 + 0.30 = 0.60 $$

Answer: 60% of the data is at most two. Notice you did not have to add anything — the cumulative column had already done the adding for you. That is exactly why the lab asks you to build it.

Try It Now 1.5.3

Camila is sure that grouping the counts cannot change any answer; her wife Rosa, who teaches the other section, thinks it can. Settle it for them. Take the same ten movie counts — 0, 1, 1, 2, 2, 2, 3, 4, 4, 6 — and group them the way the second lab table does: 0–1, 2–3, 4–5, 6–7+. Then answer "what percent of the data is more than three?" from the grouped table, and check it against the ungrouped table. Do the two answers agree?

Solution

Step 1 — count into the intervals. 0–1 holds 0, 1, 1 → 3 values. 2–3 holds 2, 2, 2, 3 → 4 values. 4–5 holds 4, 4 → 2 values. 6–7+ holds 6 → 1 value. The four counts add to 10, so nothing was lost.

Step 2 — turn the counts into shares. 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 — answer from the grouped table. "More than three" means 4 or more. The 0–3 material is covered by the first two rows, which is 0.70 of the data, so what is left is

$$ 1 - 0.70 = 0.30 $$

Step 4 — check it against the ungrouped table. In the ungrouped practice table above, the cumulative relative frequency at 3 was also 0.70, giving the same 0.30.

Answer: 30% either way — the two tables agree here, so Camila wins this round. They agree because the cut point, "more than three", happens to fall exactly on the boundary between the 2–3 and 4–5 groups. But ask instead for "more than two" and the grouped table cannot answer it: 2 and 3 are locked together in one row, so you cannot separate them without going back to the raw data. Rosa is right about the general case. That is the real lesson of this lab — grouping never changes the data, but it can hide the very question you wanted to ask.

1.5.4 Discussion Questions

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.

Is one of the tables "more correct" than the other? Why or why not?
In general, how could you group the data differently? Are there any advantages to either way of grouping the data?
Why did you switch between tables, if you did, when answering the question above?