Group numerical data into class intervals

A waiting time is exactly 3 minutes. The continuous classes are 2 ≤ x < 3 and 3 ≤ x < 4. Where does it belong?

A waiting time is 2.95 minutes. The classes are 2 ≤ x < 3 and 3 ≤ x < 4. Which class contains it?

Move the value from 2.95 to 3.00, then to 3.05 minutes. The number line zooms to the current class. Watch its included start and excluded upper endpoint

What changes when the value reaches exactly 3.00?

State what the intervals mean

These investigations use equal-width classes. For continuous data use an explicit boundary convention; for rounded records state the recording precision

1. 1Continuous measurements 2 ≤ x < 3 includes 2 and every larger value below 3, including 2.95 \(\displaystyle 2\le x<3\)
2. 2Recorded tenths A class labelled 2.0–2.9 can list the recorded tenths 2.0, 2.1, …, 2.9. It is not an unrestricted continuous interval
3. 3One observation, one class Cover all records with no overlaps or gaps. Retain every empty class in the specified scheme
4. 4Check the total Changing the grouping must not change how many observations are present

First six waits (minutes)

\(\displaystyle 0.4,1.8,2.0,2.3,3.1,4.6\)

Next six waits (minutes)

\(\displaystyle 5.9,6.0,6.2,8.4,10.1,11.8\)

These twelve waiting times will use width-2-minute classes starting at 0. What is the frequency of 0 ≤ x < 2?

The time 2.0 belongs to the next class, not this one

Predict what happens if those same twelve waits are regrouped from width 2 to width 4 minutes

Change the class width from 2 to 4 minutes. Compare the first two old rows with the first new row, then check the total

Why does the first new class have frequency 5?

Class count

\(\displaystyle 5\)

Total

\(\displaystyle 12\)

The first width-4 class contains 5 of the 12 waits. What is its percentage frequency, to one decimal place?

Use 5 ÷ 12 × 100, then round the final result

This excerpt uses width-1-minute classes for the same waits. What should happen to 7 ≤ x < 8?

Can the width-4 grouped table alone recover each person’s exact waiting time?

Recorded lengths (cm)

\(\displaystyle 2.0,2.7,2.9,3.0,3.0,3.4\)

These lengths were recorded to the nearest tenth of a centimetre. How many recorded values belong to 2.0–2.9?

Count the recorded values from 2.0 through 2.9, including both endpoints

Why is a recorded-tenths class written 2.0–2.9 compatible with the next class starting at 3.0?

Data range from 12 to 39. Use whole-number classes of width 10 starting at 10. Which scheme is valid?

Which class width is always best for showing a distribution?

Produce a complete grouped table

These invented waiting times are recorded to the nearest 0.1 minute. Group the recorded values into five classes of width 2 minutes, starting at 0 and ending at 10. Each class includes its lower boundary and excludes its upper boundary. Produce a complete count and percentage table, displaying percentages to one decimal place. Include the frequency total and the sum of the displayed rounded percentages; state separately the exact whole-sample percentage

1. 1Records 1–8 (minutes) 0.5, 1.2, 1.2, 2.0, 2.8, 3.9, 4.0, 4.6
2. 2Records 9–16 (minutes) 5.5, 5.9, 8.0, 8.4, 8.4, 9.1, 9.7, 9.9
3. 3Choose and use technology Select a statistical grouping function or spreadsheet conditional-count function. Explain why it suits this boundary convention. Record the raw-data range, all class boundaries, formula or settings and resulting frequencies. Do not round the recorded values again before grouping
4. 4Verify independently Manually list the values in each class and check that every record appears exactly once. In particular, check 2.0, 4.0 and 8.0, retain empty classes, and compare the frequency sum with the number of raw records. Calculate one percentage by hand and explain any rounded-total discrepancy
5. 5Test a boundary change Explain how the table would need to change if a new recorded wait of exactly 10.0 minutes were added. Keep the same lower-inclusive, upper-exclusive convention
6. 6Complete before continuing Save your table, explanation and technology record for self or teacher review

Model response: complete grouped table

Compare your saved table and tool record with this model. A different suitable tool is valid if it implements the stated intervals and preserves all records

1. 1Count every class 0 ≤ x < 2 contains 0.5, 1.2, 1.2: count 3. The next class contains 2.0, 2.8, 3.9: count 3. Then 4.0, 4.6, 5.5, 5.9: count 4. The class 6 ≤ x < 8 is empty. The final class contains 8.0, 8.4, 8.4, 9.1, 9.7, 9.9: count 6. The total is 16.
2. 2One technology record Enter the raw values in A2:A17. Put lower boundaries 0, 2, 4, 6, 8 in D2:D6 and upper boundaries 2, 4, 6, 8, 10 in E2:E6. In F2 use COUNTIFS($A$2:$A$17,">="&D2,$A$2:$A$17,"<"&E2), then fill down. These two conditions count each lower boundary and exclude its upper boundary. Check the software’s interval convention instead of assuming a bin-endpoint default matches this task
3. 3Percentages and precision Put COUNT(A2:A17) in F7 to count the 16 numerical records. In G2 use ROUND(100*F2/$F$7,1), then fill down; label G as Percentage (%) and display one decimal place. The percentages are 18.8%, 18.8%, 25.0%, 0.0%, 37.5%. Use SUM(G2:G6) for the displayed percentage column sum, 100.1%; the exact whole-sample percentage is 100%. The two exact 18.75% shares each round upwards to 18.8%, causing the 0.1-point difference. Do not change a correct row to force 100%
4. 4Independent checks The manual class lists give 3+3+4+0+6=16, matching the raw-record count. The values 2.0, 4.0 and 8.0 enter the classes starting at those boundaries exactly once. By hand, divide 3 by 16 and multiply by 100 to get 18.75%, which rounds to 18.8%. Matching totals alone would not detect exchanging two class counts, so check every interval label against its list
5. 5New value at 10.0 Add the class 10 ≤ x < 12 with frequency 1 and change the sample total to 17. Recalculate every percentage using 17. Putting 10.0 in 8 ≤ x < 10 or silently dropping it would violate the stated convention
6. 6Self-review checklist Check complete class coverage, boundary membership, the zero row, counts and sample size, headings/units, the stated precision, the meaning of the rounded total, your reproducible function choice and rationale, and the independent tallies and boundary test

The completed model table retains an empty class and a displayed percentage total of 100.1%. Which audit is valid?