Construct and percentage a two-way frequency table

One card is blue and round. Where should its single tally go?

Cards 1, 2 and 3

\(\displaystyle \text{Blue, Round}\)

Cards 4 and 5

\(\displaystyle \text{Blue, Square}\)

Card 6

\(\displaystyle \text{Red, Round}\)

Cards 7 and 8

\(\displaystyle \text{Red, Square}\)

Count the blue-and-round cards from these paired records

Cards 1, 2 and 3

\(\displaystyle \text{Blue, Round}\)

Cards 4 and 5

\(\displaystyle \text{Blue, Square}\)

Card 6

\(\displaystyle \text{Red, Round}\)

Cards 7 and 8

\(\displaystyle \text{Red, Square}\)

How many of the listed cards are blue?

Cards 1, 2 and 3

\(\displaystyle \text{Blue, Round}\)

Cards 4 and 5

\(\displaystyle \text{Blue, Square}\)

Card 6

\(\displaystyle \text{Red, Round}\)

Cards 7 and 8

\(\displaystyle \text{Red, Square}\)

How many of the listed cards are round?

Cards 1, 2 and 3

\(\displaystyle \text{Blue, Round}\)

Cards 4 and 5

\(\displaystyle \text{Blue, Square}\)

Card 6

\(\displaystyle \text{Red, Round}\)

Cards 7 and 8

\(\displaystyle \text{Red, Square}\)

What is the total number of cards?

Why can you not independently shuffle the colour list and the shape list before tallying?

Change some blue cards from round to square. Watch both margins and the grand total

Which total is preserved?

Use the table to find the number of square cards

The table is transposed. Find the Blue total

Row total

\(\displaystyle 17\)

Round count

\(\displaystyle 11\)

A Blue row contains 17 cards, each round or square. Eleven are round. How many are square?

Round total

\(\displaystyle 25\)

Blue-round count

\(\displaystyle 17\)

There are 25 round cards. Seventeen are blue. The remaining round cards are red. How many are red?

Find the grand total of this three-category table

The Green-and-Square count is zero. What can you conclude?

Construct, check, then interpret

A two-way frequency table preserves paired categorical records. Add margins only after every record has been assigned to one cell

1. 1Keep each pair intact One record contributes one count to the cell matching both categories
2. 2Categories must be clear Choose non-overlapping categories that account for every included record, or state how missing or other values are handled
3. 3Check both sets of margins All row totals and all column totals must give the same grand total

Row totals add to 28 but column totals add to 29. What should you do?

A record was entered twice. What repair is justified?

For a percentage within an explanatory category, which records form the denominator group?

Two-way frequency tables

Keep the two categorical values from each case together, place each case in one cell, and add margins. For a conditional percentage, divide a cell by the total for its explanatory group and multiply by 100.

1. 1Keep paired records together Place a case in the cell identified by both of its categorical values \(\displaystyle \left(\text{Group A},\text{Response 1}\right)\to\text{one cell}\)
2. 2Add every margin A row or column total is the sum of all the cell frequencies in that margin \(\displaystyle 8+12=20\)
3. 3Use the group total For a conditional percentage, divide the cell frequency by the total for its explanatory group \(\displaystyle \frac{8}{20}=40\%\)
4. 4Check the grand total The grand total must equal the sum of all row totals and the sum of all column totals \(\displaystyle 20+25=45\text{ (grand total)}\)

Given values

\(\displaystyle 8,\quad12\)

Two cells in the Group A row contain 8 and 12 records. Find the Group A row total

Given values

\(\displaystyle n_{\mathrm{cell}}=8,\quad n_A=20\)

The first cell contains 8 records and the Group A total is 20. Find its conditional percentage

A cell has 8 records, its Group A total is 20, and the whole sample has 50 records. Which calculation gives the percentage within Group A?

Name the condition before choosing the margin

“Borrow among Early participants” uses the Early total. “Early among Borrow participants” uses the Borrow total. Transposing a table changes where a margin is drawn but does not change the group you condition on

1. 1Zero and missing values A zero cell gives 0% within a positive-sized group. An empty group has denominator zero, so its conditional percentages are undefined. Missing responses are not observed “No” responses. State whether missingness is an explicit category or whether incomplete records are excluded, report how many and why, and keep the same policy throughout
2. 2Round at the end Calculate from counts using full precision. Rounded conditional percentages may sum to 99.9% or 100.1%; do not change the observed counts to force a displayed sum. Exact percentages in an exhaustive group sum to 100%

Given values

\(\displaystyle n_{\mathrm{Early,Borrow}}=2,\quad n_{\mathrm{Borrow}}=5\)

In this table, what percentage of all Borrow participants attended Early?

The same cell contains 2 Early Borrow participants. Which pair of conditional percentages is correct?

An optional session has no participants. What is the percentage selecting Repair within that session?

Construct a complete table and reverse a condition

Fifteen fictional participants each choose one reuse action. Paired records: E1: Early, Repair; E2: Early, Repair; E3: Early, Repair; E4: Early, Borrow; E5: Early, Borrow; E6: Early, Recycle; L1: Late, Repair; L2: Late, Repair; L3: Late, Repair; L4: Late, Borrow; L5: Late, Borrow; L6: Late, Borrow; L7: Late, Recycle; L8: Late, Recycle; L9: Late, Recycle. Construct a two-way count table with Early/Late rows and Repair/Borrow/Recycle columns, including every margin

1. 1Calculate and explain Convert each session’s counts to conditional percentages. Show exact fractions and percentages to one decimal place where needed. Find Early among Borrow, then compare it with Borrow among Early. Transpose the complete count table and explain why neither conditional question changes. Account for a rounded row sum below 100%
2. 2Choose and verify technology Choose a pivot-table or two-criteria counting function. Record one row per ID, both category fields, the count aggregation and totals. Then calculate cell/session total×100, protecting the intended denominator when copying. Verify at least one cell by manually listing IDs and verify both sets of margins sum to the number of records
3. 3Complete before continuing Save your complete table, labelled displays, report and technology record before opening the model response. Complete writing, drawing and external-tool execution are not automatically marked; use self or teacher review

Model response: keep the condition with its denominator

The count rows (Repair,Borrow,Recycle,Total) are Early=(3,2,1,6), Late=(3,3,3,9), Total=(6,5,4,15). Early percentages are 50%, 100/3%=33.3% rounded, and 50/3%=16.7% rounded. Late percentages are each 100/3%, shown as 33.3%,33.3%,33.3%

1. 1Verify and reverse Early Borrow records are E4 and E5. All Borrow records are E4,E5,L4,L5,L6, so Early among Borrow is 2/5×100=40%. Borrow among Early is 2/6×100=100/3%, about 33.3%. After transposition, row totals become column totals but these denominators stay 5 and 6. The Late displayed sum 99.9% is rounding; exact thirds sum to 100%
2. 2One reproducible workflow Enter ID,Session,Response in separate columns. Count ID by Session and Response, with row and column totals. Alternatively use a two-criteria count for each pair, then sum margins. In a separate percentage table divide each cell by its session total, fixing the total column while copying across. Independently tally the two Early Borrow IDs and verify 6+9=6+5+4=15. Keep complete records paired and record any future missing-response policy rather than treating missing as zero

A software table shows a Late row of 33.3%,33.3%,33.3%. Should one source record be edited to make the displayed sum 100%?