Build a frequency and percentage frequency table

Water drinks

\(\displaystyle 18\)

Juice drinks

\(\displaystyle 9\)

Milk drinks

\(\displaystyle 3\)

Drinks sold

\(\displaystyle 30\)

A canteen sold 30 drinks in one lunch break: 18 waters, 9 juices and 3 milks. What is the percentage frequency of juice?

If each percentage is rounded separately, must the displayed percentages add to exactly 100%?

Set the car count to 4, then to 7. Watch every percentage change and compare the sum of the displayed rounded percentages at those two settings

What happened to the sum when there were 4 cars and then 7 cars?

Building a frequency table

A frequency table records how often each category or value occurs. For percentage frequency p, use the row count f and total number of observations n.

1. 1List every row Include every category or every discrete value in the required range, even when a value has frequency zero \(\displaystyle 0,1,2,3,4\)
2. 2Check the total Add the frequencies and compare the result with the sample size \(\displaystyle 4+3+5+2=14\)
3. 3Use the whole sample Find each percentage frequency by dividing its frequency by the total number of observations and multiplying by 100. \(\displaystyle \frac{9}{30}\times100=30\%\)
4. 4Round only the percentage Keep the count and total exact, then round the displayed percentage as instructed; a rounded column can be just above or below 100% \(\displaystyle \frac{1}{6}\times100\approx16.7\%\)

Scores

\(\displaystyle 0,1,1,3,0,3,1,0\)

The requested table includes every score from 0 to 3. How many observations have score 0?

Count occurrences of the value 0; do not confuse a recorded zero with no observation

A frequency table for scores 0 to 3 shows frequency 0 for score 2. What does this mean?

Goals in each match

\(\displaystyle \begin{gathered}0,2,1,3,0,2,2,1\\5,2,0,1,2,3,2,2\end{gathered}\)

Matches

\(\displaystyle 16\)

A school team recorded the goals scored in each of 16 matches. The results are to be put in a discrete frequency table for 0 to 5 goals. Find the percentage frequency of the 2-goal row, to 1 decimal place

Tally how many of the 16 matches had exactly 2 goals, then divide by 16, multiply by 100, and round to one decimal place

Goals in each match

\(\displaystyle \begin{gathered}0,2,1,3,0,2,2,1\\5,2,0,1,2,3,2,2\end{gathered}\)

Matches

\(\displaystyle 16\)

Using the same 16 matches, find the percentage of matches with at least 3 goals, to 1 decimal place

Which rows are 3 or more goals? Include the empty 4-goal row in the requested range, then round the percentage to one decimal place

Juice drinks

\(\displaystyle 9\)

Largest count (water)

\(\displaystyle 18\)

Drinks sold

\(\displaystyle 30\)

Using the wrong denominator. One of these lines is right. Which one?

Categories

\(\displaystyle 6\)

Each displayed share

\(\displaystyle 16.7\%\)

Six categories have equal counts. Each percentage is displayed as 16.7%. What is the sum of the displayed percentages?

Add the six displayed values. Do not replace their sum with the exact underlying total

Six categories have equal counts, and each percentage is displayed as 16.7%. Should one category be changed to 16.5% just to force the displayed sum to 100%?

A table gives the frequency of each goal count over 16 matches. Can it tell you which match had each score?

Produce a complete frequency table

These invented records give the single club chosen by each of 24 students. Construct a table with all four clubs, frequencies, percentages to one decimal place, column headings and totals. Keep each student in exactly one category

1. 1Records 1–6 Games, Design, Garden, Music, Design, Games
2. 2Records 7–12 Garden, Games, Design, Music, Games, Garden
3. 3Records 13–18 Design, Games, Music, Design, Garden, Games
4. 4Records 19–24 Design, Music, Games, Garden, Design, Games
5. 5Use and justify technology Choose a spreadsheet counting function or a statistical frequency-table function. Explain why it suits category labels, use it, and record the input range, formula or settings and output. Calculate percentages from exact counts before rounding
6. 6Verify without repeating the same operation Manually tally Games and Music, and check that the frequency total equals the number of records. Explain what a displayed percentage total does and does not establish
7. 7Generate a counterexample Choose your own three positive category counts. Use technology to test whether their separately rounded one-decimal percentages total 100%. Find a counterexample to the claim that this always happens, and verify the exact unrounded sum by hand

Model response: Produce a complete frequency table

Compare your saved response with this model. Other correct layouts, technology functions and explanations are possible

1. 1Complete table In the order Design, Garden, Games, Music, the frequencies are 7, 5, 8, 4 and the percentages are 29.2%, 20.8%, 33.3%, 16.7%. Frequencies total 24; these displayed percentages total 100.0%. Every percentage uses 24 as its denominator
2. 2One technology record Place the raw labels in A2:A25 and the four distinct club names in D2:D5. In E2 use COUNTIF($A$2:$A$25,D2), then fill down. Put SUM(E2:E5) in E6. In F2 use ROUND(100*E2/$E$6,1), then fill down. Label F as Percentage (%) and display one decimal place; these cells already contain percentage numbers, so do not apply a percentage format that multiplies them by 100 again. COUNTIF counts matching category labels; an arithmetic mean of these labels would not answer the question
3. 3Independent checks Games occurs twice in each line, and Music occurs once in each line. The four lines contain 24 records, matching the sum of the category frequencies. A total of 100% alone cannot detect swapped category counts; compare labels and at least two tallies as well \(\displaystyle \begin{gathered}2+2+2+2=8\\7+5+8+4=24\end{gathered}\)
4. 4Counterexample and exact check One possible choice is three counts of 1. Each percentage displays as 33.3%, so the displayed sum is 99.9%. The exact sum is 100%. Do not alter a correctly rounded entry to force a displayed total of 100% \(\displaystyle \begin{gathered}\frac{100}{3}\%\approx33.3\%\\3\times\frac{100}{3}\%=100\%\end{gathered}\)
5. 5Self-review checklist Check all records and category labels, count totals, the stated denominator and precision, a reproducible tool record and rationale, two manual tallies, and your own counterexample with an exact check

This is the model table. Which independent check would detect an accidental exchange of the Games and Music frequencies?