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Tables with totals

Statistics and data · level 2, lesson 2 of 6

A question to start

Priya and Tom both counted the same tally chart of favourite seasons, from a class of 26 children. Priya says the counts add up to 26. Tom says they add up to 27.

They cannot both be right. Who made the mistake, and how could you find it without asking all 26 children again?

The answer is a table with a total. A total is like an alarm. When it does not match the number of people you asked, it rings, and it tells you to look again. In this lesson you learn to build tables, to add their totals quickly, and to use the totals as a check, even in a table with rows and columns.

The idea

From tallies to a table

A tally chart is perfect while the answers are coming in. But tally marks are slow to read, and you cannot easily add them. When the survey is finished, we turn each tally into a number and write the numbers in a table.

A table is a grid of boxes. Each line going across is a row. Each line going down is a column. The top row holds the headings, which say what each column means. And the table has a title that says what it is about.

Here is the season survey from the last lesson as a table.

Our favourite season (26 children)

Season Number of children
winter 6
spring 5
summer 11
fall 4
Total 26

The last row is the total: all the counts added together.

Adding a column quickly

To add a column of numbers, look for pairs that make 10, or that are easy to add.

  • 6 and 4 make 10 (winter and fall).
  • 5 and 11 make 16 (spring and summer).
  • 10 and 16 make 26.

You may add the numbers in any order you like, and the total is always the same. That is a very useful fact. It means you can hunt for friendly pairs instead of always adding from the top.

Why the total is a check

Every child gave one answer, and every answer went in one row. So when you add up all the rows, you should get exactly the number of children. If the total is:

  • too big, something was counted twice: maybe a child answered twice, or a tally mark was read wrong, or one number was copied wrong.
  • too small, something was missed: maybe a child was away, or a mark was skipped.
  • just right, that is good news. (It is not a full proof that every number is right: two mistakes could cancel out. But it catches most mistakes.)

Back to Priya and Tom. Tom's table said: winter 6, spring 5, summer 12, fall 4. Total 27. Priya's said summer 11. Look at the tally for summer again: two gates and one line, which is 5, 10, 11. Tom read 11 as 12. Priya is right. The total found the mistake, and the tally showed which row it was in.

A table with two columns of counts

Sometimes we ask the same question in two groups and want to compare them. Two classes, Room 1 and Room 2, each chose a snack.

Snack choices in two classes

Snack Room 1 Room 2 Total
apple 8 5 13
crackers 6 10 16
yogurt 9 7 16
Total 23 22 45

Now there are totals both ways.

  • The totals across each row (the last column) tell you how many children chose each snack in both rooms together. Apple: 8 and 5 make 13.
  • The totals down each column (the last row) tell you how many children are in each room. Room 1: 8 and 6 and 9 make 23. Room 2: 5 and 10 and 7 make 22.
  • The number in the bottom right corner is the grand total: every child in both rooms.

The corner check

Here is something beautiful. You can find the grand total in two different ways:

  • add the last column: 13 and 16 and 16 make 45;
  • add the last row: 23 and 22 make 45.

They must be the same. Why? Both ways add up every number inside the table, once each. Adding by rows first, or by columns first, just changes the order, and the order of adding never changes the answer. It is like counting the children in a classroom by rows of desks or by columns of desks: either way you count every child.

So if your two corner answers are different, there is a mistake somewhere in the table. This is the corner check, and people who keep accounts for shops and businesses have used it for hundreds of years.

What a table is good for

A table is good for exact numbers. You can look up "how many in Room 2 chose crackers?" and find 10 straight away. A table is not so good for seeing the pattern at a glance; for that we draw a graph, in the next lesson. Most good reports have both: a table for the exact numbers, and a graph for the picture.

Worked examples

Example 1. Tallies to a table. A tally chart about favourite sports shows: soccer, two gates and three lines; hockey, one gate and four lines; swimming, one gate; basketball, three lines. Twenty-seven children answered. Make the table and check it.

  1. Soccer: 5, 10, then 11, 12, 13. 13.
  2. Hockey: 5, then 6, 7, 8, 9. 9.
  3. Swimming: 5. Basketball: 3.
  4. Total: 13 and 9 make 22, and 5 make 27, and 3 make 30.
  5. But only 27 children answered! The total is 3 too many.
  6. The tallies are read again, and they are right. So the problem is in the voting: the teacher finds out that three children put up their hands twice, for two sports. The class does the survey again, more carefully.
Sport Number of children
soccer 13
hockey 9
swimming 5
basketball 3
Total 30

The table is honest about what was counted. But because 30 is not 27, it cannot be trusted as a vote.

Example 2. Filling in a missing number. A table of favourite colours from 24 children has red 7, blue 9, green ?, yellow 3. Find the missing number.

  1. The total must be 24.
  2. The numbers we know: 7 and 9 make 16, and 3 make 19.
  3. Green is what is left: from 19 count up to 24. That is 5.
  4. Check: 7 and 9 and 5 and 3 make 24. Yes.

Example 3. Reading a table with two groups. Use the snack table above.

  1. How many children in Room 2 chose crackers? Find the crackers row and the Room 2 column: 10.
  2. Which snack was most popular in Room 1? The biggest number in the Room 1 column is 9: yogurt.
  3. Which snack was most popular in both rooms together? Look at the Total column: crackers 16 and yogurt 16 are tied.
  4. How many more children are in Room 1 than in Room 2? 23 and 22: 1 more.

Notice that the "most popular" answer can change when you put groups together. Room 1 liked yogurt best, Room 2 liked crackers best, and together it is a tie.

Try it

  1. Make a table with a total for these counts of favourite pets: dog 7, cat 5, fish 3, no pet 4, rabbit 1.
  2. Add 8, 3, 7, 2 by finding pairs that make 10.
  3. A class of 25 children made a table: walk 9, car 8, bus 6, bike 3. Do the total check. What might have happened?
  4. A table of 30 children's favourite fruit has apple 12, banana ?, grapes 10. Find the missing number.

  5. Copy and finish this table.

    Drink Class A Class B Total
    milk 7 9 ?
    juice 10 ? 18
    water 4 6 ?
    Total ? ? ?

  6. In question 5, do the corner check.

  7. Why should the bottom-right corner of a two-way table be the same whether you add down or across?
Answers

  1. 7 and 5 make 12, and 3 make 15, and 4 make 19, and 1 makes 20.

    Pet Number of children
    dog 7
    cat 5
    fish 3
    no pet 4
    rabbit 1
    Total 20

  2. 8 and 2 make 10, 3 and 7 make 10, so the total is 20.

  3. 9 and 8 make 17, and 6 make 23, and 3 make 26. That is 1 more than 25, so one child was counted twice (or one count was written wrong). Count again.
  4. 12 and 10 make 22. From 22 count up to 30: 8 chose banana.
  5. Milk total: 7 and 9 make 16. Juice, Class B: from 10 up to 18 is 8. Water total: 4 and 6 make 10. Class A total: 7 and 10 and 4 make 21. Class B total: 9 and 8 and 6 make 23. Grand total: 44.
  6. Down the last column: 16 and 18 and 10 make 44. Across the last row: 21 and 23 make 44. They match.
  7. Both ways add every number inside the table exactly once. Only the order of adding is different, and changing the order of adding never changes the answer.

Common mistake

Skipping the total. The total takes a minute and catches most counting mistakes. A table without a total gives no warning when something is wrong.

Thinking a matching total proves everything is right. If one count is 1 too big and another is 1 too small, the total still matches. The total check catches most mistakes, not all of them, so read the tallies carefully too.

Where it leads