Middle Years

Lesson

Two-way tables represent data that is classified by two criteria. If the first criterion was a student's main writing hand, and the other was whether or not they enjoy English, we could produce a table like this:

Right | Left | |
---|---|---|

Enjoys English | $4$4 | $9$9 |

Doesn't enjoy English | $2$2 | $15$15 |

To read a two way table, look at the column and row that a number is in to find the right cell, the box where a single number is written. For example there are $9$9 students who are left-handed and enjoy English (top-right cell), and $2$2 students who are right handed that don't enjoy English (bottom-left cell).

Tables will often inclue totals of each column, each row, and the total sum in the corner:

Right | Left | Total | |
---|---|---|---|

Enjoys English | $4$4 | $9$9 | $13$13 |

Doesn't enjoy English | $2$2 | $15$15 | $17$17 |

Total | $6$6 | $24$24 | $30$30 |

A pet store sells three types of animals. Some of them have spots, some do not. This two-way table represents the animals they have for sale one day:

Cat | Dog | Fish | |
---|---|---|---|

Has spots | $7$7 | $4$4 | $11$11 |

No spots | $5$5 | $10$10 | $8$8 |

**Think:** We can use the following formula to find the probability:

$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Probability=Number of favourable outcomesTotal number of outcomes

We will need to find the total number of animals and the number of fish to substitute into the formula.

**Do:** The number of spotted fish is the number in the top-right cell, and the number of fish without spots is the number in the bottom-right cell. Adding these together tells us that there are $11+8=19$11+8=19 fish altogether.

The total number of animals is all the numbers added together, $7+4+11+5+10+8=45$7+4+11+5+10+8=45.

This means the probability is $\frac{19}{45}$1945.

A two way table often presents information that could also be presented with a Venn diagram. We can convert between a two way table and a Venn diagram by matching up their different parts.

This two-way table represents the handedness of students, and whether or not they entered the Talent Show:

To represent this information in a Venn diagram, we choose one row and one column to become circles. Here we chose the column marked "Left" and the row marked "Entered":

- The number that is in both chosen categories $\left(4\right)$(4) goes in the overlap of the two circles.
- The other value in the "Entered" row $\left(9\right)$(9) represents the "Entered and right-handed" students, and goes in the "Entered" circle outside the overlap.
- The other value in the "Left" column $\left(2\right)$(2) represents the "Didn't enter and left-handed" students, and goes in the "Left-handed" circle outside the overlap.
- Any cells that are in neither the highlighted row nor the highlighted column $\left(15\right)$(15)goes into the surrounding box, outside both circles.

Using this Venn diagram:

Create a two-way table representing the same information.

**Think:** Each circle represents a category, so each circle name should be a heading - one a row heading, and one a column heading.

The numbers outside the "Cut" circle are "Not cut", and the numbers outside the "Dyed" circle are "Not dyed", so they should be the other headings.

**Do:** The empty two way table will look like this:

Cut | Not cut | |
---|---|---|

Dyed | ||

Not dyed |

Cut | Not cut | |
---|---|---|

Dyed | $35$35 | $20$20 |

Not dyed | $12$12 | $3$3 |

$50$50 students were asked whether or not they were allergic to nuts and dairy. The two way table is provided below.

Allergic to Nuts | Not Allergic to Nuts | |
---|---|---|

Allergic to Dairy | $6$6 | $11$11 |

Not Allergic to Dairy | $6$6 | $27$27 |

How many students are allergic to nuts?

How many students are allergic to nuts or dairy, or both?

How many students are allergic to

**at most**one of the two things?

A student makes a Venn diagram of students who are late to school, and students who catch the bus to school.

Complete the following table:

Late Not late Caught bus $\editable{}$ $\editable{}$ Didn't catch bus $\editable{}$ $\editable{}$

Students in Irene's class were asked if they owned a dog and asked if they owned a snake. The following two way table shows that information.

Owns a dog | Doesn't own a dog | |
---|---|---|

Owns a snake | $2$2 | $3$3 |

Doesn't own a snake | $13$13 | $11$11 |

Which of the following Venn diagrams represents the information provided in the two way table?

ABCD