Crosstab software provides a versatile tool to help you glean deeper meaning from your survey results in a number of ways. And one of those ways is by determining if variables or events are independent or dependent.

**Independent vs. Dependent Events**

Independent events are those that have no bearing on each other. The probability of one event happening has no influence on the likelihood of the other event.

A prime example of two independent events is the combination of flipping a coin and rolling a die. The probability of getting a head or tail on the coin has no influence or bearing on the probability of rolling a certain number on the face of the die.

Dependent events are those that are somehow related or linked. The probability of one event happening has an impact on the likelihood of the other event occurring.

An example here can come from a deck of cards. Let’s say you were going to choose two cards, and your first card was an ace, which you kept out of the deck and kept on the side. The act of picking an ace in your first draw decreases the likelihood of picking an ace on your second draw, since there are only four aces in the deck and one is already chosen.

These events would be considered dependent.

**Crosstab Software for the Determination**

A previous post on using crosstab software to determine probability featured a table showcasing the number of males and females with a given eye color. That same table is back, but this time to help determine if specific events are independent or dependent.

Eye Color |
Black |
Brown |
Blue |
Green |
Grey |
Total |

Male |
25 | 15 | 12 | 20 | 10 | 82 |

Female |
20 | 30 | 10 | 15 | 10 | 85 |

Total |
45 | 45 | 22 | 35 | 20 | 167 |

Let’s take the variables of being male and having green eyes, determining if they are independent or dependent.

An equation can be used to prove independence of two variables:

- P (A and B) = P (A|B) * P(B) = P(A) * P(B)

Your first step is to head to the crosstab software table to gather the input needed to complete the equation.

P (A and B) refers to the person (being male AND having green eyes) = 20/167

- 20 males have green eyes
- 167 is the total number of people in the sample

P (A) refers to the person being male = 82/167

- 82 males are in the sample
- 167 is the sample total

P (B) refers to the person having green eyes = 35/167

- 35 people with green eyes are in the sample
- 167 is the sample total

P (A|B) refers to the person being male given green eyes, or the number of males with green eyes out of the total number of people with green eyes = 20/35

- 20 males have green eyes
- 35 is the total number people with green eyes in the sample

**Completing the Equation**

Once all the variables are plugged into the equation, the next step is to simply do the math.

The main equation again is:

- P (A and B) = P (A|B) * P(B) = P(A) * P(B)

Plug in the variables, adding question marks since you’re not yet sure if the equation will compute. Your equation thus becomes:

- 20/167 = ? 20/35 * 35/167 = ? 82/167 * 35/167

If the equation does pan out, and all three areas are equal, then the variables are indeed independent. Once you do the calculations, however, you find:

- .1197 = ? .5714 * .2095 = ? .4910 * .2095
- .1197 = ? .1197 = ? .1028

Since the three areas of the equation are not equal, the variables of being male and having green eyes are not independent.

While you may not need to calculate green-eyed males for your next product or promotion, you can use the equation in different ways when needed. It’s just one more way crosstab software can come to your aid to enhance your understanding and better your business.