# The Spearman-Brown Formula: Definition & Example

The Spearman-Brown formula is used to predict the reliability of a test after changing the length of the test.

The formula is:

Predicted reliability = kr / (1 + (k-1)r)

where:

• k: Factor by which the length of the test is changed. For example, if original test is 10 questions and new test is 15 questions, k = 15/10 = 1.5.
• r: Reliability of the original test. We typically use Cronbach’s Alpha for this, which is a value that ranges from 0 to 1 with higher values indicating higher reliability.

The following example shows how to use this formula in practice.

### Example: How to Use the Spearman-Brown Formula

Suppose a company uses a 15-item test to assess employee satisfaction and the test is known to have a reliability of 0.74.

If the company increases the length of the test to 30 items, what is the predicted reliability of the new test?

We can use the Spearman-Brown formula to calculate the predicted reliability:

• Predicted reliability = kr / (1 + (k-1)r)
• Predicted reliability = 2*.74 / (1 + (2-1)*.74)
• Predicted reliability = 0.85

The new test has a predicted reliability of 0.85.

Note: We calculated k as 30/15 = 2.

### Cautions on Using the Spearman-Brown Formula

Based on the Spearman-Brown formula, we can see that increasing the number of items on a test by any number will increase the predicted reliability of the test.

For example, suppose we increase the number of items on the test from the previous example from 15 to 16. Then we would calculate k as 16/15 = 1.067.

The predicted reliability would be:

• Predicted reliability = kr / (1 + (k-1)r)
• Predicted reliability = 1.067*.74 / (1 + (1.067-1)*.74)
• Predicted reliability = 0.752

The new test has a predicted reliability of 0.752, which is higher than the reliability of 0.74 on the original test.

Using this logic, we might think that increasing the length of the test by a massive amount of items is a good idea because we could push the reliability closer and closer to 1.

However, we should keep in mind the following:

1. Using too many items can cause fatigue effects.

If a test has too many questions then individuals may become fatigued as they answer more and more questions, causing them to produce less reliable answers as the test drags on.

2. The new items added to the test should be of equal difficulty to the existing items.

It’s important that if we do decide to increase the length of a test that we make sure the new items / questions we’re adding are of equal difficulty to the existing items otherwise the predicted reliability will not be accurate.