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To illustrate how to conduct rate-of-change calculations, wewill use the following example. Note that this is just an example;the data in the table below do not match the data collected in thisexperiment.

Fossil SticklebackPelvic Phenotype Totals
	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
Complete:	20	8	3	1	3	0
Reduced:	0	5	16	19	5	16
Absent:	0	7	1	0	12	4

Using these numbers, you need to calculate the rate of change inthe relative frequency of stickleback with a complete pelvisper 1,000 years.

Step 1. Calculate the relative frequency of stickleback with acomplete pelvis in each layer using this formula:

Relative frequency =

stickleback with a complete pelvis total number of stickleback analyzed in the layer

In this example, layer 1 had a total of 20 fish and 15 had acomplete pelvis; the relative frequency of fish with a completepelvis is 15/20 = 0.75. In other words, 75% of fish in that layerhad a complete pelvis.

For layer 2 the relative frequency of fish with a complete pelvisis 0.5.

Step 2. Calculate the rate of change in relative frequenciesbetween layer 1 and layer 2—a span of 3,000 years.

To do that, you subtract the number of the older layer (layer 1)from that of the more recent neighboring layer (layer 2).

Thus, the change in relative frequency of stickleback with acomplete pelvis between layer 1 and layer 2 = 0.5-0.75 = -0.25.(Note that it is a negative number because the relative

frequency of fish with a complete pelvis decreased.)

Step 3. Calculate the rate of change for 1,000-year increments. Todo this, you must divide each rate of change by 3 because there are3 1,000-year increments between layers 1 and 2, and between layers2 and 3, and so on.

So, the rate of change in relative frequency of stickleback with acomplete pelvis between layer 1 and layer 2 per 1,000 years =-0.25/3 = -0.083. In other words, for every thousand years betweenlayer 1 and layer 2 there is an average 8.3% decrease in therelative frequency of fish with the complete pelvis.

First 3,000 years
(From layer 1 to layer 2)

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Next 3,000 years
(From layer 2 to layer 3)

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Next 3,000 years
(From layer 3 to layer 4)

Next 3,000 years
(From layer 4 to layer 5)

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Next 3,000 years
(From layer 5 to layer 6)

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Rate of change per
thousand years

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