Measuring Variation

Counting repeated parts in organisms often results in demonstrating that there is variation among individuals of the same species. Snakes, lizards, and fishes have scales on their bodies, and the number of rows of body scales usually varies within the same species. The same is frequently true for the numbers of vertebrae in these same animals. In fishes, the fins also are formed of repeated parts called fin supports which support the membranes of the fins, and many species of fishes show variation in the number of fin supports.

To demonstrate the tendency to show variation in repeated parts, we will measure variation in the dorsal and anal fins a local marine fish, Oligocottus snyderi, the fluffy sculpin. As do many fishes that inhabit rocky bottom, the fluffy sculpin lacks scales, but there are generous patches of little skin flaps on the head and body that give the fish a hairy or "fluffy" appearance, and thus its name. This species lives in shallow rocky areas along the sea shore. As many sculpins do, it lies on the bottom and preys upon small invertebrate animals. It can make short swimming runs by pushing off strongly from the bottom with its huge pectoral fins, steadying itself in the water by raising its dorsal and anal fins like keels.

The object of our study today is to measure the degree of variation in the number of dorsal and anal fin rays in the fluffy sculpin, and to compare the degree of variation found in the two fins. The number of fin rays varies naturally, due to both genetic and environmental causes. By comparing the degree of variability in the two fins, we might be able to determine if the number of rays in each fin is controlled by the same or by different factors. If the number of rays is controlled by the same factor (such as the number of body segments laid down during the embryonic stage) we might expect the variation in dorsal and anal fin-ray counts to be the same, whereas if they are controlled by different factors, the variation in counts might be different.

With your laboratory partner, select five specimens of fluffy sculpin, placing them in a finger bowl covered with water. Make sure that you keep the specimens wet, as the counts are difficult to make on dried specimens.

Note that the dorsal and anal fins each are subdivided into an anterior and posterior part. For counting fin supports, however, we will consider two parts as a single fin. For each fish, one laboratory partner makes a count of the dorsal fin supports and also makes a separate count of the anal fin supports. Hold the fish up to the light, if necessary, in order to see fin supports distinguished from color pattern. With forceps, gently lift fin supports if any have folded down on top of others; try not to tear fin membranes, or it will become difficult to make recounts.

The second laboratory partner then makes the same counts, preferably without knowing the first partner's results. Whenever the two partners differ in their count, that count should be repeated until agreement is reached. Record these counts.

Place your results on the class chalk board so that all data generated in the class are available to all students. Carefully transfer all data to the data sheet provided. You should have at least 30 dorsal fin counts and 30 anal fin counts. Using all the data gathered by the entire class for the dorsal fin, calculate the variance for the dorsal fin count. Do the same for the anal fin count. When you finish calculating the variances, record them on the data sheet. The formula for variance is often given as follows:

That is, for each fin, you calculate the average count (the mean), then add up the squared differences between each count and the mean, and finally divide this sum by the number of observations minus one. Note that when the individual counts differ greatly from the mean, the variance will be larger, and when the individual counts do not differ as much from the mean, the variance will be smaller. Dividing by the sample size is like calculating the average difference from the mean. When you finish calculating the variances, record them on the data sheet.

It is likely that, at the sensitive period in larval development when the fishes' body segments are being formed, temperature and other environmental variables influence individual phenotypes differentially with the result that there will be variation among individuals in the number of body segments. Since the number of body segments approximates the number of vertebrae a fish will have, the numbers of vertebrae within a population is therefore likely to show variation. Numbers of fin supports in the dorsal and anal fins may also be related to the number of body segments. If this is true for the fluffy sculpin, then we might expect that the variance, s₂, (an estimator of variability) for the dorsal fin is about the same as the variance for the anal fin, that is, that the two fins are equally variable. To test whether this is so, we will use a statistical test, the F-test.

To find whether the two fins are equally variable, one calculates an F value, which is actually a ratio, and then compares it with a Table of Critical Values, which is available from your instructor. If the calculated F value is less than the critical F value found from the Table, then you may conclude that the two fins are equally variable. If the calculated F value is greater than the critical F value found from the Table, then you may conclude that the two fins have different degrees of variability. The calculated F value is determined by dividing the larger of the two variances by the smaller:

When you have your F value, you will need two more simple calculations in order to use the Table of Critical Values. These are the degrees of freedom of your two samples. The degrees of freedom for the numerator, v₁, is n₁-1 and for the denominator, v₂, is n₂-1. Discuss with your instructor the meaning of probability level and decide which probability level you will use. With this information, decide whether or not the dorsal fin has the same amount of variability as the anal fin in your sample of fluffy sculpin.