The use of the chi-square test to determine predicted
ratios of mixed samples of colored and white bean seeds
INTRODUCTION. The Chi-square test is a
statistic used to determine if experimentally obtained data are a satisfactory
approximation of a theoretical expected ratio. All data obtained from an
experiment deviate some degree from predicted (expected) results. If these
deviations are small, they may be attributed to chance. To determine if the
observed deviations from predicted values are due to chance alone, the
chi-square test is used. The chi-square statistic is given as
. If the
observed number (O) of individuals having a particular phenotype were exactly
equal to the corresponding theoretically expected numbers (E), the fit would be
perfect and the C 2 value
would be zero. Thus a small value for C
2 indicates that the expected and observed ratios are in close
agreement, whereas a large difference between the two indicates marked
deviation from the expected ratio. In order to determine if the observed
deviations from the predicted values are within the limits expected by chance,
statisticians have agreed on the arbitrary limits of 0.05 for a given Degrees
of Freedom. In this exercise, predicted 1:1 ratios of colored and white bean
seeds were tested using the chi-square test.
METHODS. A beaker filled with an unknown mixture of colored and white bean seeds was obtained and a hypothesis predicting the ratio of colored to white bean seeds was constructed. In this case, it was predicted that the ratio of colored to white bean seeds was 1:1. To test this hypothesis a random sample of beans were removed from the beaker and sorted by color. The observed and expected numbers (based on a 1:1 ratio) were recorded. A chi-square analysis was completed to test the hypothesized ratio. The procedure was repeated an additional two times using different random samples of bean seeds.
RESULTS. A mixture of colored and white beans was visually analyzed and hypothesized to contain a 1:1 ratio of the two types. To test this hypothesis three random samples of the beans were counted and sorted based on color. The initial sample produced 53 colored and 38 white seeds (table 1). A chi-square analysis of this sample resulted in a C2 value of 2.47.
|
Table 1. Chi-square results, sample
one. |
|||||
|
Classes (phenotypes) |
Observed (O) |
Expected (E) |
Deviation (O-E) |
(O-E)2 |
(O-E)2/E |
|
Colored |
53 |
45.5 |
7.5 |
56.25 |
1.24 |
|
White |
38 |
45.5 |
-7.5 |
56.25 |
1.24 |
|
Totals |
91 |
91 |
0 |
C 2 = |
2.47 |
The second random sample obtained from the original mixture contained 63 colored and 49 white seeds (table 2). A second chi-square analysis was used to compare these data to the hypothesized 1:1 ratio. This test resulted in resulted in a C2 value of 1.75 (table 2).
|
Table 2. Chi-square results, sample
two. |
|||||
|
Classes (phenotypes) |
Observed (O) |
Expected (E) |
Deviation (O-E) |
(O-E)2 |
(O-E)2/E |
|
Colored |
63 |
56 |
7 |
49 |
0.875 |
|
White |
49 |
56 |
-7 |
49 |
0.875 |
|
Totals |
112 |
112 |
0 |
C 2 = |
1.75 |
A third random sample contained 71 colored and 43 white seeds (table 3). A chi-square analysis of these data based on the hypothesized 1:1 ratio resulted in a C 2 value of 6.88.
|
Table 3. Chi-square results,
trial three. |
|||||
|
Classes (phenotypes) |
Observed (O) |
Expected (E) |
Deviation (O-E) |
(O-E)2 |
(O-E)2/E |
|
Colored |
71 |
57 |
14 |
196 |
3.44 |
|
White |
43 |
57 |
-14 |
196 |
3.44 |
|
Totals |
114 |
114 |
0 |
C 2 = |
6.88 |
DISCUSSION. Three random samples of beans taken from a mixture of colored and white seeds were tested in an attempt to determine the overall ratio of seeds in the mixture. Based on a visual inspection of the mixture, a 1:1 ratio was hypothesized. Chi-square analyses were used to determine if the actual data fit the predicted 1:1 ratio. In our first two trials, the C 2 value was consistent with the hypothesis of a 1:1 ratio because the calculated values of 2.47 and 1.75 are considerably less than 3.84, the maximum value for a two-term ratio (one degree of freedom) at the 0.05 level. In trials one and two, the deviations observed were attributed to chance alone. In trial three, the calculated C 2 value of 6.88 exceeded the theoretical value of 3.84 - the probability that the deviations can be attributed to chance alone is less than 0.05%. For trial three, the hypothesis of compatibility (1:1) between the observed and expected ratio is rejected. Because of our one inconsistent finding, we cannot firmly conclude that the beans are arranged in a 1:1 ratio. More trials are needed to determine if sample three was a poor sample or more accurately predicts the true ratio of colored to white bean seeds.
REFERENCES
Childers, Briana (1999) The chi-square test. Laboratory write-up for ABIO 350. Summer, 1999.