Geostatistics - A Primer of GIS: Fundamental Geographic and Cartographic Concepts - page 279

TABLE 14.3. (Observed - Expected) 2
Yearly Snowfall

Elevation Low (0-20 cm) Medium (20-60 cm) High (60+ cm)

500-1000 m 46.118 9.653 13.308

1000-2000 m 0.988 2.031 5.660

2000+ m 59.429 3.059 38.564

Expected probability (low snowfall) = 65/133 = 0.489

Second, calculate the expected probability that the observations are
between 500-1,000 m:

Expected probability (500-1,000 m elevation) = 70/133 = 0.526

Since the null hypothesis assumes they are independent variables, calcu-
late the combined expected probability by multiplying the two expected
probabilities together:

Combined expected probability = 0.489 × 0.526 = 0.162

Then multiply the total number of observations (133) by the combined
expected probability to determine the expected frequency of low snowfall at
500-1,000 m elevation:

Expected frequency = 133 × 0.257 = 34.209

Repeat these four steps for each relationship (the other eight cells)
between snowfall and elevation to complete the expected frequencies table.
Remember that the expected counts in each table square are not the
actual observations, but are based on the assumption that there is no rela-
tionship between the two variables. If we even visually compare the two
tables, we can see differences suggesting that the idea that more snow falls at
higher elevations is probably right and that the null hypothesis will be
disproven by the chi-square statistic. To calculate the chi-square statistic and

TABLE 14.4. (Observed - Expected) 2 /Expected
Yearly Snowfall

Medium
(20-60 cm) Elevation Low (0-20 cm) High (60+ cm) Totals

500-1000 m 1.348 2.086 3.379 6.813

1000-2000 m 2.195 3.397 5.504 11.097

2000+ m 4.750 7.378 12.168 24.296

Total 8.294 12.861 21.051 42.206