SME
Geographic Information Systems (GIS) · Problem type
Clustering from quadrat counts
Use this for “Shrubs are counted in 40 quadrats: mean 2.5 a quadrat, variance 6. Find the variance-to-mean ratio and the chi-square. Are the shrubs clustered?”
Diagram
40 quadrats with mean 2.5 and variance 6 (whole counts made to s² = 6).
VMR
=
s
2
x̄
=
2.4;
χ
2
=
(m
−
1)
×
VMR
=
39
×
2.4
=
93.6
with
39
df.
It is past the upper 2.5 % point, 58.12: the points are clumped.
This is an example. Type your own values to start a new problem.
VMR
=
s²
x̄
m
Number of quadrats
example
χ²
Chi-square
worked out (not typed)
Formulas
VMR
=
s
2
x̄
2.4
=
6
2.5
χ
2
=
(
m
−
1)
×
VMR
93.6 = (40 − 1) × 2.4
Assumptions
•
Quadrats are equal squares laid over the whole study area, and every point is counted once.
•
Variance-to-mean ratio (VMR) near 1 is random (a Poisson pattern); above 1, clustered; below 1, dispersed.
•
Compare χ² with the chi-square table at m − 1 degrees of freedom: for 40 quadrats, above 54.6 is clustered and below 25.7 dispersed (5% in each tail).
Step-by-step
WE KNOW
m = 40, x̄ = 2.5, s² = 6
FIND
variance-to-mean ratio (VMR), chi-square (χ²)
Step 1 · Find variance-to-mean ratio (VMR)
Use VMR = s² ÷ x̄
Counts that bunch up spread more than their mean; random counts spread about as much as their mean.
VMR
=
6
2.5
VMR = 2.4
Step 2 · Find chi-square (χ²)
Use χ² = (m − 1) × VMR
The ratio scaled by the degrees of freedom, one fewer than the quadrats.
χ
2
=
(40
−
1)
×
2.4
χ
2
=
39
×
2.4
χ
2
=
93.6
Check
Put the numbers back in. Both sides should match.
2.4
=
6
2.5
✓
93.6 = (40 − 1) × 2.4 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.