SME
Geographic Information Systems (GIS) · Problem type
Mean center and standard distance
Use this for “Three clinics sit at (2, 1), (4, 5) and (9, 3) km on a grid. Find their mean center and standard distance.”
Chart
x̄
=
2 + 4 + 9
3
=
5;
ȳ
=
1 + 5 + 3
3
=
3.
SD
=
√(Σ(
(x − x̄)
2
+
(y − ȳ)
2
)
÷
3)
=
√(
34
3
)
=
3.367
km:
about
two-thirds
of
a
spread-out
set
lie
within
it.
This is an example. Type your own values to start a new problem.
x₁
Point 1 x
example
y₁
Point 1 y
example
x₂
Point 2 x
example
y₂
Point 2 y
example
x₃
Point 3 x
example
y₃
Point 3 y
example
x̄
Mean center x
worked out (not typed)
ȳ
Mean center y
worked out (not typed)
SD
Standard distance
worked out (not typed)
Formulas
x̄
=
x₁
+
x₂
+
x₃
3
5
=
2 + 4 + 9
3
ȳ
=
y₁
+
y₂
+
y₃
3
3
=
1 + 5 + 3
3
SD
=
√((
(
x₁
−
x̄
)
2
+
(
y₁
−
ȳ
)
2
+
(
x₂
−
x̄
)
2
+
(
y₂
−
ȳ
)
2
+
(
x₃
−
x̄
)
2
+
(
y₃
−
ȳ
)
2
)
÷
3)
3.3665
=
√((
(2 − 5)
2
+
(1 − 3)
2
+
(4 − 5)
2
+
(5 − 3)
2
+
(9 − 5)
2
+
(3 − 3)
2
)
÷
3)
Assumptions
•
Projected grid coordinates in kilometers, so distances are straight lines on the map; each point counts the same.
•
The mean center is the average x and the average y; it need not lie on any point.
•
The standard distance is the root of the points’ mean squared distance from the mean center: the spatial standard deviation.
Step-by-step
WE KNOW
x₁ = 2 km, y₁ = 1 km, x₂ = 4 km, y₂ = 5 km, x₃ = 9 km, y₃ = 3 km
FIND
mean center x (x̄), mean center y (ȳ), standard distance (SD)
Step 1 · Find mean center x (x̄)
Use x̄ = (x₁ + x₂ + x₃) ÷ 3
Add the x-coordinates and divide by the number of points.
x̄
=
2 + 4 + 9
3
x̄
=
15
3
x̄
=
5
km
Step 2 · Find mean center y (ȳ)
Use ȳ = (y₁ + y₂ + y₃) ÷ 3
Add the y-coordinates and divide by the number of points.
ȳ
=
1 + 5 + 3
3
ȳ
=
9
3
ȳ
=
3
km
Step 3 · Find standard distance (SD)
Use SD = √(((x₁ − x̄)² + (y₁ − ȳ)² + (x₂ − x̄)² + (y₂ − ȳ)² + (x₃ − x̄)² + (y₃ − ȳ)²) ÷ 3)
Square each point’s distance from the mean center, average the squares, and take the root.
SD
=
√((
(2 − 5)
2
+
(1 − 3)
2
+
(4 − 5)
2
+
(5 − 3)
2
+
(9 − 5)
2
+
(3 − 3)
2
)
÷
3)
(2 − 5)
2
+
(1 − 3)
2
=
9
+
4
=
13
(4 − 5)
2
+
(5 − 3)
2
=
1
+
4
=
5
(9 − 5)
2
+
(3 − 3)
2
=
16
+
0
=
16
13 + 5 + 16 = 34
34
3
=
11.3333
SD = 3.3665 km
Check
Put the numbers back in. Both sides should match.
5
=
2 + 4 + 9
3
✓
3
=
1 + 5 + 3
3
✓
3.3665
=
√((
(2 − 5)
2
+
(1 − 3)
2
+
(4 − 5)
2
+
(5 − 3)
2
+
(9 − 5)
2
+
(3 − 3)
2
)
÷
3)
✓
Topic check
See what you know: 5 questions on this topic
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