[Workshop] Practical Basic Statistics

Area Measuring
Let’s learn basic statistics
through various “Area Measuring”

Issei Jinguji
Former JICA Expert
AGRICULTURAL STATISTICS PLANNING AND DESIGNIGJICA Tsukuba International Center
14 August 2018

1

Contents
1. Introduction ....................................................................................................................... 1
2. Measuring for geometric shapes ...................................................................................... 1(1) Squares and Rectangles .............................................................................................. 1(2) Triangles ....................................................................................................................... 2
3. Measuring for complicated shapes .................................................................................. 3(1) Rectangular offset method .......................................................................................... 3(2) Crossing offset method ................................................................................................ 3(3) Triangular offset method ............................................................................................. 4(4) Section sheet method ................................................................................................... 4(5) Dot-grid sheet method ................................................................................................. 5(6) Comparison of the results obtained five area calculation methods ......................... 6
4. Measuring for complicated shapes by category .............................................................. 7(1) Section sheet method ................................................................................................... 7(2) Dot-grid sheet method ................................................................................................. 8
5. Dot-grid method as an area sampling survey ............................................................... 10(1) Change from a measuring method to a sampling survey method .......................... 10(2) High density dot-grid method for improving accuracy ............................................11(3) Comparison of the four results and analyses........................................................... 12(4) Operation on the GIS such as Google Earth ............................................................ 14
6. Afterword ......................................................................................................................... 16
7. Quiz [Basic]: Calculate the area and precision of the following figures by color on thetables. ................................................................................................................................... 18(1) Calculation with a low density dot-grid plate.......................................................... 18(2) Calculation with a high density dot-grid plate ........................................................ 19
8. Summary of the Area measuring ................................................................................... 20

2

1. Introduction
Yesterday, you learned basic statistics through “Stone Cutting” which
is an interesting experiment on statistical survey. Today, you learn area
calculations, or area measuring surveys. In general, when you need data onplanted area, you distribute a printed questionnaire of a certain format tolocal offices, which are asked to fill it out and report. In this case, it does notmatter how they fill out the questionnaire. Therefore they often fill in items
without properly researching or investigating, such as a reliable survey
based on the random sampling and actual measuring method. And thoseresults are often unreliable.
As you have already experienced through the stone cutting survey, if
you want to obtain reliable statistics, you should adopt a reliable method
based on the statistical manner with measuring. But it seems very difficult
to apply such a method to a planted area survey in a big target region.
Recently, a simple method for planted area surveys was developed inJapan. It is called “Dot sampling method”. The sampling is conducted on
Google Earth. The detail will be shown by Mr. Kenji Kamikura later.
Here, before you will learn the Dot sampling method, you learn
various traditional area measuring methods. It might be roundabout, but it
will bring you a lot of fruits in the near future.

2. Measuring for geometric shapes
First we would like to show you a review lesson. Do you remember
the following calculation methods which you learned a long years ago when
you were students? It might be hard work to apply those methods to an
actual planted area survey. But let’s learn step by step
As you know, the area of a shape is a number that tells how manysquare units are needed to cover the shape. This is the basic concept of area.
You can find an area by drawing a shape on graph paper, and counting thesquares inside the shape. But it is not efficient when you calculate the areaof geometric shapes such as a rectangle or a triangle. Therefore you can use
area formulas instead.
(1) Squares and Rectangles
The area of a squares (or a rectangle) is the length (a) times the width(b).
1S = a×ｂ
a
2.7cm

b
4.7cm

Figure 4: Rectangle
Area calculation S = a×ｂ=2.7 ×4.7=12.7cm2(2) Triangles
The area of a triangle is the side (a) times the height (h) and divided
by two. The formula is shown below. When you use this formula, you have to
make a perpendicular line (h) against a side (a).
S = (a×h)/2.
2.3cm
h
a
5.2cm

Figure 5: Triangle-1
Area calculation: S =(5.2×2.3)/2=6.0cm
Next one is another formula to calculate the area of triangle, which iscalled “Heron’s formula”. When you calculate the area using the formula, you
don’t have to draw a perpendicular line. You just measure the lengths ofthree sides.

S  s(s  a)(s  b)(s  c)

Where s  (a  b  c) / 2

b 5.2cm

a
4.1cm

4.7cm

c

2

Figure 6: Triangle-2
Area calculations=(4.1+5.2+4.6)
2

=6.85

S = √6.85 × (6.85 − 3.9) × (6.85 − 5.2) × (6.85 − 4.6) = √75.02 = 8.67cm2

3. Measuring for complicated shapes
When you meet such complicated or twisted shapes which are shown
below, you should use offset methods. The methods consist of basically three
types, namely, a rectangular offset method, a crossed offset method andtriangular offset method. Those methods are used not only a sheet of paper
but also on the grounds.
(1) Rectangular offset method
The rectangular offset method is to draw a rectangle on the target
figures. The rectangle should be drawn on the target figure just to be equal
size with the area of the target figure. The offset lines are drawn with aneye-adjusting method.

a

S = a × b = 4.2 × 4.5 = 18.9cm2

4.2cm

b

4.5cm

Figure 7: Rectangular offset method(2) Crossing offset method
Next example shows a crossing offset method which is a kind of therectangular offset method shown above.

3

a
4.2cm
a
S = a × b = 4.2 × 4.5 = 18.9cm2b
4.5cm

Figure 8: Cross method(3) Triangular offset method
Next example shows a triangular offset method with Heron’s formula
4.4

S2
4.1

4.4

5.9
S1

4.7

Figure 9: Triangular offset method
Area calculation
S1 : s =(4.1+4.7+5.9)
2= 7.35

𝑆1 = √7.35 × (7.35 − 4.1) × (7.35 − 4.7) × (7.35 − 5.9) = √91.79 = 9.58cm2
S2 : s =(4.4+4.4+5.9)
2= 7.35

𝑆1 = √7.35 × (7.35 − 4.4) × (7.35 − 4.4) × (7.35 − 5.9) = √92.75 = 9.63cm2
S = 𝑆1 + 𝑆2 = 9.58 + 9.63 = 19.21cm2(4) Section sheet method
When the area of a complicated shape is drawn on a sheet of paper,
it can be measured putting a section sheet on it, which is made of4

transparent plastic. This method attempts to count the number of square
units within the complicated target shape dividing two groups, namely
complete squares and incomplete squares groups. For this reason, it can besaid that the method is faithfully based on the concept of area (“the area of ashape is a number that tells how many square units needed to cover the
shape”) as we mentioned above. Therefor this method is taught at a primaryschool to understand the basic concept of area.
The formula for the area calculation is shown as follows:
Total number of square units = (Number of complete square units×1) +((Number of incomplete square units×0.5)
Area of the target figure = Area per square (unit) × Total number ofsquare units.

Figure 10: Complicated shape and section sheet method
Area calculation
Area per square (unit) = 0.64×0.64 = 0.4096cm2
Total number of square units= (30×1) + (33×0.5) = 46.5
Area of the target figure = 0.4096×46.5 = 19.0cm2(5) Dot-grid sheet method
The area of a complicated shape can be measured with a dot-gridscale instead of the section paper method. When you use this method, first,
you count the number of dots within the target figure, after that you
multiply the area per dot by the number of dot.
This is a traditional area measuring or calculation method, but it is
not known well, because it might be used in a special field, such as
5

“measuring survey” on maps. But this method is powerful and useful for thearea calculations. If you use this method, you don’t have to measure length,
width, height (Compare to Figure4, 5), or three sides (Compare to Figure 6)
with scale or have to draw offset lines (See Figure 7, 8, 9). You just count the
number of dots in a target figure.

Figure 11: Complicated shape and dot-grid sheet method
Area calculation
Area per dot:= 0.64×0.64 = 0.4096cm2
Number of sample dot within the figure=45
Area of the target figure =0.4096×45=18.4cm2(6) Comparison of the results obtained five area calculation methods
The graph below shows the comparison of the results obtained theabove five area calculations (See Figure 7, 8, 9, 10 and 11). The result showseach method is good.
cm2
25
20

18.9

18.9

19.2

19

18.4

18.9

RectangularoffsetCrossed offset

TriangularoffsetSection Paper(Dot-grid(Average)

15
10
5
0

methodFigure 12: Comparison of the five area calculation methods
64. Measuring for complicated shapes by category(1) Section sheet method
The figure shown below is more complicated than the figure shown
above since it is divided into some segmentations. When you need to
calculate the area by color (category/attribute), you can calculate the areas
applying above method. But, it seems difficult and boring a bit.

Figure 13:

More complicated shape and section sheet method

Area calculation
Area per square (unit):= 0.64×0.64 = 0.4096cm2
Number of squares=47
Area of the target figure =0.4096×47=19.3cm2
The calculation table for the number of squares is shown below.
Table 2:Category
Green

Yellow

Area calculation table for the section sheet method
Number ofcomplete square(Weight:1)

Number ofincomplete
square(Weight: 0.5)

1

16

1

15

0

6

7

13

0

7
7

Adjusted
number ofcompletesquarePercentage
(%)

17.5

37.2

20

42.6

Blue

3

13

9.5

20.2

Total

12

70

47.0

100.0

(2) Dot-grid sheet method
If you adopt the dot-grid method, you can measure the area by color
easily. The method is the same as the former method (Figure 11). You will beable to calculate the area by category quickly. You will find out it is like tomeasure with “a two dimensional scale”. This is a strong method especiallyfor the area calculation of complicated figures as you have known already.

Figure 14: More complicated shape and dot-grid sheet method
An example of the area calculation by color is shown on the tablebelow.
Table 3:Category

Area calculation table for dot-grid sheet method
Number ofsample dots
𝑛𝑖

%

Green

19

42.2

Yellow

20

44.5

Blue

6

13.3

Total

45

100.0

Area per dot

Area by colork𝑇̂ = 𝑛1 × 𝑘
7.8

0.40968.2
2.5

0.40968

18.4

Note 1: Special area measuring tools
We don’t explain those methods in detail, just introduce a bit.
(1) Planimeter
A planimeter is a machine to measure complicated figures by tracing
the edge (border line) of those on maps.

Figure 15: Area calculation with a planimeter(2) GIS (Geographic Information System)
GIS has a function to measure the area of land on electrical maps,
which is like the planimeter explained above. Google Earth Pro is a kind of
GIS.

2,240m2
882m2

Figure 16: Area calculation on Google Earth Pro

95. Dot-grid method as an area sampling survey(1) Change from a measuring method to a sampling survey method
1) Theoretical background
By the way, above explanations are how to measure the size of area
on various figures. You might have found out already that those were notstatistics, but just actual measuring surveys using various measuring tools
and methods. It is true in a sense.
However, as you have already found this area measuring method
can be changed as a sampling survey. This method is called “Attribute survey”
to estimate proportion by category. The attribute survey is one of the mostimportant methods to generate reliable statistics. The dot grid plate is a typeof sampling method. The description of below quotation is in case of thesample dots which are selected with a simple random sampling method. The
location of each dot-grid point can be used as a sample point on a figure,which is selected at random using the systematic sampling method. The
difference of the both should be notable.
In this connection, you should pay attention that the dot-grid
method can estimate area “proportions” by counting the number of dots by
category (color) only judging “yes (true)” or “no (false)”. In this case, thefigure as your target can be a binomial population in statistics.
For instance, a statistic text (“BASIC THEORY OF SAPMLINGSURVEYS” MAFF of Japan 2001) which we learned says as follows:“The binomial population is a distribution in which the values of avariable can be either 0 or 1. If “1” is assigned to those which meet a certain
condition and “0” is assigned to those which do not, a distribution called abinomial population will be obtained. For example, the number males andfemales and the probability of getting “1” or not by throwing a die belong tothis category. When n sample (sample size = n) are sampled from a binomial
population, the distribution of the sum of sample value is called the binomial
distribution. If, it is assumed that the probability that variable x is 1 in abinomial population is P and 1 − P = Q the mean μ of this distribution will
be P , the variance 𝜎 2 will be PQand the standard deviation σ(the squareroot of the variance) will be √PQ.”

102) Adaptation for area survey
Assuming the area of target figure (𝑊) will be given, the area by
category (𝑇̂) is estimated by the proportion { 𝑃̂ = (𝑛𝑖 ⁄𝑛)} times W. The
formula is shown below.
𝑇̂ = (𝑛𝑖 ⁄𝑛) × 𝑊
Based on this idea, the area and precision of above figure (See Figure14) can be calculated by color like Table 4. In this calculation, the area oftarget figure is 18.1 cm2, which was calculated with planimeter measuring.
Table 4: Area calculation table for the dot sampling method
Numb
Catego
ry

er of
sampl
e dotsPercenta
ge(%)

Precision
Area
(cm2)

SE

CV

Compleatesurvey

Diff
ere
nce

Area
by
planimeter

Percen
tage
(P)%

D
= 𝑃̂
−𝑃

17.4

7.1

39.2

3.0

7.4

16.6

8.0

44.2

0.3

2.4

6.3

27.0

3.0

16.6

-3.3

18.1

-

-

18.1

𝑛𝑖

𝑃̂
= (𝑛𝑖 ⁄𝑛) × 100

𝑇̂= (𝑛𝑖 ⁄𝑛)
×𝑊

Green

19

42.2

7.6

7.4

Yellow

20

44.5

8.1

Blue

6

13.3

Total

45

100.0

CV
= (𝑆𝐸 ⁄𝑃̂ )√𝑃̂ (100 − 𝑃̂ )⁄𝑛 × 100
SE =

100.
0

0.0

W: 18.1cm2 is calculated by a planimeter.

(2) High density dot-grid method for improving accuracy
It is known that a bigger sample size makes a more reliable estimate.
It means the estimate becomes close to the true value, and the sampling
error becomes smaller. Therefore, we tried to put a higher density dot-gridplate on the figure, and we counted the number of dots by color as follows.
The sample size was 179 as the result.

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Figure 17: More complicated shape by a high density dot-grid sheet method
The result under the sample size is 179 is shown below. As the former
sample size was 45, the new sample proportion is much closer to true value
which is measured by a planimeter, and precision (SE, CV) is much better
than when sample size was 45.
Area calculation by color is shown on the table below.
Table 5:

Area calculation table for the high density dot sampling method

CategoNumberof sample
dots

Precision
Percentage
(%)

Area
(cm2)

Comparison
CompletSECV

e-survey

Differe
nce

(%)

ry

𝑆𝐸

𝑇̂
= (𝑛𝑖 ⁄𝑛)×𝑊
= √𝑃̂ (100 − 𝑃̂ )⁄𝑛CV= (𝑆𝐸 ⁄𝑃̂ )
× 100

Perce
ntage

D = 𝑃̂ − 𝑃

3.6

9.3

39.2

-0.1

8.0

3.7

8.4

44.2

-0.1

16.8

3.0

2.8

16.6

16.6

0.2

100.0

18.1

-

-

100.0

0.0

𝑛𝑖

𝑃̂
= (𝑛𝑖 ⁄𝑛) × 100

Green

70

39.1

7.1

Yellow

79

44.1

Blue

30

Total

179

(3) Comparison of the four results and analyses
Next table shows comparison of the four results shown above,
namely the section sheet method, dot sampling method, high density dot
sampling method and complete survey method.
The estimate of proportion with the high density dot sampling
method is closer to the value of the complete survey than other results.
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Table 6:

Comparison of the high density dot sampling method and others
Section
Dot
High
Completesheetsampling
density dot
survey(%)
method (%)
method (%) sampling (%)(Figure13)(Figure14)(Figure17)(Table4)

Green

37.2

42.2

39.1

39.2

Yellow

42.6

44.5

44.1

44.2

Blue

20.2

13.3

16.8

16.6

Total

100.0

100.0

100.0

100.0

It is interesting to try analyses using results of various surveys. Thisis also a field of basic statistics. From the data shown above table, you candraw graphs as follows.

Blue,
16.8

Green,
39.1

Yellow, 44.1Figure18: Share of Color by the high density dot sampling survey
50 %40

39.1 39.2

44.1 44.2

30
16.8 16.6

20
10
0
Green
Yellow
High density dot sampling (%)
CategoryBlue
Complete survey(%)(Collor)Figure19: Comparison of high density dot sampling survey
and complete survey
Furthermore, you will find out that the each difference ( 𝐷 = 𝑃̂ − 𝑃 )is quite smaller than the SE (Standard error) which shows the simple
random sampling error (See Table 5). There are two reasons there, we think.
The first reason is that the dot-grid sample is selected by such asystematic random sampling method, which is a regular interval sampling13

method in area (or in distance). From a practical point of view, the method isstable, and it has a characteristic such as an actual measuring method witha two dimensional scale, as we mentioned above. You can say the first reason
comes from the difference of sampling method.
The second reason is that each color part as an attribute in the figure
is gathered into certain sizes. If the each part by color is divided into more
small pieces and if they are completely scattered independently, as the result,even the systematic sampling method cannot exert its power, and bring inthe same results as the simple random sampling method. You can say thesecond reason comes from the difference of population’s characteristic.
Therefore, it suggests that you can share the systematic random
sampling method as a dot grid method is better than the simple random
sampling method under such conditions like Figure 17. Of course, in addition,
you should not forget the systematic random sampling method is moreconvenient than the simple random sampling method from the practical view
point.
Memo: Concerned this matter, please remember the experiment of the stone cutting
survey to estimate the average weight per stone using sampling methods. The standard
error (SE) of the systematic sampling method closes to the SE of the simple random
sampling method. You found that the systematic sampling method is not superior to thesimple random sampling method. The reason why is the arrangement of stones in the
population was done without considering the size (weight) or color of each stone.
Therefore, unfortunately, the systematic sampling method could not show its power inthe case.

(4) Operation on the GIS such as Google Earth
You have learned various area measuring methods through thoseexperiments. We think that you might have been interested in the dot-grid
method, especially.
Dot-grid sampling method is an old method. It was known that themethod could be applied to an area survey on aerial maps. But those maps
were so expensive. Therefore, it was difficult to use such expensive maps fora planted survey. But, the advent of “Google Earth” made it possible. Google
Earth is a kind of GIS. Anyone can use it on the Internet.
Connected an old method to the latest technology, a new area survey
14method was born. We call the method “Dot sampling method”, when the
dot-grid method is used on Google Earth. And the method is developing day
by day.
Note 2: Theoretical back ground (1)Inferences for Population Proportions
A

statistics

book

(NEIL

A.

WEISS,“ELEMENTARYSTATISTICS”.

Addison-Wasley Publishing Company USA 1996) explains the attribute survey as
follows:
DEFINITION: Population proportion and sample proportion
Consider a population in which each member is classified as either having or not having
a special attribute, a so-called two category population. Then we use the following
notation and terminology.
Population proportion, 𝑷: The proportion (percentage) of entire population that has
the specified attribute.
̂ : The proportion (percentage) of a sample from the population
Sample proportion, 𝑷that has the specified attribute
FORMULA: Sample proportion
̂ is computed using the formula
A sample proportion, 𝑷
𝑝̂ =

𝑛1
𝑛

Where 𝑛1 denotes the number of members sampled that have the specified attributeand 𝑛 denotes the sample size..
Correspondence between notations for means and proportions
Parameter

Statistics

Means

μ

𝑋̅

ProportionP̂𝑷

̅ , can be used to make inferences about a population mean,
As we know, a sample mean, 𝑋
μ. Similarly, a sample proportion, 𝑃̂, can be used to make inferences about a population

proportion 𝑃.
15

KEY FACT: The sampling distribution of the proportion
Suppose a large random sample of size 𝑛population with population 𝑃̂ .

is to be taken from a two-category

Then the random variable 𝑃̂ is approximately

normally distributed and has mean 𝜇𝑃̂ = 𝑃 and standard deviation 𝜎𝑃̂ = √𝑃(1 − 𝑃)/𝑛.
In other word, probabilities for 𝑃̂ are approximately equal to areas under the normal
curve with parents 𝑃 and √𝑃(1 − 𝑃)/𝑛.

Note 3: Theoretical back ground (2)The dot sampling method originates in the point sampling method.
Dr. Frank Yates describes the point sampling method as follows (SAMPLINGMETHODS for CENSUS AND SURVEYS, London, 1949)
3.9 Sampling with probabilities proportional to size of units“If we have areas demarcated on a map, such as fields, and a point is located at
random on the map, the probabilities of the point falling within the boundaries of thedifferent fields are clearly proportional to the areas of the field. Consequently, areas canbe selected at random with probabilities proportional to their size by the simple
procedure of taking random points on the map.” (P.35)“The principle has applications in agricultural surveys designed to determine
the acreage and yield of different crops, total cultivated area, etc. All that is required for
acreage is to determine the proportion of points which fall in areas of the given type.
The method therefore particularly attractive when carrying out surveys of the area
crops, etc., by aerial survey, by provided the different crops can be recognized on thephotograph, since it avoids all the measurement of area which would be required if an
ordinary random sample of areas were taken.” (P.35)

6. Afterword
Under the title of “Practical Basic Statistics”, we have shown you
some interesting statistical experiments.
The first one is called “Stone cutting survey”. You have found that
random sampling method with weighing is really reliable through theexperiment. Furthermore, you have learned how to calculate statisticalvalues using the data (a sample variance) you got out from the experiment.
16

The second one is on the area measuring experiments. You havelearned various measuring and calculation methods. The measuring methodwith dot-grid plate was changed to a sampling survey as an attribute survey.
It can be a sampling survey method which is used on even Google Earth.
Therefore, the dot-grid method has a possibility to make an actual land use
surveys easy dramatically. You are going to learn the method practically soon.
It will resolve a lot of problems which you face on the carrying to improve
agricultural statistics.
I hope that those experiences on basic statistics for two days would
be useful for you to make preparations of latter lectures and to improve
agricultural statistics in the future.

17

7. Quiz [Basic]: Calculate the area and precision of the following
figures by color on the tables.
(1) Calculation with a low density dot-grid plate.

Number ofsampleCategodotsry𝑛𝑖Percenta
ge
(%)

Area
(km2)

𝑃̂
= (𝑛𝑖 ⁄𝑛) × 100

𝑇̂= (𝑛𝑖 ⁄𝑛)×𝑊

PrecisionComparisonSECV= √𝑃̂ (100 − 𝑃̂ )⁄𝑛CV= (𝑆𝐸 ⁄𝑃̂)
× 100

SE

Complete

Differsurveyence

PD= 𝑃̂ − 𝑃

Green

39.2

Yellow

44.2

Blue

16.6

Total

100.0

Note: W : Each of your country’s area is tentatively assigned to W.

18

(2) Calculation with a high density dot-grid plate

Number ofsampleCategodotsry𝑛𝑖Percenta
ge
(%)

Area
(km2)

𝑃̂
= (𝑛𝑖 ⁄𝑛) × 100

𝑇̂= (𝑛𝑖 ⁄𝑛) × 𝑊

Precision
SE

Comparison
CV

Complete

Differsurveyence

PD= 𝑃̂ − 𝑃CV= (𝑆𝐸 ⁄𝑃̂)= √𝑃̂ (100 − 𝑃̂ )⁄𝑛 × 100
SE

Green

39.2

Yellow

44.2

Blue

16.6

Total

100.0

Note: W : Each of your country’s area is tentatively assigned to W.

19

8. Summary of the Area measuring
1. Topic:
2. Date:
3. Reporter:
4. Purpose

5. Activities

6. Output

7. Impressions and suggestions

20

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