Variance and Standard Deviation 

are the two important measurements in statistics. Variance is a measure of how data points vary
from the mean, whereas standard deviation is the measure of the distribution of statistical data. The basic difference between both
is standard deviation is represented in the same units as the mean of data, while the variance is represented in squared units. Let
us learn here more about both the measurements with their definitions, formulas along with an example.

Read more:

 Variance

Variance and Standard Deviation are the two important measurements in statistics. Variance is a measure of how data
points vary from the mean, whereas standard deviation is the measure of the distribution of statistical data. The basic
difference between both is standard deviation is represented in the same units as the mean of data, while the variance is
represented in squared units. Let us learn here more about both the measurements with their definitions, formulas along
with an example.

Read more:

 Variance

Variance
According to layman’s words, the variance is a measure of how far a set of data are dispersed out from their mean or average value.
It is denoted as ‘σ2’.

Properties of Variance

 It is always non-negative since each term in the variance sum is squared and therefore the result is either positive or zero.
 Variance always has squared units. For example, the variance of a set of weights estimated in kilograms will be given in
kg squared. Since the population variance is squared, we cannot compare it directly with the mean or the data
themselves.

Standard Deviation
The spread of statistical data is measured by the standard deviation. Distribution measures the deviation of data from its mean or
average position. The degree of dispersion is computed by the method of estimating the deviation of data points. It is denoted by the
symbol, ‘σ’.

Properties of Standard Deviation

 It describes the square root of the mean of the squares of all values in a data set and is also called the root-mean-square
deviation.
 The smallest value of the standard deviation is 0 since it cannot be negative.
 When the data values of a group are similar, then the standard deviation will be very low or close to zero. But when the
data values vary with each other, then the standard variation is high or far from zero.

Variance and Standard Deviation Formula

As discussed, the variance of the data set is the average square distance between the mean value and each data value. And
standard deviation defines the spread of data values around the mean.
The formulas for the variance and the standard deviation for both population and sample data set are given below:
Variance Formula:
The population variance formula is given by:
σ2=1N∑i=1N(Xi−μ)2
Here,
σ2 = Population variance
N = Number of observations in population
Xi = ith observation in the population
μ = Population mean
 
The sample variance formula is given as:
s2=1n−1∑i=1n(xi−x―)2
Here,
s2 = Sample variance
n = Number of observations in sample
xi = ith observation in the sample
x―
= Sample mean
Standard Deviation Formula
The population standard deviation formula is given as:
σ=1N∑i=1N(Xi−μ)2
Here,
σ = Population standard deviation
Similarly, the sample standard deviation formula is:
s=1n−1∑i=1n(xi−x―)2
Here,
s = Sample standard deviation

Variance and Standard deviation Relationship

Variance is equal to the average squared deviations from the mean, while standard deviation is the number’s square root. Also, the
standard deviation is a square root of variance. Both measures exhibit variability in distribution, but their units vary: Standard
deviation is expressed in the same units as the original values, whereas the variance is expressed in squared units.

Example
Question: If a die is rolled, then find the variance and standard deviation of the possibilities.
Solution: When a die is rolled, the possible outcome will be 6. So the sample space, n = 6 and the data set = { 1;2;3;4;5;6}.
To find the variance, first, we need to calculate the mean of the data set.
Mean, x̅ = (1+2+3+4+5+6)/6 = 3.5
We can put the value of data and mean in the formula to get;
σ2 = Σ (xi – x̅)2/n
σ2 = ⅙ (6.25+2.25+0.25+0.25+2.25+6.25)
σ2 = 2.917
Now, the standard deviation,σ = √2.917 = 1.708

Variance and Standard Deviation Formula

As discussed, the variance of the data set is the average square distance between the mean value and each data value.

And standard What is a least-squares adjustment?

ArcGIS Pro 3.0|Other versions| Help archive

A least-squares adjustment uses statistical analysis to estimate the most likely coordinates for connected
points in a measurement in a network. In ArcGIS Pro, a least-squares adjustment can be run on a parcel fabric
using the following tools:

 Analyze Parcels By Least Squares Adjustment tool.

  Apply Parcel Least Squares Adjustment tool.

Learn more about running a least-squares adjustment on the parcel fabric

Least-squares adjustment and best-estimate coordinates

The coordinates of a new point can be uniquely computed by a bearing and a distance from an existing point.
However, relying only on the results of a single set of coordinates is risky, since there is no way to tell whether
the measurements defining those coordinates are correct. Coordinates computed from measurements from
other existing points can be compared with the coordinates computed by the first set of measurements.
Generally, the more measurements defining a single point, the more reliable its coordinates are and the more
confidence there is in detecting mistake measurements. These additional measurements are called redundant
measurements.

All measurements contain some degree of error. Redundant measurements will compute slightly different
coordinates for the same point. Since there can only be one coordinate location for a point, best-estimate
coordinates for the point can be derived by computing a weighted average of the redundant measurements,
with each weight defined by the measurement accuracy. The higher the accuracy of the measurement, the
higher its weight and the more influence it will have in computing the best-estimate coordinates of the point.

The coordinates for point Sp5 are computed using a weighted average.

Although the weighted average approach works for computing a single point, it is not sufficient to compute the
coordinates for multiple points in a network such as the parcel fabric. The techniques and algorithms in a least-
squares adjustment provide the most rigorous and widely accepted solution for computing coordinates in a
network of weighted measurements.

To summarize, a least-squares adjustment works as follows:

 Estimates the statistical best-fit solution for coordinates of points in a weighted measurement network.
 Computes a solution by finding a minimum for the sum of the squares of the measurement residuals. A
measurement residual is the amount needed to correct a measurement for it to fit into the best-fit
solution.
 Is a mathematical procedure based on the theory of probability; estimated coordinates are computed
with varying levels of uncertainty.
 Includes statistical tests to analyze and verify adjustment results.
 Constrained and free network least-squares adjustments
There are many types of least-squares adjustment. The parcel fabric can be adjusted by both constrained and
free network least-squares adjustments.

Constrained adjustment
A constrained least-squares adjustment is run on a measurement network that is constrained by control points.
Control points are points that have known x,y,z coordinates and can be completely constrained (do not move in
the adjustment) or weighted (some movement allowed based on accuracy). Control points can represent
accurate, surveyed coordinates for physical features on the surface of the earth. Control points are added to a
measurement network to place the network in a coordinate system and to detect measurement mistakes known
as blunders.

Learn more about points in a least-squares adjustment

Free network adjustment

A free network adjustment is run on measurements only, and the network is not constrained by control points.
A free network adjustment is run to test the network for measurement errors before connecting the
measurements to control points.

Learn more about constrained and free network adjustments in the parcel fabric

Feedback on this topic?

In this topic

1. Least-squares adjustment and best-estimate coordinates

2. Constrained and free network least-squares adjustments



WHAT IS EDM IN SURVEY?

Electronic Distance Measurement

Electronic distance measurement (EDM) is a method of determining the length between two points using electromagnetic
waves. EDM is commonly carried out with digital instruments called theodolites.

EDM instruments are highly reliable and convenient pieces of surveying equipment and can be used to measure distances
of up to 100 kilometers. Each piece of EDM equipment available at Engineer Supply provides dependably accurate distance
measurements displayed on an easy-to-read digital screen.

Devices known as total stations share similarities with theodolites and can be used to measure distances as well as angles.

deviation defines the spread of data values around the mean.

ow does electronic distance measurement work?
Electronic distance measurement (EDM) is a way of determining the length between two points by looking a phase changes
that occur as electronic energy waves, which travel from one end of a straight line to the other. But when large variations
occur in the terrain or when there's a great deal of obstruction, this method isn’t as effective. So, this method of
measurement is avoided in difficult terrain.

What are the types of EDM instruments?

Here are the three types of instruments that are used for electronic distance measurement, which are based on the methods
being used:
 Microwave Instruments — Also called tellurometers, these instruments use microwaves. And they have been around
since the 1950’s.

 Infrared Wave Instruments — Uses prism reflectors that pick up amplitude modulated infrared waves at the end of a
line.

 Visible Light Wave Instruments — Uses modulated light waves to measure up to a specific range.
If you’re looking for a place to buy a broad range of theodolites, total stations, or other tools used for electronic distance
measurement, be sure to look at what we have at Engineering Supply.

How does a total station measure distance?

Total stations use the principle of a theodolite and combine it with electronic distance measurement. They also come with a
microprocessor and electronic data collector, which is combined with a storage system. Like theodolites, they can be used to
measure the sloping distance of certain objects from the instrument, as well as horizontal and vertical angles. Once the data
has been collected, it can be uploaded to a computer or laptop for further processing.

Does a transit measure distance?

A transit is a telescope that moves around horizontal and vertical axes, and it’s used to measure angles. The readings will
allow you to map a specific site by measuring the position of all of its features, which you can do with a great deal of
precision.

How do I measure dimensions?

Three-dimensional measurement is similar to measuring in two dimensions. But in 3-D, the boundaries of a solid object are
called the surface area and not the perimeter (which is measured in square inches or feet). The volume refers to the space
that it occupies, which is measured in cubic inches or feet. To measure a specific dimension (whether it’s length, width, or
height), simply measure any side of a specific surface. If you need to measure two or three dimensions, you’ll need to
measure the distance of each side that relates to that specific property.
If you’re looking for a place that has a broad selection of total stations, theodolites, or other electronic distance measurement
tools, be sure to look at what we have at Engineering Supply.

ow does electronic distance measurement work?

Electronic distance measurement (EDM) is a way of determining the length between two points by looking a phase changes
that occur as electronic energy waves, which travel from one end of a straight line to the other. But when large variations
occur in the terrain or when there's a great deal of obstruction, this method isn’t as effective. So, this method of
measurement is avoided in difficult terrain.

What are the types of EDM instruments?

Here are the three types of instruments that are used for electronic distance measurement, which are based on the methods
being used:
 Microwave Instruments — Also called tellurometers, these instruments use microwaves. And they have been around
since the 1950’s.

 Infrared Wave Instruments — Uses prism reflectors that pick up amplitude modulated infrared waves at the end of a
line.

 Visible Light Wave Instruments — Uses modulated light waves to measure up to a specific range.
If you’re looking for a place to buy a broad range of theodolites, total stations, or other tools used for electronic distance
measurement, be sure to look at what we have at Engineering Supply.

How does a total station measure distance?

Total stations use the principle of a theodolite and combine it with electronic distance measurement. They also come with a
microprocessor and electronic data collector, which is combined with a storage system. Like theodolites, they can be used to
measure the sloping distance of certain objects from the instrument, as well as horizontal and vertical angles. Once the data
has been collected, it can be uploaded to a computer or laptop for further processing.

Does a transit measure distance?

A transit is a telescope that moves around horizontal and vertical axes, and it’s used to measure angles. The readings will
allow you to map a specific site by measuring the position of all of its features, which you can do with a great deal of
precision.

How do I measure dimensions?

Three-dimensional measurement is similar to measuring in two dimensions. But in 3-D, the boundaries of a solid object are
called the surface area and not the perimeter (which is measured in square inches or feet). The volume refers to the space
that it occupies, which is measured in cubic inches or feet. To measure a specific dimension (whether it’s length, width, or
height), simply measure any side of a specific surface. If you need to measure two or three dimensions, you’ll need to
measure the distance of each side that relates to that specific property.

If you’re looking for a place that has a broad selection of total stations, theodolites, or other electronic distance measurement
tools, be sure to look at what we have at Engineering Supply.

ow does electronic distance measurement work?

Electronic distance measurement (EDM) is a way of determining the length between two points by looking a phase changes
that occur as electronic energy waves, which travel from one end of a straight line to the other. But when large variations
occur in the terrain or when there's a great deal of obstruction, this method isn’t as effective. So, this method of
measurement is avoided in difficult terrain.

What are the types of EDM instruments?

Here are the three types of instruments that are used for electronic distance measurement, which are based on the methods
being used:
 Microwave Instruments — Also called tellurometers, these instruments use microwaves. And they have been around
since the 1950’s.

 Infrared Wave Instruments — Uses prism reflectors that pick up amplitude modulated infrared waves at the end of a
line.

 Visible Light Wave Instruments — Uses modulated light waves to measure up to a specific range.
If you’re looking for a place to buy a broad range of theodolites, total stations, or other tools used for electronic distance
measurement, be sure to look at what we have at Engineering Supply.

How does a total station measure distance?

Total stations use the principle of a theodolite and combine it with electronic distance measurement. They also come with a
microprocessor and electronic data collector, which is combined with a storage system. Like theodolites, they can be used to
measure the sloping distance of certain objects from the instrument, as well as horizontal and vertical angles. Once the data
has been collected, it can be uploaded to a computer or laptop for further processing.

Does a transit measure distance?

A transit is a telescope that moves around horizontal and vertical axes, and it’s used to measure angles. The readings will
allow you to map a specific site by measuring the position of all of its features, which you can do with a great deal of
precision.

How do I measure dimensions?

Three-dimensional measurement is similar to measuring in two dimensions. But in 3-D, the boundaries of a solid object are
called the surface area and not the perimeter (which is measured in square inches or feet). The volume refers to the space
that it occupies, which is measured in cubic inches or feet. To measure a specific dimension (whether it’s length, width, or
height), simply measure any side of a specific surface. If you need to measure two or three dimensions, you’ll need to
measure the distance of each side that relates to that specific property.

If you’re looking for a place that has a broad selection of total stations, theodolites, or other electronic distance measurement
tools, be sure to look at what we have at Engineering Supply.

ow does electronic distance measurement work?

Electronic distance measurement (EDM) is a way of determining the length between two points by looking a phase changes
that occur as electronic energy waves, which travel from one end of a straight line to the other. But when large variations
occur in the terrain or when there's a great deal of obstruction, this method isn’t as effective. So, this method of
measurement is avoided in difficult terrain.

What are the types of EDM instruments?

Here are the three types of instruments that are used for electronic distance measurement, which are based on the methods
being used:
 Microwave Instruments — Also called tellurometers, these instruments use microwaves. And they have been around
since the 1950’s.

 Infrared Wave Instruments — Uses prism reflectors that pick up amplitude modulated infrared waves at the end of a
line.

 Visible Light Wave Instruments — Uses modulated light waves to measure up to a specific range.
If you’re looking for a place to buy a broad range of theodolites, total stations, or other tools used for electronic distance
measurement, be sure to look at what we have at Engineering Supply.

How does a total station measure distance?

Total stations use the principle of a theodolite and combine it with electronic distance measurement. They also come with a
microprocessor and electronic data collector, which is combined with a storage system. Like theodolites, they can be used to
measure the sloping distance of certain objects from the instrument, as well as horizontal and vertical angles. Once the data
has been collected, it can be uploaded to a computer or laptop for further processing.
Does a transit measure distance?
A transit is a telescope that moves around horizontal and vertical axes, and it’s used to measure angles. The readings will
allow you to map a specific site by measuring the position of all of its features, which you can do with a great deal of
precision.

How do I measure dimensions?

Three-dimensional measurement is similar to measuring in two dimensions. But in 3-D, the boundaries of a solid object are
called the surface area and not the perimeter (which is measured in square inches or feet). The volume refers to the space
that it occupies, which is measured in cubic inches or feet. To measure a specific dimension (whether it’s length, width, or
height), simply measure any side of a specific surface. If you need to measure two or three dimensions, you’ll need to
measure the distance of each side that relates to that specific property.

If you’re looking for a place that has a broad selection of total stations, theodolites, or other electronic distance measurement
tools, be sure to look at what we have at Engineering Supply.ppppppp

Trigonometric leveling is so named because it uses a total station instrument's (TSI) slope distance and zenith angle
measurements to mathematically compute an elevation difference which, with a few more bits of information, can be used to
determine a point's elevation. Using appropriate procedures, and controlling errors, elevation accuraciy can be better than 0.1 ft.
Because trigonometric leveling is not limited to a horizontal line of sight, it is more flexible and provides faster elevation data
collection than differential leveling.

a. Elevation determination
TSI slope reduction is discussed in the Electronic Distance Measurement topic. Slope reduction geometry is shown in Figure F-
1 and accompanying Equations F-1 and F-2.
 
Trigonometric leveling is so named because it uses a total station instrument's (TSI) slope distance and zenith angle
measurements to mathematically compute an elevation difference which, with a few more bits of information, can be used
to determine a point's elevation. Using appropriate procedures, and controlling errors, elevation accuraciy can be better
than 0.1 ft. Because trigonometric leveling is not limited to a horizontal line of sight, it is more flexible and provides faster
elevation data collection than differential leveling.

a. Elevation determination
TSI slope reduction is discussed in the Electronic Distance Measurement topic. Slope reduction geometry is shown in
Figure F-1 and accompanying Equations F-1 and F-2.
 
 Trigonometric Relationships
 

  Equation F-1

  Equation F-2

H: Horizontal distance V: Vertical distance

  Z: Zenith angle R: Earth radius
S: slope distance k: Refraction constant
The terms in brackets in both equations account for earth curvature and atmospheric refraction. These are systematic errors which
are compensated mathematically. The refraction constant is generally taken as a percentage of earth curvature. At low to medium
altitudes k = 0.14 (14%), at higher altitudes k=0.07 due to a thinner atmosphere. Earth radius, R, is 20.906x10 6 ft
For short distances, earth curvature and refraction drop out and the equations become simplified. Most TSI have the option to
turn on the corrections.
Vertical distance, V, is the elevation difference between the TSI and the reflector. Adding V to the TSI elevation gives us the
reflector elevation. What we want, however is the elevation of the ground point at the reflector location. To determine that, we
need two additional pieces of information: (1) TSI elevation, and, (2) height of the reflector above the ground point. Once we
have those, then the elevation of any observed point, i, is computed from:

Elevi = ElevTSI + Vi - HRi   Equation F-3

ElevTSI: Elevation of the TSI
HRi: Height of reflector at point i

b. Sideshots
Besides the way the elevations are determined, another major difference between trigonometric
and differential leveling is point connectivity. In differential leveling we normally have one BS
and one FS at each set up - we survey in then out of each elevation point. Trigonometric leveling
is used when a number of elevations are measured from a single instrument set up. All those
points are surveyed into, but not out of. Each elevation point determined by trigonometric
leveling is an open link, also known as a sideshot.
In a closed differential level network, Figure F-2(a), each point has a BS and FS; each is
connected to another point and their elevations are based on the BM.
Figure F-2(b) depicts a trigonometric network referenced to the differential network. Points B, C,
and D serve as control for trigonometric leveling. The green shots at points B, C, and D are all
sideshots. Because sideshots are not connected to other points their elevations cannot be
checked. An elevation error could be in an individual sideshot measurement or a control point
elevation error which affects all the sideshots from that point

igure F-2(b) depicts a trigonometric network referenced to the differential network. Points B, C, and D serve as control for
trigonometric leveling. The green shots at points B, C, and D are all sideshots. Because sideshots are not connected to other
points their elevations cannot be checked. An elevation error could be in an individual sideshot measurement or a control point
elevation error which affects all the sideshots from that point

(a) Control Network   (b) Sideshots

Figure F-2

2. Determining TSI elevation

a. TSI set up over known elevation
In this scenario the TSI is set up over a point whose elevation is known. The elevation must be transferred to the TSI by
measuring the Height of the Instrument, HI, Figure F-3.
 Instrument Height
Figure F-3

 
The TSI's elevation is calculated from:
ElevTSI = Elev + HI   Equation F-4

Elev: Point reference elevation

 
The HI is the vertical distance from the reference elevation to the TSI's Horizontal Axis (HA). Because the TSI is over the point,
it is not possible to directly measure the HI. Most TSIs have an HA index mark on one or both standards which can be used to get
a close value for the HI. There are two simple ways to measure the HI:

(1) Using a tape or pocket-rod

A cloth tape or pocket-rod can be used to measure the distance from the elevation reference to the HA index mark, Figure F-4.
 Figure F-4
Measuring HI with Tape

(2) Using a prism pole

Another method is to place the prism pole next to the instrument and raise or lower the reflector so its center lines up with the HA
index mark, Figure F-5. The reflector pole reading is recorded as the HI.

Figure F-5
Measuring HI with Prism Pole

 
Each method is subject to errors which will be discussed in a following section.
 

b. Sight to known elevation

The TSI is set up at a location from which a reference elevation or benchmark (BM) is visible. The vertical distance is measured
to a reflector held on the BM, Figure F-6.

Figure F-6
 Using Benchmark to Establish TSI Elevation

 
The elevation of the TSI is determined from:

ElevTSI = ElevBM + HRBM-VBM   Equ

2. Determining TSI elevation

a. TSI set up over known elevation
In this scenario the TSI is set up over a point whose elevation is known. The elevation must be transferred to the TSI by
measuring the Height of the Instrument, HI, Figure F-3.

Instrument Height
Figure F-3

 
The TSI's elevation is calculated from:
ElevTSI = Elev + HI   Equation F-4

Elev: Point reference elevation

 
The HI is the vertical distance from the reference elevation to the TSI's Horizontal Axis (HA). Because the TSI is over the
point, it is not possible to directly measure the HI. Most TSIs have an HA index mark on one or both standards which can
be used to get a close value for the HI. There are two simple ways to measure the HI:

(1) Using a tape or pocket-rod

A cloth tape or pocket-rod can be used to measure the distance from the elevation reference to the HA index mark, Figure
F-4.
 Figure F-4
Measuring HI with Tape

(2) Using a prism pole

Another method is to place the prism pole next to the instrument and raise or lower the reflector so its center lines up with
the HA index mark, Figure F-5. The reflector pole reading is recorded as the HI.

Figure F-5
Measuring HI with Prism Pole

 
Each method is subject to errors which will be discussed in a following section.
 
b. Sight to known elevation
The TSI is set up at a location from which a reference elevation or benchmark (BM) is visible. The vertical distance is
measured to a reflector held on the BM, Figure F-6.

Figure F-6
Using Benchmark to Establish TSI Elevation

 
The elevation of the TSI is determined from:
ElevTSI = ElevBM + HRBM-VBM   Equ