PHYSICS

2.2. THEORY OF ERRORS
2.2.1. Types of Errors
If the measurements are physically derived using measuring
tools, they are referred to as direct; if derived using a formula,
they are indirect. For example, determining the length of an
object with a metric ruler is a direct measurement, while determining
the moment of inertia using the formula I = mr', is an indirect
measurement.
The measured value for a quantity always differs from the true
value for the quantity. The reasons for the difference are instrumental
measurement errors (due to the imperfection of the measuring
instrument) and personal errors (due to measuring errors by
the individual). Another method for classifying errors is based on
their properties. An error is systematic if it is constant over several
measurements or random if it changes during the measurements.
The following definitions more precisely describe these
terms.
Error - the difference between the determined value of a physical
quantity and the true value.
The foregoing classification of measurement errors is based on
the cause of the errors.
Systematic errors are caused by the imperfection of measuring
methods and inaccuracy of instruments. These errors remain constant
or change in a regular fashion in repeated measurements of
one and the same quantity.
Random errors mean the individual errors that are given rise to
the person performing the measurements; they include errors
owing to incorrect reading of the tenth graduation of an instrument
scale, small changes of the measurement conditions, asymmetric
placement of the indicator mark etc. These errors are changing
in an irregular fashion in repeated measurements of one and
the same quantity.
2.2.2. Errors in Direct Measurements
Let x I' X 2'
X 3'
x n
denote the raw data derived from experimental
observations. The arithmetic mean «x» is the sum of observations
divided by the number (n) of observations:
n
LXi
Y. + Y- + Y- +...+ X . I
< X > = 0'1 OV2 oVJ n =-'-=-- (2.1)
n
n
The difference between a data point and the mean is referred to
as a deviation (8):
8. = x. - < x > (2.2)
I
I
Dispersion or variance (0'2) is defined as the sum of the
squared deviations divided by n - I:
n
2
LD j
a 2 = ..i..::L-­ (2.3)
n-l
Sample standard deviation (0') is defined as the square root of
the dispersion by the following formula:
0'=
n
LS;
i;1 (2.4)
n-l
Confidence interval is the difference between the largest and
smallest observations in a sample.
Confidence interval of systematic error (6) is the smallest division
of the measuring instrument.
Confidence interval of random error (~) is defined by the following
formula:
o
.!1 = t
n
:L S;
i;1 (2.5)
n(n-l)
10
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