### DATA ANALYSIS AND ERROR ESTIMATION

```DATA ANALYSIS AND ERROR ESTIMATION
Purpose
To learn how to analyze experimental data and to practice error analysis.
Error Estimation
Suppose that a quantity X is measured with an uncertainty in the measurement ∆X. Then the
measurement is recorded as X ± ∆X.
∆X is called the maximum possible error (MPE) or uncertainty inX.
(∆X/X) is called the fractional error (or relative error) in X and
(∆X/X)×100 % is the percent error.
Now, suppose a quantity Z is a function of two independent quantities (variables) X and Y. The
expressions for ∆Z for some common relations between X, Y and Z are given in the following
table:
Relation between X, Y and Z
Relation between errors
Z = X + Y or Z = X − Y
∆Z = ∆X + ∆Y
Z = XY or
Z = X/Y
Z = Constant. XmYn
(∆Ζ/Ζ)=(∆X/X) + (∆Y/Y)
(∆Ζ/Ζ)= m(∆X/X) + n(∆Y/Y)
or Z = Constant. Xm/Yn
Exercise (1)
Two lengths X and Y are measured with a meter ruler as follows:
X = 35.5 ± 0.1 cm and Y = 67.3 ± 0.1 cm. Find the percent error in
a) Z1 = X + Y
b) Z2 = Y – X
c) Z3 = XY
d) Z4 = X/Y
e) Z5 = X2/Y3
Exercise (2)
The data in the following table relates to measurements made of the period of oscillation T of a
simple pendulum of variable length L. Theory predicts the following relationship:
where g is the acceleration due to gravity.

= 2�

1. Calculate the value of g for each data point using the above equation. Observe the rules of
significant figures when doing the calculation.
Table 1
Length (L) cm
± 0.5 cm
Period (T) sec
± 0.01 sec
57.5
50.0
41.5
35.0
29.0
22.5
16.5
1.51
1.42
1.28
1.19
1.06
0.93
0.81
g (cm/s2)
T2 (sec2)
2. Write down the general expression for the fractional error (∆g/g) in terms of (∆T/T) and
(∆L/L).
3. Calculate ∆g for the last data point in Table 1.
4. Record the value of g from the last data point with its error as g1 ± ∆g1
5. Find the average value of g for all data points in Table 1. Call this value gav.
6. Look at the g values in Table 1 and identify the maximum and minimum values of g (gmax
and gmin). Make an estimate for the maximum possible error in g (call it ∆gav) from the
following formula:
∆ =
−
2
7. Record your experimentally determined value of g asgav ± ∆gav
8. Calculate T2 for each data point in Table 1, and plot T2 versus L. Determine g from the slope
of the graph.
9. The accepted value of g in Dhahran area is 980 ± 1 cm/s2. Fill the table below and observe
how the three different values of g agree (or disagree) with the accepted value and with each
other within their uncertainty limits.
Table 2
Accepted value
From Step 4
From Step 7
From Step 8
g
(cm/s2)
∆g
(cm/s2)
% error
(∆g/g)×100
Lower bound
(g−∆g)
(cm/s2)
Upper bound
(g+∆g)
(cm/s2)
980
1
0.1
979
981
```