1. Department of Physics 1st year Laboratory
Introduction to Errors
Dr S P Tear

2. Aims To introduce: The concept of random and systematic errors
The distinction between precise and accurate measurements
Measurements of error: standard deviation and standard error
Combining errors
Introduction to Errors

3. Example
Consider the results of your measurement of the resistance of a coil at different temperatures:
Ω at 10°C
Ω at 20°C
Is the difference between these two values significant?
We can’t say, unless we know the error on the measurements
Yes, if the error in each measurement is Ω
No, if the error in each measurement is 0.01 Ω
Introduction to Errors

4. Systematic and random errors
Systematic errors:
Generally constant throughout a set of readings
Constant offset, constant factor, or both
Caused by factors which are not accounted for in the measurements or theory
Examples:
Offset error: voltmeter not zeroed properly
Gain error: clock running fast or slow
Introduction to Errors

5. Random errors
Vary arbitrarily (randomly) from one measurement to the next.
Equally likely to be positive or negative
Always present in an experiment
True value
Random only
Random &
systematic
Introduction to Errors

6. Precision and accuracy
A result is precise if the random error is small
And accurate if it is free from systematic and random errors
Thus, it is possible to have a inaccurate but precise result
Introduction to Errors

7. Reducing systematic errors
Careful experimental design (reduce in expt or taken account of)
Calibrate measuring instruments (most digital instruments in the lab. will have negligible systematic errors)
Maybe the theory is not quite right!
No easy way to detect systematic errors
correlate with other experiments?
discrepancy with theory?
Introduction to Errors

8. Reducing random errors
Repeating the measurements and take the mean (detects random errors too)
The more readings taken, the closer the mean will approach the true mean (in limit of  readings)
Suppose you have a set of n successive measurements of the same quantity: x1, x2, …xn, then the best estimate of the measured value is the arithmetic mean:
Introduction to Errors

9. Error in the mean? What is the error in the mean?
If we only have a finite number of measurements, can we determine how close the mean of our sample of n measurements is to the true mean?
Standard
Deviation
Standard
Error
Introduction to Errors

10. Standard deviation & standard error
Measures the spread in the random errors in the measurement
Tells us the error in a single measurement
Doesn’t get smaller the more readings taken
Standard Error
Error in the mean of n measurements
Gets smaller the more readings you take, because the mean becomes more precise as more measurements are included in it.
Introduction to Errors

11. Statistically speaking
Standard deviation: there is a ~67% chance that the next single measurement I make will lie within 1 s.d. of the true mean (~95% within 2 s.d.)
Standard error: there is a ~67% chance that the measured mean of the readings lies within 1 s.e. of the true mean (~95% within 2 s.e.)
Note: increasing the number of readings reduces the standard error but only at the rate of 1/n, so time and resources often dictate the optimum number of readings to take.
Introduction to Errors

12. Combining errors: sum & difference
Refer to section 4 of the Laboratory Handbook
Combining errors - in quadrature
Sum and Difference: Z = A + B or Z = A - B
Add the squares of the ‘absolute errors’:
The absolute error in Z, Z is:
where A and B are the errors in A and B.
Introduction to Errors

13. Combining errors: product & quotient
Combining errors - in quadrature
Products and Quotients: Z = AB or Z = A/B
We add the squares of the ‘fractional errors’:
The fractional error in Z, is:
where A/A and B/B are the fractional errors in A and B
Introduction to Errors

14. Today’s experiments Refer to section 4 of the Laboratory Handbook
Circus of 3 activities, 55 minutes each
Measurement of time
Measurement of length & angle
Measurement of voltage and current
20 minute break at 3:10
Notebooks marked at end of each expt. RHS!
Introduction to Errors