Estimating and Reporting Measurement Errors

Question 1

Improvement of measurment accuracy and repeating of measurements ________ uncertainty.

Answer 1

reduces

Question 2

True or false:

Two values V₁±ΔV₁ and V₂±ΔV₂ which overlap in their error range ±ΔV_i are two distinguishable results.

Answer 2

False

Question 3

A rule of thumb for estimating uncertainty of digital and analog equipment is:

Answer 3

• ± half the smallest division (in analog devices)
• ± the last significant digit unless otherwise quoted (in digital devices)

Question 4

What are random errors? How can they be controlled?

Answer 4

Random Errors: measurement errors that fluctuate due to the unpredictability or uncertainty inherent in the measuring process.

Can be minimalized through control of experiment conditions.

e.g. The rate of liquid vaporization depends on the humidity and pressure of air, which can fluctuate during measurement. Controlling ambient conditions helps control random uncertainties caused by these factors.

Question 5

What are systematic errors?

Answer 5

A systematic error is an additive source of error that results from a persistent issue.

Leads to a consistent error in measurements.

e.g. a source of systematic error can be uncalibrated measuring equipment or instrumental errors

Question 6

Errors introduced by incorrect model assumptions?

Answer 6

A model which oversimplifies a complex system or ignores key variables can introduce errors to measurements based on this model’s assumptions.

e.g. Varying diameter values (“uncertainty in diameter value”) measured for a spherical looking object under the assumption of a perfect sphere.

Question 7

To find the number of significant digits, display number in ________ with one ________ before decimal point.

Answer 7

scientific notation
nonzero digit

e.g. 0.0025 kg will be converted to 2.5x10^-3kg

Question 8

Trailing zeros after decimal point are ________.

Significant figures\insignificant figures

Answer 8

significant figures

e.g. 22.0 has 3 significant figures

Question 9

Leading zeros before decimal point are ________.

Significant figures\insignificant figures

Answer 9

insignificant figures

e.g. 0.0025 has 2 significant figures (not 5)

Question 10

Countable numbers (e.g. 12 eggs) are always ________ values.

Answer 10

Precise numbers

Question 11

Final or trailing zeroes are significant only after ________.

Answer 11

a decimal point

e.g. 400.00 has 5 significant figures whereas 400 has only 1.

Question 12

Numbers are rounded ________ when the digit to the right is greater than 5 and ________ when it is smaller than 5.

Answer 12

up
down

e.g. 2.36 is rounded to 2.4 and 2.34 is rounded to 2.3

Question 13

When the digit to the right is 5, number is rounded ________.

Answer 13

to the nearest even number

e.g. 2.35 will be rounded to 2.4 and 2.65 will be rounded to 2.6

Rounding should be done only in the final phase of the calculations.

Question 14

Number of digits after decimal point in addition\substraction result?

Answer 14

1. Add up the number of significant figures to the right of the decimal part of each number used in the calculation.
2. Perform the calculation (addition or subtraction) as usual.
3. The answer must not contain more significant figures to the right of the decimal point than the fewest of any of the figures worked out in part 1.

e.g. 0.10 + 0.024 = 0.12

Question 15

Number of significant figures in multiplication\division result?

Answer 15

Same number of significant figures as in the smallest total of significant figures in the initial numbers.

e.g 3.10⋅ 3.50 = 10.85≈10.9, because the smallest total of significant figures in the initial numbers is 3.

Question 16

Number of significant figures in results obtained by operations between measured values and exact numbers?

Answer 16

Determined by number of significant figures in measurement result.

Question 17

Number of significant figures in results obtained by operations between measured values and conversion constants?

Answer 17

Determined by number of significant figures in measurement result.

e.g. 100.000 °C⋅273.15 [K/(°C)]=273.150K
* Measured value (100.000 °C) has 6 significant figures, as in the result.
* Conversion constants are considered exact values.

Question 18

Number of significant figures in the mantissa of log(x)?

Mantissa - the fractional part of the common (base-10) logarithm

Answer 18

Same as number of significant figures in x.

Question 19

Number of significant figures in 10^y?

y=log x (10^y is the antilogarithm of y)

Answer 19

Same as the number of significant figures in the mantissa of y

e.g. 10^12.389=2.45×10¹²

Question 20

Explain why the following measurement result is reported incorrectly:
9.82±0.02385 mL

Answer 20

Too many digits after decimal point in measurement error.

Correct form is:
9.82±0.02 mL

Question 21

Explain why the following measurement result is reported incorrectly:
6051.78±20 g

Answer 21

Too many digits in measured value.

Correct form is:
6050±20 g

Question 22

Report the following result as Value±Error:

Measured value: 92.81
Error: 0.3

Answer 22

92.8±0.3

Question 23

Report the following result as Value±Error:

Measured value: 93
Error: 3

Answer 23

93±3

Question 24

What is a representative value for a set of measurements?

Give 2 examples of common representative values.

Answer 24

A single value representing a set of measurements of the same variable (e.g. weight of the same object).

A represaentative value can be the average or the median of all measured values.

Question 25

What is the mean (average) of N measured values (x₁…x_N)

Answer 25

Question 26

What is the median for a set of values (x₁…x_N )?

Answer 26

The median is the middle number in a group of numbers when they are put in order from smallest to biggest.

Question 27

What is the median for a set of values (x₁…x_N ) when N is an **even** number?

Answer 27

The average between the two numbers in the middle. ## Footnote e.g. for the set (0.1006, 0.0990, 0.0997, 0.1010) the median will be (0.0990+0.0997)/2

Question 28

What does the standard deviation represent?

Answer 28

The standard deviation is a measure of the amount of variation of the values of a variable about its mean. ## Footnote (A measure for the **precision** of the measurement)

Question 29

Given a set of N measurements (x₁…x_N ) of the variable x and its mean value, what is its the standard deviation s?

Answer 29

## Footnote (x_i-x ̅) is the deviation of meaurement i from the average value (the representataive value)

Question 30

What is precision? What is the impact of random errors on precision?

Answer 30

* Precision measures the degree to which repeated (or reproducible) measurements under unchanged conditions show the **same results**. * Random errors reduce precision. ## Footnote The repeating results do not necessarily match the theoretical value, unless results are also **accurate**.

Question 31

Random errors can be evaluated and handled using \_\_\_\_\_\_\_\_.

Answer 31

Statistical tools ## Footnote e.g. finding a representative value using a median or mean value or evaluating precision using standard deviation.

Question 32

The presence of \_\_\_\_\_\_\_\_ errors can account for reduced accuracy.

Answer 32

Systematic errors

Question 33

**Accuracy** is how close a given set of measurements are to \_\_\_\_\_\_\_\_.

Answer 33

their true value

Question 34

Can you eliminate the impact of systematic errors on measurement results?

Answer 34

Yes, but only if the nature of this impact is well characterized. ## Footnote e.g. If the source of the error is an uncalibrated scale, one must substract the weight displayed on the empty scale from the measurement results.

Question 35

The measurement results in the table below were obtained for the same variable. 1. What can be a possible cause of the error responsible for the difference in measured values (the value before ±)? 2. How can this error be reduced?

Answer 35

1. Difference in ruler material. Expansion/contraction as a result of temperature differences. 2. By modeling and taking into account effects of thermal expansion or similar effects caused by variations in air-humidity. ## Footnote Notes: * the uncertainties 0.05 and 0.08 are probably temperature independent. * Effects of humidity and temperature that are taken into account will also change (increase) the error (value after ±), because now it is the accumulated error.

Question 36

If the uncertainty of a single measurement x is δx (e.g uncertainty value noted on measurement equipment), the uncertainty for the **average value** x ̅ of N consecutive measurement of the same variable will be:

Answer 36

## Footnote For this reason, precision is improved with N.

Question 37

2 ways to decrease random errors? ## Footnote And improve precision

Answer 37

1. Take a large number of measurements (large N). 2. Control experimental environment so as to decrease potential sources for random errors.

Question 38

One possible way to reduce systematic errors? ## Footnote (and improve accuracy)

Answer 38

Calibrate equipment

Question 39

Formula for relative standard deviation (RSD)?

Answer 39

## Footnote Relative standard deviation - standard deviation relative to measurement average.

Question 40

Formula for coefficient of variation (CV)?

Answer 40

CV=100×(s/x ̅ )=100×S_r ## Footnote RSD in %

Question 41

When calculating x ̅, RSD or systematic error, the number of significant figures after a decimal point is determined according to \_\_\_\_\_\_\_\_.

Answer 41

The number of significant figures after the decimal point in the original measured values (x₁…x_N)

Question 42

Given an average of measurements x ̅ and the theoretical value V of variable x, what is the systematic error?

Answer 42

Systematic error = (average value) - (theoretical value)

Question 43

An outlier is \_\_\_\_\_\_\_\_.

Answer 43

a data point that differs significantly from other measured values.

Question 44

\_\_\_\_\_\_\_\_, is used for identification and rejection of outliers.

Answer 44

Dixon's Q test

Question 45

Dixon's Q-test determines if a value is an outlier within a \_\_\_\_\_\_\_\_, assuming data is \_\_\_\_\_\_\_\_.

Answer 45

within a given confidence (90%, 95%...) normally distributed

Question 46

Steps in applying a Q test for potentially bad data?

Answer 46

1. Arrange the data in order of increasing values. 2. calculate Q as: Q=|Δ₁|/Δ₂ ## Footnote |Δ₁| - absolute difference between the outlier in question and the closest number to it (gap). Δ₂ - difference between largest value and smallest value in data set, **including outlier** (range).

Question 47

# Dixon's Q-test: If the Q value for a certain set of data is greater than \_\_\_\_\_\_\_\_ of Q for the desired confidence. The value suspected as an outlier is \_\_\_\_\_\_\_\_.

Answer 47

the critical value rejected

Question 48

# True or false: Dixon's Q-test can be used only once for a given set of data.

Answer 48

True

Question 49

Q=0.722 was obtained for a data point suspected as an outliner in a set of 5 measurements. Can we reject it with a confidence of 90%? 95%? 99%

Answer 49

90%, 95% - yes 99% - no (critical value is 0.821)

Question 50

Rejection of outliners must be fully recorded \_\_\_\_\_\_\_\_.

Answer 50

In your lab report ## Footnote alongside a rationalization of the rejection.

Question 51

Calculations made from measurement results have \_\_\_\_\_\_\_\_ accumulated from \_\_\_\_\_\_\_\_.

Answer 51

Errors Individual measurement results

Question 52

Given a function q(x…z) of measurement results x...z, what is the accumulated error δq of q?

Answer 52

## Footnote * Equation holds true only for sufficiently small errors δx, δy, δz. * General formulae for δq of simple functions such as addition, multiplication and raising to power are given in table 4 pg 30 of intruducion pdf.

Question 53

If the accumulated error Sy of calculation result y has an uncertainity in the order of 10^-2, the uncertainity in calculated result must also be \_\_\_\_\_\_\_\_.

Answer 53

in the order of 10^-2

Question 54

Given a calculation performed on measurement of r: V=(4/3)πr³=(4/3)π(1.2578/2)³=1.041917646 cm³ with an error: Sv=0.0005 cm³ how should one display the calculation result with it's error?

Answer 54

V=1.**0419**±0.0005 cm³ ## Footnote Number of digits after the decimal point is determined by the order of the error. Error is in the order of 10^-4 so calculation result is accurate up to the fourth digit after the decimal point.