Measurement and Data Processing

Qualitative data 

Includes all non-numerical information obtained from observations not from measurement. 

Example Qualitative

color change, bubbling,... 

Quantitative data 

Numerical information obtained from measurements 

Quantitative data  is affected by

the type of instrument used and if it was correctly used or not, human limitations,... 

 

Precision is assessed by

by repeating measurements under the same conditions. 

Precise values are

close to each other. 

Accuracy is how

close a measured or processed value is to the true value.

Validity

Is concerned with whether something measures what it is meant to measure 

Reliability 

Is connected to whether doing the same investigation again produces the same results. 

repeatability: same experimenter + repeated trials + same method and conditions --->

Agreement of results 

When conducting several trials, mean is

calculated, outliers are identified and cleared. 

 

Reproducibility 

Different experimenter + repeated trials + same method and conditions ---> Agreement of results 

 

Explain why measurements taken with a pH probe are more precise than pH values determined using pH paper.  

• PH probe give qualitative data, pH paper indicates color, a qualitative data. 

• PH probe can be calibrated, unlike pH paper.  

• PH probe give more precise values (ex. 7.45), unlike pH paper that give a range of possible values (7-8) 

• pH probes provide consistent measurements when used correctly and properly maintained. pH paper can vary in results due to differences in paper batches, storage conditions, and subjective interpretation of the color change. 

 

Can experimental outcomes be reliable but not valid?  

Yes. Reliability is connected to whether doing the same investigation again produces the same results whereas validity is concerned with whether something measures what it is meant to measure.  

Uncertainties (random errors) 

Measured and processed values are always inexact.  

Repeating a measurement under the same conditions and

with the same instrument produces randomly varying values. 

Factors influencing this random variation, known as random error, include

instrument imprecision, fluctuations during readings, and human reaction time.  

Random error

described by the uncertainty, and the probability of the reading being too high or too low from a measurement to the other.  

Impact of Random Error

Reduce Precision

Measurements should be recorded along with

their associated uncertainty and unit. 

Instruments with greater precision -->

measurements that have lower uncertainties. 

1- Based on the display or

scale on the instrument you are using: 

Digital instrument:  

The uncertainty of this instrument will be its least count, that is, the lowest value above zero that the instrument can register (usually the reading of “1” in the lowest decimal place on the display) 

Analog instrument:  

Analogue instruments have scales marked on them.  

Analog: The uncertainty is estimated as

half the smallest scale division

2- uncertainty stated by the

 

manufacturer of the measuring instrument 

3-estimating the range

over which a value fluctuates. 

Sometimes the measured value fluctuates over time. In these cases, you can try to

estimate the uncertainty based on the variation in values you observe.

Another way is by finding the furthest reading from the mean value: the uncertainty will be

the higher between (highest value-mean) and (mean-lowest value). 

Expressing uncertainties 

Absolute uncertainty, Percentage uncertainty, Relative or fractional uncertainty 

Absolute uncertainty 

Has the same unit as the value that it is associated with. 

Percentage uncertainty 

Equal to the absolute uncertainty expressed of a % of the vale it is associated with. 

%uncertainty =

(absolute uncertainty of the value/the value)x100 

It's a percentage so %! 

Relative or fractional uncertainty 

Ratio comparing the absolute uncertainty to the associated value. 

Relative uncertainty =

(absolute uncertainty of the value/the value) 

Often uncertainties are

stated to 1 significant figure.  

The value and the uncertainty associated to it should have the

same number of decimal places.  

Absolute uncertainty is

ALWAYS reported to 1 significant figure.  

Percentage uncertainty can be

reported to a max of 2 sig fig in case it less than 2%. 

If % uncertainty is 2% or more,

percentage uncertainty is reported to 1 sig fig. 

All measurements are

approximate and have significant figures. The last digit given for any measurement is the uncertain digit. 

 

Significant figures

are the valid digit in a measurement. 

The number of significant figures depends on

the precision of the measuring device. 

Rules of Significant Figures 

1.Nonzero digits are significant. 

2. Final zeros after a decimal point are significant. 

3.Zeros between two significant figures are significant. 

4.Zeros used only as placeholders are not significant. 

 

Nonzero digits are

significant. 

2. Final zeros after a decimal point are 

significant

.Zeros between

two significant figures are significant. 

Zeros used only as

placeholders are not significant. 

.Counting numbers have an

infinite number of significant figures. 

Conversion factors

have an infinite number of significant figures. 

When adding or subtracting, the answer should have the same number of 

decimal places as the limiting term. The limiting term is the number with the least 

decimal places.

When multiplying or dividing, the answer

should have the same number of significant figures as the limiting term. The limiting term is the number with the least number significant figures 

Keep the rounding to the desired number of significant figures or decimal places to

the final step, and don't rush to round the intermediate values because rounding too early could lead to incorrect answers! 

Propagating uncertainties: The overalll uncertainty of a calculated result can be

estimated by propagating the uncertainty introduced by each of the measurements. 

The way this is done depends on the type of calculation: 

Addition or subtraction Uncertainties are

propagated by adding the absolute uncertainties of the processed values. 

Multiplication and division: Uncertainties are  

propagated by adding the percentage uncertainties of the processed values. 

Exponents  The percentage uncertainty of the raw data is

multiplied by the value of the exponent. 

 

Systematic Error

Errors affecting the results in the same direction (too high or too low by a fixed amount)

Impact of Systematic Error

Reduce Accuracy but the impact is low if we are using the difference between values

Random

Also known as uncertainties, lead to the same probability of reading being too high or too low.

Estimates of random errors, obtained by uncertainty propagation, are

useful in the analysis of errors in experimental results. 

Comparisons between percentage uncertainties, and, where available, theoretical values and percentage errors can

give a basis for evaluating the precision and accuracy of a processed result. 

Percentage error (not to be confused with percentage uncertainty) is a way of quantifying how far an experimental value is from the theoretical value: 

equation

Note that the theoretical value is  

sometimes called the accepted value or literature value. 

Interpretation of features of graphs 

Correlation  and Gradient

Positive Correlation

if IV increases, DV increases

Negative Correlation

IV increases and DV decreases

No Correlation

points randomly scattered in the graph

Linear and nonlinear functions 

= Linear relationship: y=mx + c, with m : slope ; c : y-intercept. 

Direct Proportionality

Linear function, when a variable increases, the other increases by the same rate. The straight line goes by the origin.  

Inverse proportionality:  

when a variable increases, the other decreases by the same rate. Nonlinear graphs (exponential, logarithmic, quadratic,…) 

Gradient (Slope) 

• Measure of a line's steepness 

• Rate of change of one variable with respect to the other. 

• Calculated by choosing 2 points on the line that are ideally well separated from each other.  

• Can be positive or negative 

• Straight lines have constant slopes, curves have changing ones.  

•• To determine the slope in one point of a curve, we draw a tangent line to the curve in this point and we calculate its slope.