Error Analysis

1. Error Analysis

This page contains information that will prove useful in writing the “Error Analysis” section of your report.

  1. Errors in Measurement: A measurement or experimental result is of little value if nothing is known about the probable size of its error. There are many types of errors, which affect measured quantities, and there are many ways to classify them. The most general classification is determinate (systematic) errors and indeterminate (random) errors.

1)      Indeterminate (random) errors are random fluctuations and cannot be corrected for. It is this type of error that our intuition suggests that repeated measurements allow us to “average” out.

2)      Determinate (systematic) errors are frequently constant and can normally be corrected for if the systematic effect is identified.

3)      Illegitimate errors: this type of error accounts for inexplicable “events”, which results in a measurement whose value deviates significantly from what it is “expected. Eg. An error reading a number.

 

A measurement with relatively small indeterminate error is said to have high precision.

A measurement with relatively small determinate error is said to have high accuracy.

A measurement, which has both high precision and high accuracy, is sometimes called highly reliable.

 

B.     Error Analysis: This section describes equations that you will use in most experiments to determine the reliability of your data.

    1. Percent Difference (%D): This is used to compare 2 experimental measurements.

; Where E1 corresponds to experimental value 1 and E2 experimental value 2.

    1. Percent Discrepancy (% Disc.): This is used to compare an experimental measurement versus a theoretical/table value.

 

; Where T corresponds to the theoretical value and E corresponds to the experimental value.

 

2. Graphical Representation of Experimental Data

From examination of tabulated values of a number of measurements of related quantities, t is often difficult to grasp the relationship existing between the numbers. A method widely used to discover such relationships is the graphical method, which gives a pictorial view of the results. Visual results provide very valuable information about the dependencies of measured and calculated quantities.

A.     Independent & Dependent Variables

Ø      In any experimental study of cause and effect the aim is to vary one condition at a time (the cause) and to observe the corresponding values of another quantity (the effect), which is suspected of being related to the first.

Ø      This existing relationship is most easily interpreted from the graph if the first of these quantities, the independent variable (cause) is plotted on the abscissa scale (X axis) and the dependent variable (effect) is plotted on the ordinate sale (Y axis).

 

B.     Choice of Scale

Ø      Note the range of values of the independent variable (X quantity), and the number of spaces along the X-axis. Choose a scale for the main divisions on the graph paper that are easily subdivided and such that the entire range of values may be included.

Ø      Use the same procedure for the ordinate scale, but the subdivisions on the ordinate and the abscissa scales need not be alike.

Ø      In many cases it is not necessary that the intersection of the two axes represent the zero values of both variables.

Ø      If the values to be plotted are exceptionally large or small, use some multiplying factor that permits using a maximum of two or three digits to indicate the value of the main division. A multiplying factor such as x 102 placed at the right of the units may be used.

 

C.     Labeling

Ø      After you have decided which variable is t be plotted on which axis, neatly letter the name of the quantity being plotted together with the proper unit.

Ø      Abbreviate units in standard form e.g. meters (m)

Ø      Then write the numbers along main divisions on the graph paper, using an appropriate scale.

Ø      The title should be neatly lettered on the body of the graph paper, but it is usually best to do this after the points have been plotted so the title will not interfere with the curve.

Ø      Explanatory legends and scales should also be shown.

 

D.     Plotting & Drawing the Curve

Ø      In drawing the graph it is not always possible to make all the points lie on a smooth curve. In such cases, a smooth curve should be drawn through the series of points to follow the general trend and thus represent an average. One should look first for a possible straight line fit since it is the simplest.

Ø      Sometimes a data point appears to have no relationship to the rest of the data. This point should not be over weighted. The graph points out immediately that this data point may not be consistent with the other measurements. Explanation and/or re-evaluation (if time permits) of inconsistent data points should be considered.

Ø      When more than one curve is drawn on a graph crosses (x), triangles, squares, and circles should be used.

 

E.      Analysis & Interpretation of Graph

Ø      One of the principal advantages afforded by graphical representation is the simplicity with which new information can be obtained directly from the graph by observing its shape and intercepts.

Ø      The shape of a graph immediately tells one whether the dependent variable increases or decreases with an increase of the independent variable. It also shows something of the rate of change.

Ø      If the points lie on a straight line, there is a linear relationship between the variables.

Ø      If the variables are directly proportional to each other, they approach zero simultaneously, and the line passes through the origin.

Ø      Curves which are straight lines and do not pass through the origin do not indicate direct proportion.

 

F.      Straight Line Graphs

Ø      The standard form for a straight line is given by: y = mx + b      

Ø      Where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the intercept of the line with the y-axis.

 

G.     Slope

Ø      The slope of a graph is found by dividing DY by DX using for each the scales and units that have been chosen for those axes.

Ø      The unit of the slope will be the ratio of the units on the respective axes.

 

3. Units

Ø      A great deal of time and frustration can be avoided if care is taken to always use correct units in laboratory calculations.

Ø      These calculations will give meaningful results only when all physical constants ad measured quantities are in a consistent set of units.

Ø      If you are uncertain of the various units associated with the quantities in a given experiment, consult your manual and find out what they are before beginning.

Ø      The student should always insure that correct units are used in experimental work.