1. Introduction to Data Analysis and Processing www.cma-science.nl Technology Enhanced Inquiry Based Science Education

2. Data analysis and processing Once data is collected, you need tools to analyse or process the data. Data analysis usually refers to getting information from the data, for instance by looking in more detail (zooming), reading co- ordinates of points in the graph, or determine a slope of a tangent line to the graph. Data processing means that the data are worked in one way or another to produce new data. www.cma-science.nl

3. Overview of tools Processing Select/Remove data Smooth Derivative Integral Function fit Signal Analysis www.cma-science.nl Other tools Using formulas Analysis Zoom Scan Slope Area Statistical tools Statistics Histogram

4. Zoom www.cma-science.nl If you want to see the data in more detail, you can use the Zoom function. An area of the diagram will be enlarged for closer inspection.

5. Scan www.cma-science.nl If you need to read the co-ordinates of points on graphs (or other points in the diagram), use the option Scan. The co-ordinates are displayed in the box (top right corner). Example: scanning a position vs. time graph

6. Slope www.cma-science.nl Slope gives the slope of the tangent at any point of a displayed graph. This is a measure of the rate with which a quantity changes, e.g. the speed of an object. Example: the rate at which a capacitor discharges at t=0,29s.

7. Area www.cma-science.nl With Area you can determine the area between a displayed graph, the horizontal axis and two boundary lines. An area below the axis is negative. Example: Area under the graph of an induced EMF by a falling magnet. The area is a measure of the magnetic flux B.

8. Select/Remove data www.cma-science.nl If your data set has spikes (erroneous measurements) or if part of the data is irrelevant, with Select/Remove Data you can both select single points or a range of data for removal or retention. Examples: Range: discharging a capacitor (cut off first horizontal part) (top). Points: keeping the lower envelope of the pressure variation in a blood pressure measurement (bottom).

9. Smooth www.cma-science.nl If you have a rough or limited set of measured points you can use Smooth to approximate your data with a smooth curve or with a dataset consisting of more points. There are three smoothing methods: Moving average, Bezier, Spline. The smoothed graph can successively be processed further.

10. Smooth: Moving average www.cma-science.nl Use Moving average to reduce noise and eliminate fluctuations in the graph. The smoothed graph has the same number of points and each point is replaced by the average of a number of neighbouring points. The Filter width parameter determines this number of points. Moving average is often used to highlight long-term trends and cycles. Example: CO 2 -measurement, Filter width determines the degree of filtering. Original graphFilter width = 1Filter width = 2 Original graphFilter width = 1Filter width = 10

11. Smooth: Bézier www.cma-science.nl Use Bezier to create a smooth curve with more points then the original data set. The smoothed graph is forced through the first and the last original point. The intermediate points determine the degree of curvature of the smooth graph.

12. Smooth: Spline …..…. www.cma-science.nl Use Spline to smooth a graph by means of a polynomial approximation of 5 th degree. A smoothing factor controls the trade off between fitting the raw data and minimizing the roughness of the approximation. For a lower value of smoothing factor the spline curve gets closer to the raw data. When its value is 0 the smoothing curve is a natural quintic spline curve through all original points. Spline is a powerful tool to deal with noisy data and for computation of smooth derivatives. Smoothing factor of resp. 0, 0.05, 5.154 (auto) and 10,000.

13. Smooth: Derivative www.cma-science.nl Derivatives are very important in science, as they are a measure of the rate of change of a quantity. They are used often to calculate the speed of processes, or the point where change is maximum. Differences method: direct calculation via differences between successive points (often noisy). Smooth method: the derivative is applied to a smooth spline function from the data. Example: determining the equivalence point of an acid-base titration (top: differences; bottom: smooth).

14. Function fit www.cma-science.nl For verifying data against theory one usually wants to approximate the data with a standard mathematical function. Function fit is a procedure to make such approximation. A large number of standard mathematical functions are available. The coefficients of the fit function are determined using a least-squares method.

15. Function fit : linear fit www.cma-science.nl Position vs. time data of a motion of a cart. A linear fit on the straight part of the graph gives the speed during this phase of the motion (v=14.7 cm/s).

16. Function fit : exponential fit www.cma-science.nl A radioactive decay can be described by an exponential function a*exp(b*x)+c. By fitting the data to this function, one gets ‘a’ - the initial number of atoms and ‘b’ the parameter which is related to the speed of the decay process (the half-life time of the source).

17. Function fit : exponential fit www.cma-science.nl The process of discharging of a capacitor can be described by an exponential function. The coefficient ‘a’ of the fit-function is related to the begin voltage of the capacitor, while ‘b’ is related to the ‘RC-time’, the time interval it takes to reach half of the begin voltage of the capacitor.

18. Signal Analysis www.cma-science.nl If you have a sound signal consisting of a number of frequencies, e.g. a tone of a musical instrument, or a spoken or sung vowel, Signal Analysis can help you to analyse which frequencies are present or, in case of speech, which formants are present in the signal. There are four methods: Fourier Transform, Linear Prediction, R-ESPRIT, Prony.

19. Signal Analysis of sound beats www.cma-science.nl R-Esprit Original waveform Prony Fourier transform

20. Signal Analysis: linear prediction www.cma-science.nl Linear Prediction (LP) is suitable for analysing sound vibrations of the human voice. Example: two spectra from the vowel ‘a’ (as in ‘Cake’) sung one fifth apart (same male voice). The overall shape of the graphs share some characteristics resulting from resonances from the shape and cavities of the singer.

21. Statistics www.cma-science.nl Use Statistics to display statistical information about your data. Example: statistical data from a data set of the EMF induced in a coil by a falling magnet. From the symmetry in the signal the statistical data confirm that the average value lies near 0.

22. Histogram www.cma-science.nl With Histogram you can get a graphical representation of the distribution of your data. It indicates the number of data points that lie within a range of values called bins. The height of the bar equals the number of times the data point value fell within the bin. Example: The subject got 100 times a sound signal to react to as fast as possible. The histogram shows how the reaction times are distributed in time (e.g. 19 times between 0.2 and 0.3 s).

23. Using Formulas www.cma-science.nl It is also possible to do calculations on the data by using formulas. You can use different kinds of arithmetical and mathematical functions. Data ranges with formulas are automatically recalculated. During a new run, the newly calculated values appear in real-time. Example: measuring p and V, and creating a diagram of V vs. 1/p (Boyle’s law).

24. www.cma-science.nl Centre for Microcomputer Applications