LABS ARE NOT MEANT TO BE FUN, OR EASY, OR TO BOOST YOUR GRADES.

(though, I do hope you enjoy it, and I always always hope you earn a good grade!)

We do labs so that you learn how to investigate a problem and how to analyze data. By 11th grade, you should know how to investigate a problem fairly well.

One of our big focuses this year, then, is how to analyze data in a rigorous fashion.

(so, use this ppt as a reference AND DO YOUR BEST WORK on the upcoming lab!!)

Independent & Dependent Variable

» In order to create a valid experiment, the experimenter can only change ONE variable at a time

» The variable that is being changed by the experimenter is called the ______________________ variable.

» The variable that changes as a result of the independent variable is called the ________________ variable.

» In physics there is no CONTROL variable, instead there are» CONTROLED variables is what is kept the same throughout the

experiment because it also affects the dependent variables being tested, thus affecting the outcome of the experiment

INDEPENDANT

DEPENDENT

Your knowledge and grade in physics is related to the amount of quality time you spend studying..

independent variable: ? dependent variable: ?

Graphing the Independent and Dependent Variables» If you are measuring the amount of bacteria over time, what is the

independent and dependent variable?

Independent Variable on the x-axis!

Dependent Variable on the y-axis!

The graph shown here is ok. Why only ok?

The graph shows us the overall trend (average value). In science we care about BOTH the average value and some measure of the variation/uncertainty in the data.

When we make a measurement, we generally assume that some exact or true value exists based on how we define what is being measured. While we may never know this true value exactly, we attempt to find this ideal quantity to the best of our ability with the time and resources available. As we make measurements by different methods, or even when making multiple measurements using the same method, we may obtain slightly different results. So how do we report our findings for our best estimate of this elusive true value? The most common way to show the range of values that we believe includes the true value is:

measurement = best estimate ± uncertainty

Measurement = (mean value ± uncertainty) unit of measurementIf the best estimate is the average or mean of independent measurements

uncertainty

best estimate

uncertainty

Measurement

valuesThe uncertainty shows the area around the average value where the true value of the measurement is likely to be found.

For example, the result (20.1 ± 0.1) cm basically communicates that the person making the measurement believes the value to be closest to 20.1 cm but it could have been anywhere between 20.0 cm and 20.2 cm. 

» In ANY science, we care about the variation in the data nearly as much as the overall outcome / trend.

Why?

» Because the variation gives us some (NOT perfect) idea of how close our measurement is to the TRUE value.

» Variation also limits our ability to make a conclusion about an overall trend or difference in our data.

This is real data, btw!

We use the synonymous terms uncertainty, error, or deviation to represent the variation in measured data. Data nearly always has uncertainty. There are many causes of uncertainty (different results when doing the same measurements), including:

No measuring instrument is fully precise. Each instrument has an inherent amount of uncertainty in its measurement. Even the most precise measuring device cannot give the actual value because to do so would require an infinitely precise instrument.

Its nearly impossible to fully control all variables. Un- or poorly- controlled variables will cause variation in your results. For example, changes in wind speed and direction would effect the results of a projectile launching experiment.

Sometimes you are measuring a small number of individuals from a larger population who differ in some trait. This is very common in biology but is NOT usually relevant to physics.

instrument uncertainty

uncontrolled variables

Sampling from a population

Random and systematic errors

11th grade

Precision indicates the quality of the measurement, without any guarantee that the measurement is "correct." Accuracy assumes that there is an ideal value, and tells how far your answer is from that ideal, "right" answer.

Random errors are statistical fluctuations (in either direction) in the measured data due to the precision limitations of the measurement device. Random errors can be evaluated through statistical analysis and can be reduced by averaging over a large number of observations. It can be observed in the graph and has to be discussed in the lab report.Systematic errors are reproducible inaccuracies that are consistently in the same direction. These errors are difficult to detect, cannot be analyzed statistically or reduced by increasing the number of observations. It might be observed in the graph from the shift of the best fit line and has to be discussed in the lab report.

12th grade

Common sources of error in physics laboratory experiments:

Incomplete definition - the measurement is not always clearly defined. If two different people measure the length of the same rope, they would probably get different results because each person may stretch the rope with a different tension. Specify the conditions that could affect the measurement. Change instrument uncertainty, maybe?

Failure to account for a factor (usually systematic) The most challenging part of designing an experiment is trying to control or account for all possible factors except the one independent variable that is being analyzed. For instance, you may ignore air resistance when measuring free-fall acceleration, or you may fail to account for the effect of the Earth’s magnetic field when measuring the field of a small magnet. The best way to account for these sources of error is to brainstorm with your peers about all the factors that could possibly affect your result. This brainstorm should be done before beginning the experiment so that arrangements can be made to account for the confounding factors before taking data. Sometimes a correction can be applied to a result after taking data to account for an error that was not detected. This is done from the graph!!!!!

Environmental factors (systematic or random) - Be aware of errors introduced by your immediate working environment. You may need to take account for or protect your experiment from vibrations, drafts, changes in temperature, electronic noise or other effects from nearby apparatus.

Failure to calibrate or check zero of instrument (systematic)

Parallax (systematic or random) - This error can occur whenever there is some distance between the measuring scale and the indicator used to obtain a measurement. If the observer's eye is not squarely aligned with the pointer and scale, the reading may be too high or low (some analog meters have mirrors to help with this alignment).

12th grade

A major part of any lab is evaluating the strength of your data, identifying the possible sources of error, and suggesting improvements or further lines of inquiry for the future.

DO NOT ANSWER: “I think my data are good because I worked hard and didn’t mess up. I could do better if I had more time and better equipment.”

Instead, » use your uncertainties to comment meaningfully on whether or not you can

really make a conclusion or find a trend.» Think carefully about sources of error (often, these are poorly controlled

variables that are affecting your results, but sometimes your whole approach may be flawed).

» If your uncertainties are really large you should explain why and suggest how to reduce them in the future.

» If you obtained the expected results then think of a follow up question you could study. If the results were surprising, think of a way to figure out why you got those results.

Human error (messing up) is not a valid type of experimental error, thus the term “HUMAN ERROR“ should NEVER be used in a lab report!!! Instead the two acceptable types of experimental error explored in physics labs are systematic and random error.

11th grade

The uncertainty of a single measurement is limited by the precision and accuracy of the measuring instrument, along with any other factors that might affect the ability of the experimenter to make the measurement.

1. Reporting Uncertainty for a Single Measurement – Instrument uncertainty or estimated uncertainty

No measuring instrument is fully precise. Each instrument has an inherent amount of uncertainty in its measurement. Even the most precise measuring device cannot give the actual value because to do so would require an infinitely precise instrument.

Generally we report the measured value of something with the decimal place or precision going not beyond the smallest increment on the instrument.

If not stated differently by manufacturer, these uncertainties are:

Analog instrument (the one with a scale): ½ of the smallest increment (precision) Digital instrument : the whole smallest increment

Uncertainties are given to 1 significant figure.

L = (10.66 ± 0.05) cm (length is anywhere btw 10.61 & 10.71 cm)

# of SF is 4: 3 known with certainty and one estimate

measurement = (best estimate ± uncertainty) unit

EX: The length of a rod is measured using part of a metre rule that is graduated in millimetres, as shown below.

Which one of the following is the measurement, with its uncertainty, of the length of the rod?  A.5 0.1 cm B. 5 0.2 cm C. 5.0 0.1 cm D. 5.0 0.2 cm

Example 1: Using a ruler

A measurement of length is, in fact, a measure of two positions and then a subtraction, even if the first position is 0.

reading ± the smallest division on the measuring instrument

So, although instrument uncertainty is half the smallest division on the ruler, because there are two uncertainties in the game, the uncertainty in the case of reading with a ruler is

Different sites report different rules for ruler. Both have its own logic. You can use either ½ of the smallest increment or the whole one if you explain

CONCLUSION: The experimenter can determine the error to be different from instrument uncertainty provided some justification can be given. For example, mercury and alcohol thermometers are quite often not as accurate as the instrument uncertainty says. Instrument uncertainty when measuring time with stopwatch is certainly not the one stated by manufacturer – usually 0.01 s. It is ridiculous since you could never, ever move your thumb that fast! It doesn’t take into account human reaction time. More realistic would be ± 0.3 s.Just justify it.

Example 2: Using a stop – watch

Consider using a stop-watch which measures to 1/100 of a second to find the time that is about 1s.

t = 1s ± 0·01s

which is equivalent to saying that the time t is between 0·99s and 1·01s

This sounds quite good until you remember that the reaction-time of the person using the watch might be about 0·15s. Now considering the measurement again, with a possible 0·15s at the starting and stopping time of the watch, we should now state the result as

t = 1s ± (0·01+ 0·3)s In other words, t is between about 0·7s and 1·3s.

Angle of launch (+ 5o)

Horizontal Distance travelled (m) (+ 1 cm)

Trial 1 Trial 2 Trial 3 Trial 4 Trial 5

20 6.2 6.6 6.7 6.2 6.4

40 9.9 9.6 9.5 10.1 9.8

60 8.7 9.1 8.6 8.3 8.6

80 3.1 3.5 3.7 3.2 3.3

Example of the data table: Distance of horizontal travel of a projectile launched at different angles.

When someone else reads this report they would know experimental uncertainties.You have to specify whether this is estimated or instrument uncertainty.

If you add the sentence like: Although measurement tape had instrument uncertainty of 0.5 mm, due to set up of experiment (….) estimated uncertainty is 1 cm (reason).

2. Reporting Best Estimate and Uncertainties in Repeated Measurements

Average or mean is the best available estimate of the "true" value.

uncertainty

mean value

uncertainty

Measurement

values

Measurement is not one particular value, rather it is a range of values. That range represents the measurement result at a given confidence.

Unfortunately, there is no general rule for determining the uncertainty.

Measurement = (mean value ± uncertainty) unit

One way to express the variation among the measurements is to use the average deviation.

This statistic tells us that the additional measurement taken will lie in the range (mean value ± average deviation) with 50% confidence.

Most common way to characterize the spread of data: STANDARD DEVIATION as uncertainty then Measurement = (mean value ± standard deviation)indicates approximately a 68% confidence interval

Meaning:a) if you make one more measurement using the same instrument and method, you can reasonably expect (with about 68% confidence) that the new measurement will be within 1 s.d. of the estimated average.

b) for a large enough sample, approximately 68% of the readings will be within 1 s.d. of the mean value, 95% of the readings will be in the interval ±2s.d., and nearly all (99.7%) of readings will lie within 3 standard deviations from the mean.

Usually done in fields where a sample represents the whole population(biology, sociology, psychology …), although it is very often done in physics too, but not IB

The standard deviation is always slightly greater than the average deviation, and is used because of its association with the normal distribution that is frequently encountered in statistical analyses.

In PHYSICS, Chemistry there is no sample representing the whole population , so we chose in IB PHYSICS uncertainty : next page

For example, result (20.1 cm ± 0.1) cm basically communicates that the person making the measurement believe the value to be closest to 20.1 cm but it could have been anywhere between 20.0 cm and 20.2 cm.

The following is the IB guideline (at the moment) for uncertainties in IA

1. Taking several measurements of something (x) leads to distribution of values (x1, x2, x3,…)

2. Each individual measurement has experimental uncertainty

1. Maximum deviation in the measurements from the mean (we did it last year in IB physics) We claimed that measurement lied in the range (mean value ± maximum deviation) with “reasonable” confidence.

3. Measurement uncertainty is Absolute uncertainty that is either

2. If range of the measurements is we define absolute uncertainty as range/2 (Easier, so we are going to o that this year)

3.

4.𝑹𝒂𝒏𝒈𝒆𝒐𝒇 𝒕𝒉𝒆𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒎𝒆𝒏𝒕𝒔=𝑀𝑎𝑥𝑖𝑚𝑢𝑚𝑣𝑎𝑙𝑢𝑒−𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑣𝑎𝑙𝑢𝑒=𝑥𝑚𝑎𝑥−𝑥𝑚𝑖𝑛

5. 𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆𝒖𝒏𝒄𝒆𝒓𝒕𝒂𝒊𝒏𝒕𝒚 ∆ 𝒙= 𝑟𝑎𝑛𝑔𝑒2

=𝑥𝑚𝑎𝑥−𝑥𝑚𝑖𝑛

2

6.𝑭𝒓𝒂𝒄𝒕𝒊𝒐𝒏𝒂𝒍𝒖𝒏𝒄𝒆𝒓𝒕𝒂𝒊𝒏𝒕𝒚=±∆𝑥𝑥

7.𝑷𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆𝒖𝒏𝒄𝒆𝒓𝒕𝒂𝒊𝒏𝒕𝒊𝒆𝒔=±∆ 𝑥𝑥

100%

Measurement = (average value ± absolute uncertainty) unit𝑥=(𝑥¿¿ 𝑎𝑣𝑔± ∆𝑥 )𝑢𝑛𝑖𝑡𝑠¿Always round your stated uncertainty up to match the number of decimal places of your measurement, if necessary. In the IB Physics laboratory, you should take 5 measurements if you can manage it.

(h𝑜𝑤𝑢𝑠𝑒𝑓𝑢𝑙𝑖𝑠 𝑦𝑜𝑢𝑑𝑎𝑡𝑎 :0.5𝑚𝑒𝑎𝑛𝑠h𝑢𝑔𝑒𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦)

1. Taking several measurements of something (x) leads to distribution of values (x1, x2, x3,…)

IB REQUIREMENTS for IA – ALL TOGETHER

2. Each individual measurement has experimental uncertainty𝑥=(𝑥¿¿ 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒±∆ 𝑥)𝑢𝑛𝑖𝑡𝑠¿

EX: The Six students measure the resistance of a lamp. Their answers in Ω are: 609; 666; 639; 661; 654; 628. What should the students reports as the resistance of the lamp?  Average resistance = 643 Ω Range = Largest - smallest resistance: 666 - 609 = 57 Ω Absolute uncertainty: dividing the range by 2 = 29 Ω So, the resistance of the lamp is reported as: R = (640 ± 30) Ω

0.05

This tells you immediately that resistance is anywhere btw 613 and 673 Ω

When taking time measurements, the stated uncertainty cannot be unreasonably small – not smaller than 0.3 s, no matter what the range. For example if the range is 0.1s, then absolute uncertainty is 0.05 s which is highly unreasonable. You have to change it and explain the reasons.

When taking several measurements, it should be clear if you have a value with a large error. Do not be afraid to throw out any measurement that is clearly a mistake. You will never be penalized for this if you explain your rationale for doing so. In fact, it is permissible, if you have many measurements, to throw out the maximum and minimum values.

Remember that the manner in which you report measured and calculated values is entirely up to you, as the experimenter. However, be realistic in your precision and be able to fully justify reported measurements and calculated values, showing all of your work in doing so.

If data are to be added or subtracted, add the absolute uncertainty:

C = A + B C = (A + B) ± (A + B) C = A – B C = (A – B) ± (A + B)

If data are to be multiplied or divided, add the fractional or percentage uncertainty:

𝑃=𝑄𝑅→𝑃𝑎𝑣𝑔=

𝑄𝑎𝑣𝑔

𝑅𝑎𝑣𝑔

𝑎𝑛𝑑 ∆ 𝑃𝑃𝑎𝑣𝑔

= ∆𝑄𝑄𝑎𝑣𝑔

+ ∆𝑅𝑅𝑎𝑣𝑔

𝑃=𝑄𝑅→𝑃𝑎𝑣𝑔=𝑄𝑎𝑣𝑔𝑅𝑎𝑣𝑔𝑎𝑛𝑑∆𝑃𝑃𝑎𝑣𝑔

=∆𝑄𝑄𝑎𝑣𝑔

+∆𝑅𝑅𝑎𝑣𝑔

If data are raised to a power, multiply the fractional or percentage uncertainty by that power:

𝑃=𝑅𝑛→𝑃 𝑎𝑣𝑔=𝑅𝑎𝑣𝑔𝑛𝑎𝑛𝑑

∆𝑃𝑃𝑎𝑣𝑔

=𝑛∆𝑅𝑅𝑎𝑣𝑔

3 C DETERMINING FINAL UNCERTAINTIES IN STATED RESULTS

EX: A cylinder has a radius of 1.60 ± 0.01 cm and a height of 11.5 ± 0.1 cm. Find the volume.

V = π r2 h = π (1.60) 2 x 11.5 = 92.488 cm2 = 92 cm2

The fractional uncertainty in the radius is 0.01/1.60= 0.00625. The fractional uncertainty in r2 is thus 2 x 0.00625 = 0.01250 The fractional uncertainty in the height is 0.1/11.5 0 = 0.00870.  Fractional uncertainty in the volume is: 0.01250 + 0.00870 = 0.02120 Absolute uncertainty in V is 0.02120 x 92.488 cm2 = 1.96075 cm2

V = 92 ± 2 cm2

For all other mathematical functions (trigonometic, logarithmic, etc), you need to determine the |max value - mean value| and the |min value – mean value|. Whichever is greater, that becomes the ± reported uncertainty value. This can be time-consuming and cumbersome as it does not involve percentage uncertainties in the final uncertainty determination.

If k = 4.78 ± 0.35 cm-1 and x = 23.5 ± 0.1 cm, find sin(kx). kx = 112.33 (kx)max = 5.13 x 23.6 = 121.068 (kx)min = 4.43 x 23.4 = 103.662 sin(112.33) = 0.92501 (mean value) sin(121.068) = 0.85656 (min value) sin (103.662) = 0.97171 (max value) |max value - mean value| = |0.97171 - 0.92501| = 0.04670 |min value – mean value| = |0.85656 - 0.92501| = 0.06845

The final answer is therefore 0.925 ± 0.068 , better stated as 0.925 ± 0.070 (no units)

A student measures a distance several times. The readings lie between 49.8 cm and 50.2 cm. This measurement is best recorded asA.49.8 ± 0.2 cm. B. 49.8 ± 0.4 cm C. 50.0 ± 0.2 cm D. 50.0 ± 0.4 cm.

The power dissipated in a resistor of resistance R carrying a current I is equal to I2R. The value of I has an uncertainty of ±2% and the value of R has an uncertainty of ±10%. The value of the uncertainty in the calculated power dissipation is

A. ±8%. B. ±12%. C. ±14%. D. ±20%.

Percentage discrepancy or percentage difference is an indication of how much your experimental answer varies from the known accepted value of a quantity. For example, experiment is to determine the value of gravitational acceleration.

𝑝𝑟𝑒𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒=accepted   value  −  experimenatal   valueaccepted   value

100% 

4. DRAWING A GRAPH

In many cases, the best way to present and analyze data is to make a graph. A graph is a visual representation of 2 things and shows nicely how they are related. A graph is the visual display of quantitative information and allows us to recognize trends in data. Graphs also let you display uncertainties nicely.

When making graphs:

1. The independent variable is on the x-axis and the dependent variable is on the y-axis. 2. Every graph should have a title that this concise but descriptive, in the form ‘Graph of (dependent variable) vs. (independent variable)’. 3. The scales of the axes should suit the data ranges. 4. The axes should be labeled with the variable, units, and instrument uncertainties. 5. The data points should be clear. 7. Error bars should be shown correctly (using a straight-edge). Error bars represent the uncertainty range 8. Data points (average value ONLY!) should not be connected dot-to-dot fashion. A line of best fit should be drawn instead. The best fit line is not necessarily the straight line and should pass through all of the crosses created by the error bars. Approximately the same number of data points should be above your line as below it. 9. Each point that does not fit with the best fit line should be identified. If you have an outlier, leave it as it is, discuss it, and the draw a second graph omitting the outlier and discuss it again.10. Think about whether the origin should be included in your graph (what is the physical significance of that point?) Do not assume that the line should pass through the origin.

▪ Rather than working out error bars for each point – use the worst value and assume that all of the other errors bars are the same. In the report include explanation, a sentence like: “Taking the highest uncertainty we are reasonably sure that result is ….”

A nice way to show uncertainty in data is with error bars. These are bars in the x and y directions around each data point that show immediately how big or small the uncertainty is for that value. Plotting a graph allows one to visualize all the readings at one time. Ideally all of the points should be plotted with their error bars. In principle, the size of the error bar could well be different for every single point and so they should be individually worked out. In practice, it would often take too much time to add much time to add all the correct error bars, so some (or all) of the following short cut could be considered.

You should discuss the experiment results at the end. What type of dependence you discovered,….For example when graphing experimental data, you can see immediately if you are dealing with random or systematic errors (if you can compare with theoretical or expected results).

LINEARIZATION The area where students struggle is linearizing data. Linearizing a graph means to adjust the variables so that a curved graph turns into a straight-line graph. This does not mean to fit the curved data points with a straight line. Rather, it means to modify one of the variables in some manner such that when the data are graphed using this new data set, the resulting data points will appear to lie in a straight line.

The significance of “linearizing” dataA linear best fit to the data can give information from the slope and the intercept open to the physical interpretation for the students who are not in calculus.

EX: The time period T of oscillation of a mass m suspended from a vertical spring is given by the expression

𝑇=2𝜋 √𝑚𝑘Where k is a constant. Which one of the following will give rise to a straight-line graph? A. against m B. against C. T against m D. against m

EX: A particle is moving in a circular path of radius r. The time taken for one complete revolution is T. bbbThe acceleration a of the particle is given by expression:

𝑎= 4𝜋 2

𝑇2

A. against T B. against C. against D. against

EX: The data from an experiment of the spectral lines of the hydrogen atom is given on the table below. Derive an equation in terms of the energy and the quantum numbers for the hydrogen atom.

Quantum what's? I don't know what the heck they are talking about!It doesn't matter, you may have no clue what they are talking about, however, this problem can be done without knowing any information about atomic physics, simply analyze the data using a graph. The graph suggests that the energy varies inversely with the quantum number, therefore, we can try to linearize the graph using... E vs. 1/n. But I know that this will not lead to linear graph. So after few unsuccessful attempts ( ) I realized that the graph of E vs.1/n2 should yield a straight line

We can now use the graph to find the slope:

𝑠𝑙𝑜𝑝𝑒=13.6−0.81−0.06

=13.6𝑒𝑉 𝐸=13.6

𝑛2 𝑒𝑉

We have just derived the equation for the energy of any electron state in the H atom!

WHAT ARE MAX AND MIN LINES and what to do with them?

Remember that most Physics experiments lead to graphical analysis of data. We can use graphs to express and find uncertainties. You must include graphs of your data in aspect 3 in DCP. Let’s say you have found the line of best fit and the slope of this line (linearization)

Therefore, you have found a value of the slope that corresponds to some physical quantity (13.6 eV in previous example which is maximum energy electron can have in H atom: n = 1). Now you must use the maximum and minimum best-fit lines to determine the final uncertainty in the stated value of the slope of your best-fit line. Here’s how:

3. Determine the slopes of these two lines.

4. Your final uncertainty in the stated value of the slope of your best fit line is:

1. Draw a straight line with the least slope possible (minimum best-fit line) that connects corners of your first and last error boxes.

2. Draw a straight line with the greatest slope possible (maximum best-fit line) that connects corners of your first and last error boxes.

( max  slope  − min   slope )2

Lines of Best Fit and Max/Min Lines for ‘Graph of Quantity a vs. Quantity b’

Note that by using this technique, you may get max and min lines that do not go through the error boxes of every data point. This is ok and you will not be penalized for it.