Main Page/PHYS 4210/Muon Lifetime

Introduction

The muon is one of nature’s fundamental “building blocks of matter” and acts in many ways as if it were an unstable heavy electron, for reasons no one fully understands. Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F. Rasetti. The instrument described in this manual permits you to measure the charge averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons as a function of height above sea-level and to demonstrate the time dilation effect of special relativity. The instrument also provides a source of genuinely random numbers that can be used for experimental tests of standard probability distributions.

Our Muon Source

The top of earth's atmosphere is bombarded by a flux of high energy charged particles produced in other parts of the universe by mechanisms that are not yet fully understood. The composition of these "primary cosmic rays" is somewhat energy dependent but a useful approximation is that 98% of these particles are protons or heavier nuclei and 2% are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.

The primary cosmic rays collide with the nuclei of air molecules and produce a shower of particles that include protons, neutrons, pions (both charged and neutral), kaons, photons, electrons and positrons. These secondary particles then undergo electromagnetic and nuclear interactions to produce yet additional particles in a cascade process. Figure 1 indicates the general idea. Of particular interest is the fate of the charged pions produced in the cascade. Some of these will interact via the strong force with air molecule nuclei but others will spontaneously decay (indicated by the arrow) via the weak force into a muon plus a neutrino or antineutrino:

The muon does not interact with matter via the strong force but only through the weak and electromagnetic forces. It travels a relatively long instance while losing its kinetic energy and decays by the weak force into an electron plus a neutrino and antineutrino. We will detect the decays of some of the muons produced in the cascade. (Our detection efficiency for the neutrinos and antineutrinos is utterly negligible.)

Figure 1- Cosmic ray cascade induced by a cosmic ray proton striking an air molecule nucleus.

Not all of the particles produced in the cascade in the upper atmosphere survive down to sea-level due to their interaction with atmospheric nuclei and their own spontaneous decay. The flux of sea-level muons is approximately 1 per minute per cm² (see for more precise numbers) with a mean kinetic energy of about 4 GeV.

Careful study shows that the mean production height in the atmosphere of the muons detected at sea-level is approximately 15 km. Travelling at the speed of light, the transit time from production point to sea-level is then 50 μsec. Since the lifetime of at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel muon flux is qualitative evidence for the time dilation effect of special relativity.

Muon Decay Time Distribution

The decay times for muons are easily described mathematically. Suppose at some time t we have N(t) muons. If the probability that a muon decays in some small time interval dt is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays, then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt. Integrating, we have N(t) = N₀exp(−λ t), where N(t) is the number of surviving muons at some time t and N₀ is the number of muons at t = 0. The "lifetime" τ of a muon is the reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.

Now, we do not have a single clump of muons whose surviving number we can easily measure. Instead, we detect muon decays from muons that enter our detector at essentially random times, typically one at a time. It is still the case that their decay time distribution has a simple exponential form of the type described above. By decay time distribution D(t), we mean that the time-dependent probability that a muon decays in the time interval between t and t + dt is given by D(t)dt. If we had started with N₀ muons, then the fraction −dN/N₀ that would on average decay in the time interval between t and t + dt is just given by differentiating the above relation:

The left-hand side of the last equation is nothing more than the decay probability we seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N₀. That is, the distribution of decay times, for new muons entering our detector, is also exponential with the very same exponent used to describe the surviving population of muons. Again, what we call the muon lifetime is τ = 1/λ.

Because the muon decay time is exponentially distributed, it does not matter that the muons whose decays we detect are not born in the detector but somewhere above us in the atmosphere. An exponential function always “looks the same” in the sense that whether you examine it at early times or late times, its e-folding time is the same.

Detector Physics

The active volume of the detector is a plastic scintillator in the shape of a right circular cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing together one or more fluors with a solid plastic solvent that has an aromatic ring structure. A charged particle passing through the scintillator will lose some of its kinetic energy by ionization and atomic excitation of the solvent molecules. Some of this deposited energy is then transferred to the fluor molecules whose electrons are then promoted to excited states. Upon radiative de-excitation, light in the blue and near-UV portion of the electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of deposited energy. The properties of the polyvinyltoluene-based scintillator used in the muon lifetime instrument are summarized in table 1.

To measure the muon's lifetime, we are interested in only those muons that enter, slow, stop and then decay inside the plastic scintillator. Figure 2 summarizes this process. Such muons have a total energy of only about 160 MeV as they enter the tube. As a muon slows to a stop, the excited scintillator emits light that is detected by a photomultiplier tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the electronics section below for more detail.) A stopped muon, after a bit, decays into an electron, a neutrino and an anti-neutrino. (See the next section for an important qualification of this statement.) Since the electron mass is so much smaller that the muon mass, m_μ/m_e ~ 210, the electron tends to be very energetic and to produce scintillator light essentially all along its pathlength. The neutrino and anti-neutrino also share some of the muon's total energy but they entirely escape detection. This second burst of scintillator light is also seen by the PMT and used to trigger the timing clock. The distribution of time intervals between successive clock triggers for a set of muon decays is the physically interesting quantity used to measure the muon lifetime.

Figure 2- Schematic showing the generation of the two light pulses (short arrows) used in determining the muon lifetime. One light pulse is from the slowing muon (dotted line) and the other is from its decay into an electron or positron (wavey line).

Table 1- General Scintillator Properties.

Interaction of μ⁻’s with matter

The muons whose lifetime we measure necessarily interact with matter. Negative muons that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in much the same way as electrons do. Since the muon is not an electron, the Pauli exclusion principle does not prevent it from occupying an atomic orbital already filled with electrons. Such bound negative muons can then interact with protons

before they spontaneously decay. Since there are now two ways for a negative muon to disappear, the effective lifetime of negative muons in matter is somewhat less than the lifetime of positively charged muons, which do not have this second interaction mechanism. Experimental evidence for this effect is shown in figure 3 where “disintegration” curves for positive and negative muons in aluminum are shown . The abscissa is the time interval t between the arrival of a muon in the aluminum target and its decay. The ordinate, plotted logarithmically, is the number of muons greater than the corresponding abscissa. These curves have the same meaning as curves representing the survival population of radioactive substances. The slope of the curve is a measure of the effective lifetime of the decaying substance. The muon lifetime we measure with this instrument is an average over both charge species so the mean lifetime of the detected muons will be somewhat less than the free space value

τ_μ = 2.19703 ± 0.00004 μsec.

The probability for nuclear absorption of a stopped negative muon by one of the scintillator nuclei is proportional to Z⁴, where Z is the atomic number of the nucleus . A stopped muon captured in an atomic orbital will make transitions down to the K-shell on a time scale short compared to its time for spontaneous decay . Its Bohr radius is roughly 200 times smaller than that for an electron due to its much larger mass, increasing its probability for being found in the nucleus. From our knowledge of hydrogenic wavefunctions, the probability density for the bound muon to be found inside the nucleus is proportional to Z³. Once inside the nucleus, a muon’s probability for encountering a proton is proportional to the number of protons there and so scales like Z. The net effect is for the overall absorption probability to scale like Z⁴. Again, this effect is relevant only for negatively charged muons.

Figure 3 - Disintegration curves for positive and negative muons in aluminum. The ordinates at t = 0 can be used to determine the relative numbers of negative and positive muons that have undergone spontaneous decay. The slopes can be used to determine the decay time of each charge species.

μ⁺/μ⁻ Charge Ratio at Ground Level

Our measurement of the muon lifetime in plastic scintillator is an average over both negatively and positively charged muons. We have already seen that μ−’s have a lifetime somewhat smaller than positively charged muons because of weak interactions between negative muons and protons in the scintillator nuclei. This interaction probability is proportional to Z⁴, where Z is the atomic number of the nuclei, so the lifetime of negative muons in scintillator and carbon should be very nearly equal. This latter lifetime τ_c is measured to be τ_c = 2.043 ± 0.003 μsec..

It is easy to determine the expected average lifetime τ_obs of positive and negative muons in plastic scintillator. Let λ⁻ be the decay rate per negative muon in plastic scintillator and let λ⁺ be the corresponding quantity for positively charged muons. If we then let N^- and N⁺ represent the number of negative and positive muons incident on the scintillator per unit time, respectively, the average observed decay rate <λ> and its corresponding lifetime τ_obs are given by

where ρ ≡ N⁺/N⁻, τ⁻≡(λ⁻)⁻¹ is the lifetime of negative muons in scintillator and τ⁺≡(λ⁺)⁻¹ is the corresponding quantity for positive muons.

Due to the Z⁴ effect, τ⁻= τ_c for plastic scintillator, and we can set τ⁺ equal to the free space lifetime value τ_μ since positive muons are not captured by the scintillator nuclei. Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the scintillator.

We can measure ρ for the momentum range of muons that stop in the scintillator by rearranging the above equation:

Backgrounds

The detector responds to any particle that produces enough scintillation light to trigger its readout electronics. These particles can be either charged, like electrons or muons, or neutral, like photons, that produce charged particles when they interact inside the scintillator. Now, the detector has no knowledge of whether a penetrating particle stops or not inside the scintillator and so has no way of distinguishing between light produced by muons that stop and decay inside the detector, from light produced by a pair of through-going muons that occur one right after the other. This important source of background events can be dealt with in two ways. First, we can restrict the time interval during which we look for the two successive flashes of scintillator light characteristic of muon decay events. Secondly, we can estimate the background level by looking at large times in the decay time histogram where we expect few events from genuine muon decay.

Fermi Coupling Constant G_F

Muons decay via the weak force and the Fermi coupling constant G_F is a measure of the strength of the weak force. To a good approximation, the relationship between the muon lifetime τ and G_F is particularly simple:

where m is the mass of the muon and the other symbols have their standard meanings. Measuring t with this instrument and then taking m from, say, the Particle Data Group produces a value for G_F.

Electronics

A block diagram of the readout electronics is shown in figure 4. The logic of the signal processing is simple. Scintillation light is detected by a photomultiplier tube (PMT) whose output signal feeds a two-stage amplifier. The amplifier output then feeds a voltage comparator (“discriminator”) with adjustable threshold. This discriminator produces a TTL output pulse for input signals above threshold and this TTL output pulse triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA input within a fixed time interval will then stop and reset the timing circuit. (The reset takes about 1 msec during which the detector is disabled.) The time interval between the start and stop timing pulses is the data sent to the PC via the communications module that is used to determine the muon lifetime. If a second TTL pulse does not arrive within the fixed time interval, the timing circuit is reset automatically for the next measurement.

Figure 4- Block diagram of the readout electronics. The amplifier and discriminator outputs are available on the front panel of the electronics box. The HV supply is inside the detector tube.

The front panel of the electronics box is shown in figure 5. The amplifier output is accessible via the BNC connector labeled Amplifier output. Similarly, the comparator output is accessible via the connector labeled Discriminator output. The voltage level against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe leads. The toggle switch controls a beeper that sounds when an amplifier signal is above the discriminator threshold. The beeper can be turned off.

Figure 5- Front of the electronics box.

The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside the power switch.

Figure 6- Rear of electronics box. The communications ports are on the left. Use only one.

Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the detector tube is supplied from the electronics box through the connector DC Power. The high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the top of the detector tube. The HV level can be measured by using the pair of red and black connectors that accept standard multimeter probes. The HV monitor output is 1/100 times the HV applied to the PMT.

Figure 7- Top view of the detector lid. The HV adjustment potentiometer and monitoring ports for the PMT are located here.

A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse pairs at a fixed repetition rate of 100 Hz while the time between the two pulses comprising a pair is adjusted by the knob labeled Time Adj. The pulser output voltage is accessible at the connector labeled Pulse Output.

Software and User Interface

Software is used to both help control the instrument and to record and process the raw data. There is also software to simulate muon decay data. All software is contained in a folder labelled Muon Decay located on the c:\ drive of the computer. There is a shortcut to the main program you will be using located on the desktop called Muon.

Figure 8- User Interface.

There are 5 sections to the main display panel:

• Control
• Muon Decay Time Histogram
• Monitor
• Rate Meter
• Muons through detector

Control

The Configure button allows setting of the communications port. Click on Configure and select com4 located in the Select port section. The maximum x-axis value for the histogram of the muon decay times and the number of data bins is also set here. The default values for these parameters will be used. There are also controls for reading back all ready collected data.

The blue colored Save/Exit switch is used to finalize all your communication and histogramming selections.

Figure 9- Configure Sub-Menu.

The Start button in the user interface initiates a measurement using the settings selected from the configure menu. After selecting it, you will see the “Rate Meter” and the “Muons through detector” graphs show activity.

The Pause button temporarily suspends data acquisition so that the three graphs stop being updated. Upon selection, the button changes its name to Resume. Data taking resumes when the button is selected a second time.

The Fit button will not be used by you. You will fit your data using your favorite fitting package (Mathematica, Matlab, Fortran, Excel, etc...).

The View Raw Data button opens a window that allows you to display the timing data for a user selected number of events, with the most recent events read in first. Here an event is any signal above the discriminator threshold so it includes data from both through going muons as well as signals from muons that stop and decay inside the detector. Each raw data record contains two fields of information. The first is a time, indicating the year, month, day, hour, minute and second, reading left to right, in which the data was recorded. The second field is an integer that encodes two kinds of information. If the integer is less than 40000, it is the time between two successive flashes, in units of nanoseconds. If the integer is greater than or equal to than 40000, then the units position indicates the number of “time outs,” (instances where a second scintillator flash did not occur within the preset timing window opened by the first flash). See the data file format below for more information. Typically, viewing raw data is a diagnostic operation and is not needed for normal data taking.

The Quit button stops the measurement and asks you whether you want to save the data. Answering No writes the data to a file that is named after the date and time the measurement was originally started, i.e., 03-07-13-17-26.data. Answering Yes appends the data to the file muon.data. The file muon.data is intended as the main data file.

Data file format

Timing information about each signal above threshold is written to disk and is contained either in the file muon.data or a file named with the date of the measurement session. Which file depends on how the data is saved at the end of a measurement session.

The first field is an encoded positive integer that is either the number of nanoseconds between successive signals that triggered the readout electronics, or the number of “timeouts” in the one-second interval identified by the corresponding data in the second column. An integer less than 40000 is the time, measured in nanoseconds, between successive signals and, background aside, identifies a muon decay. Only data of this type is entered automatically into the decay time histogram.

An integer greater than or equal to 40000 corresponds to the situation where the time between successive signals exceeded the timing circuit’s maximum number of 40000 clock cycles. A non-zero number in the units place indicates the number of times this ‘timeout” situation occurred in the particular second identified by the data in the first field. For example, the integer 40005 in the first field indicates that the readout circuit was triggered 5 times in a particular second but that each time the timing circuit reached its maximum number of clock cycles before the next signal arrived.

The second field is the number of seconds, as measured by the PC, from the beginning of 1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer programming.

Monitor

This panel shows rate-related information for the current measurement. The elapsed time of the current measurement is shown along with the accumulated number of times from the start of the measurement that the readout electronics was triggered (Number of Muons). The Muon Rate is the number of times the readout electronics was triggered in the previous second. The number of pairs of successive signals, where the time interval between successive signals is less than the maximum number of clock cycles of the timing circuit, is labeled Muon Decays, even though some of these events may be background events and not real muon decays. Finally, the number of muon decays per minute is displayed as Decay Rate.

Rate Meter

This continuously updated graph plots the number of signals above discriminator threshold versus time. It is useful for monitoring the overall trigger rate.

Muons through Detector

This graph shows the time history of the number of signals above threshold. Its time scale is automatically adjusted and is intended to show time scales much longer than the rate meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking, it includes signals from not only through going muons but any source that might produce a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding so that the far left-hand side always corresponds to the start of the measurement session. The bin width is indicated in the upper left-hand portion of the plot.

Muon Decay Time Histogram

This plot is probably the most interesting one to look at. It is a histogram of the time difference between successive triggers and is the plot used to measure the muon lifetime. The horizontal scale is the time difference between successive triggers in units of microseconds. Its maximum displayed value is set by the Configure menu. (All time differences less than 20 μsec are entered into the histogram but may not actually be displayed due to menu choices.) You can also set the number of horizontal bins using the same menu. The vertical scale is the number of times this time difference occurred and is adjusted automatically as data is accumulated. A button (Change y scale Linear/Log) allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal error bars for the data points span the width of each timing bin and the vertical error bars are the square root of the number of entries for each bin.

The upper right hand portion of the plot shows the number of data points in the histogram. Again, due to menu selections not all points may be displayed. If you have selected the Fit button then information about the fit to the data is displayed. The muon lifetime is returned, assuming muon decay times are exponentially distributed, along with the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit. (For more details.)

Muon Decay Simulation

Simulated muon decay data can be generated using the program muonsimu found in the muon_simu folder. Its interface and its general functionality are very similar to the program muon in the muon_data folder. The simulation program muonsimu lets you select the decay time of the muon and the number of decays to simulate. Simulated data is stored in exactly the same format as real data.

Utility Software

The folder muon_util contains several useful programs that ease the analysis of decay data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that describe timing data inconsistent with actual muon decay.

The executable file merge merges two data files of your choosing into a single file of your choosing. The data records are time ordered according to the date of original recording so that the older the record the earlier it occurs in the merged file.

The executable file ratecalc calculates the average trigger rate (per second) and the muon decay rate (per minute) from a data file of your choosing. The returned errors are statistical.

The executable freewrap is the compiler for any Tcl/Tk code that your write or modify. If you modify a Tcl/Tk script, you need to compile it before running it. On a Windows machine you do this by opening a DOS window, and going to the muon_util directory. You then execute the command freewrap your_script.tcl, where your_script.tcl is the name of your Tcl/Tk script. Do not forget the tcl extension!

Exercises

Apparatus

1. "Muon Physics" Scintillator
2. "Muon Physics" Control Unit
3. GW Function Generator (Model: GFG-8016G)
4. Digital oscilloscope
5. 50-Ω terminators[9]
6. Control computer and software
7. assorted cables

Testing the Electronics

You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator. Log in to the local account on the computer to access the Muon software, using the username and password provided on the top right of the monitor.

1. Measure the gain of the 2-stage amplifier using a sine wave: Use the oscilloscope to setup a 100 kHz, 100 mV peak-to-peak sine wave on the function generator (FG). (The amplitude knob of the FG should be in the pulled out position to activate the -20 dB internal attenuator. This will allow you to generate the small 100 mV amplitude of the sine wave.)

   Connect the output of the FG to the input of the electronics box. Measure the amplifier output and take the ratio V_out/V_in. Due to attenuation resistors inside the electronics box inserted between the amplifier output and the front panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to determine the real amplifier gain.

   Q: Increase the frequency. Over what frequency range does the amplifier operate?

   Q: Estimate the maximum decay rate you could observe with the instrument.

2. Measure the saturation output voltage of the amp.

   Increase the magnitude of the input sine wave and monitor the amplifier output.

   Q: Does a saturated amp output change the timing of the FPGA? What are the implications for the size of the light signals from the scintillator?

3. Examine the behavior of the discriminator by feeding a sine wave to the PMT input and then adjusting the discriminator threshold. You can monitor the discriminator output directly on the oscilloscope. Make sure to terminate this connection with a 50Ω terminator as well. Describe the shape of the Discriminator output and explain the behaviour of the discriminator threshhold.

4. Now we will start looking at actual signals from the PMT. Connect the output of the PMT to the PMT input on the control box. Set the High Voltage Adjust to 10. Run the program Muon on the desktop. Click on Configure and select port com4, then click save & exit. Once you click Start from the Control panel, you will be able to observe the muon count rate in the Monitor Panel.

   Adjust (or misadjust) discriminator threshold. Observe the raw muon count rate and the spectrum of "decay" times as a function of the threshold level.

5. What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know that good signals need to be at about 200 mV or so before discriminator, so set discriminator before hand.) With a fixed discriminator threshold, alter the HV and observe the raw muon count rate and decay spectrum.
6. Connect the output of the detector (PMT output) to the PMT input of the electronics box. Look at the amplifier output using the scope.Be sure that the scope input is terminated at 50Ω. What do you see? Now examine the discriminator output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do you see?

Muon Lifetime Measurement

1. Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment. Using the Digital Multimeter (DMM) and the HV 1:100 Monitoring ports on the detector lid set the HV between -1100 V and -1200 V. Next, set the threshold on the electronics box between 180 and 220 mV.

   Start and observe the decay time spectrum for several minutes. The average muon rate should be around 6 Hz. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.

   Q: The muons whose decays we observe are born outside the detector and therefore spend some (unknown) portion of their lifetime outside the detector. So, we never measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of muons. How can this be?

2. Use Ctrl-Alt-PrtScn to obtain a screen capture of the muon main display panel which will include the Muon Decay Time Histogram (you can paste in paint and save file) prior to saving the data. Present this image in your report.
3. Using the saved data, recreate your decay time histogram and fit the decay time histogram with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for.
4. From your measurement of the muon lifetime and a value of the muon mass from some trusted source, calculate the value of Fermi coupling constant G_F. Compare your value with that from a trusted source.
5. Using the approach outlined above, measure the charge ratio ρ of positive to negative muons at ground level.
6. Once the muon lifetime is determined, compare the theoretical binomial distribution with an experimental distribution derived from the random lifetime data of individual muon decays. For example, let p be the (success) probability of decay within 1 lifetime, p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63 with a variance σ² = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent with theory?