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BME MEASUREMENTS AND ANALYSIS LABORATORY
FALL 2004
LABORATORY #1:
MEASUREMENT AND UNCERTAINTY
1
I.
Objectives
The objective of this laboratory exercise is to:
1. Learn about measurement errors and managing variation.
2. Introduce engineering software tools, Labview and Matlab for data analysis.
3. Provide a quantitative means for interpreting measurements of biological properties and variability.
II.
Introduction
II.1 Measurements: Error and Uncertainty
Measurements are made using instruments that are imperfect and thus contain errors due to the measuring
device. The 'truth' of a measurement is that value that a measurement would have were it free of the error
introduced by the measurement instrument.
measurement = truth + measurement error
There is no true value in biological systems. Here the 'truth' of a measurement takes on a somewhat
different meaning.
truth = average value + biological variability.
measurement = average value + biological variability + measurement error
After we identify the measurement errors, we can proceed to characterize and quantify the biological
variability. Sometimes we find that we cannot isolate the measurement error by quantifying this error from
the measurements. Measurement errors can classified as either systematic or random.
Systematic errors cause a measurement to be skewed in a certain direction, i.e., to be consistently larger
or consistently smaller.
For example, weighing yourself repeatedly on a bathroom scale that has has an initial reading of 20 lbs
will result in the reading of your weight to be greater than your 'true' weight. The addition of 20 lbs is a
systematic error. This form of error can be reduced once identified – in this case you can zero the scale
using the little ridged knob (calibration). A systematic measurement error may also be called a
measurement bias. Other sources of bias include one-sided error arising from various sources such as
flaws in study design or data collection methods.
Random errors are statistical fluctuations in the measured data due to limitations on the precision of the
measurement device. They are bi-directional (as opposed to the unidirectional nature of a bias). If you
weighed yourself on ten different bathroom scales (or weighed yourself ten times on one bathroom scale)
over the period of a few minutes, you would record a range of values even though you know your weight is
essentially not changing. This sort of error cannot be eliminated, but can be managed by taking many
measurements to get a closer estimate of the mean.
Precision is the degree of agreement among a series of individual measurements or values. If you
weighed yourself 10 times on a bathroom scale, the readings may vary by as much as 5 pounds. However,
on a scale at the doctor's office, the same measurement may only vary by half a pound; we say that the latter
measurement is more precise. Precision is limited by random errors. It can be expressed as standard
deviation.
Accuracy is the degree of conformity of a measured value to the 'truth' (as defined above). Measurement
A is more accurate than Measurement B if the value of A is closer to the 'truth' than B is. When you zeroe
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your bathroom scale, you improve the accuracy of the measurement. Accuracy is limited by systematic
errors.
Note that you can have a high-precision, low-accuracy measurement, such as a doctor's scale that hasn't
been calibrated. Conversely, you can have a low-precision, high accuracy scale, such as a cheap bathroom
scale that happens to yield an average weight close to your true weight. Figure 1.1 below is a representation
of the permutations of accuracy and precision:
Figure 1.1
Representation of the permutations of accuracy and precision.
The reproducibility of an experiment is the measure of the ability to yield the same or consistent results
in different (statistical) trials. This means that when different people with different measuring equipment
(or with the same equipment at different times) perform the same measurement, they get values that are
within the range of the measurements of previous trials.
The resolution is the minimum difference between two measurements that can be distinguished by a
measuring device.
II.2
The Language of Statistics
This is the mathematics used for the quantitative description of variation. Following are some key
concepts used in statistical analysis.
The mean x is the average value of a set of measurements:
N
x
x
i 1
N
i
(1.1)
N
This formula tells us to add the N measurements, x1 , x2 , x3 ... to xN . This sum is written as
i 1
xi . Now
divide by N to get the mean value of x.
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The median is the center value of a set of measurements. If we list a set of N measurements from least
value to greatest value, the median (if the number of measurements is odd) is the n th number in the list,
N 1
1 . If the number of measurements is even, the median is the average of the two center
where n
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numbers, located at position n1 and n2, where n1 = N/2 and n2= N/2 +1.
Example: 1 4 5 7 9 (5 is the median.)
Example: 3 4 5 7 7 8 (select 5 and 7 and average, the median is 6)
The mode is the most frequently occuring magnitude. 4 7 7 7 7 4 5 2 8 7 7 7 Can you find the mode?
The standard deviation, is a measure of the spread of a set of repeated individual measurements,
,xN for N number of measurements made). A large value forimplies a large spread in
xi ( x1 ,x2 ,x3 ,...
the values whereas a smaller value implies a small or narrow spread (Equation 1.2).
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The variance is the standard deviation squared,
. In other words it is a measure of the deviation
of the measurements from the mean before they have been “normalized” by taking the square root. The
variance equals the average of the squared deviations from the mean.
The variation about the mean or standard deviation
of the mean x of a series of measurements
gives the variation in the mean itself (Equation 1.3). The more measurements made (the greater N is), the
narrower
will be.
Standard Deviation:
N

 (x
1
i
 x) 2
(N  1)
(1.2)
Standard Deviation of the Mean:
N


N
 (x
i
 x) 2
1
N  (N  1)
(1.3)
t-Test:
A t-test is a statistical method to compare two groups of data by assessing whether the means of two
groups are statistically different from each other. To apply the t-test, you have to consider if the sample sizes
are equal and the groups are independent. The different t-tests are:
t test of difference between means of independent groups
t test of difference between means with correlated observations
t test of linear combination of means
t test of Pearson's correlation
t test of the mean standard deviation estimated
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The most common t-test is a test for a difference between two means.
 Null-Hypothesis: The null hypothesis is a hypothesis about a population parameter. The purpose of
hypothesis testing is to test the feasibility of the null hypothesis given the experimental data.
Depending on the data, the null hypothesis either will or will not be rejected as a viable possibility.
 Significance Level: In hypothesis testing, the significance level is the criterion used for rejecting the
null hypothesis. The significance level is used in hypothesis testing as follows: First, the difference
between the results of the experiment and the null hypothesis is computed. Next, assuming the null
hypothesis is true, the probability of a difference that large or larger is calculated(??). Finally, this
probability is compared to the significance level. If the probability is less than or equal to the
significance level, then the null hypothesis is rejected and the outcome is said to be statistically
significant. Conventionally, experimenters use either the .05 level (sometimes called the 5% level)
or the .01 level (1% level), however the choice of levels is largely subjective.
 Independent and Dependent Variables: When an experiment is conducted, some variables are
manipulated by the experimenter and others are measured from the subjects. The variables that are
manipulated are called "independent variables"; or "factors," whereas variables that are measured
from the subjects are called "dependent variables" or "dependent measures."
Example 1: Calculating the mean and standard deviation of measurements of length
To illustrate the procedure we will work out the mean value x and the standard deviation  of a set of 21
individual data points, and then the predicted uncertainty (or error),
, of the set’s mean.
xi
(data point: length in
meters)
xi x
(deviation of data point
from mean)
(square of the deviation)
x1 = 15.68
x2 =15.42
x3 = 15.03
x4 =15.66
x5 =15.17
x6 =15.89
15.35
15.81
15.62
15.39
15.21
15.78
15.46
15.12
15.93
15.23
15.62
15.88
15.95
15.37
x21 = 15.51
SUM = 326.08 m
0.15
-0.11
-0.50
0.13
-0.36
0.36
-0.18
0.28
0.09
-0.14
-0.32
0.25
-0.07
-0.41
0.40
-0.30
0.09
0.35
0.42
-0.16
-0.02
SUM = -0.05 m
0.0225
0.0121
0.2500
0.0169
0.1296
0.1296
0.0324
0.0784
0.0081
0.0196
0.1024
0.0625
0.0049
0.1681
0.1600
0.0900
0.0081
0.1225
0.1764
0.0256
0.0004
SUM = 1.6201 m2
xi x
2
From the above table we can make the following calculations for N = 21 measurements
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N
Mean  x 
x
i 1
N
i

326.08
 15.53m
21
N
S tan dard Deviation   
(x
i
 x)2

i 1
N 1
1.6201
= 0.29m
20
N
 (x
S tan dard Deviation of the Mean= 
i 1
i
 x)2
N(N  1)

1.6201
 0.062m
(21)(20)
Hence
x    15.53  0.29m : This describes the distribution of the data.
x    15.53  0.062m : This describes the variation in the mean itself.
Increasing the number of individual measurements will give us a better estimate of where the mean is by
reducing the variation of the distribution, although it cannot reduce the random errors of the individual
measurements themselves. On the other hand, more measurements do not diminish systematic errors in the
mean because these are always in the same direction.
Often we must compare different measurements. Consider two measurements
A
A
B
B
If, for example, you want to calculate their difference, the error  of the combined population is given by
2
2
2
A
B
(Eq. 1)
We should expect that A  B   , if the two data sets are statistically the same. (i.e. the distributions are so
similar that they cannot be distinguished).
If the data sets diverge greatly from the expected standard deviation,
A  B   , then they are statistically different. Often, a value of 2 is used as a simplified reference for
being significantly different.
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Example 2: Peforming the t-test (From http://davidmlane.com/hyperstat ):
An experiment was conducted comparing the memory of expert and novice chess players. The mean
number of pieces correctly placed across several chess positions was computed for each subject and the
scores recorded for each subject.
Novice Players
Expert Players
32.5
37.1
39.1
40.5
45.5
51.3
52.6
55.7
55.9
57.7
Mean: MN = 46.79
 N = 9.03
40.1
45.6
51.2
56.4
58.1
71.1
74.9
75.9
80.3
85.3
Mean: ME =63.89
 E = 15.62
1. The first step is to identify the null hypothesis and an alternative hypothesis. For experiments testing
differences between means, the null hypothesis is that the difference between means is some specified
value. Usually the null hypothesis is that the difference is zero.
For this example, the null ( H 0 ) and alternative hypotheses ( H 1 ) set the difference between
populations means (I). They are respectively:
H 0 : 1   2  0
H1 : 1   2  0
2. After that, the second step involves choosing a significance level. For this experiment, we assume that
the significance level is 0.05.
3. Next, the difference between the sample means ( M D ) is calculated:
M D  M E  M N  63.89  46.79  17.10
4. The fourth step is to find the probability p (or probability value) of obtaining a difference between
statistic and the value specified by the null hypothesis (0) as large or larger than the difference obtained
in the experiment. The general formula for testing the hypothesis is:
t
Statistic  Hypothesized value
Standard error of the statistic
Applying the general formula:
t
M D  (1   2 )
SM D
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where,
MD
1   2
SM D
= the difference between sample means,
= the difference between population means specified by the null hypothesis (usually zero)
= the estimated standard error of the difference between means
The estimated standard error, SM D is computed assuming that the variances in the two populations are equal.
If the two sample sizes are equal such that n1 = n2 then the population variance σ2 (it is the same in both
populations) is estimated by as follows:
First MSE = Mean square error is calculated:
MSE 
(  N 2   E 2 ) ((9.03)2  (15.62) 2 )

 162.79
2
2
where
 E = Standard deviation of the expert players
 N = Standard deviation of the novice players
Next, estimated standard error, SM D can be calculated by the following formula when the variance is known:
SM D 
2  MSE
2 162.9

 5.706
n
10
where n = n1 = n2
Using the value for SM D , t is equal to:
t
M D  (1   2 ) 17.10  (0)

 2.99
SM D
5.706
The probability value for t can be
http://davidmlane.com/hyperstat/t-table.html.
determined
using
a
t
table
which
is
given
at:
For the t value: 2.99 at 0.05 significance level, the probability is found out to be: p = 0.008.
The degrees of freedom for t (df) is equal to the degrees of freedom for MSE:
df  n1  1  n 2  1  18
df  N  2
where N = n1 + n 2
5. When we compare the probability calculated in Step 4 to the significance level chosen in Step 2, we can
see that the probability value (0.008) is less than the significance level (.05); therefore the effect is
significant.
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5. Because the effect is significant, the null hypothesis is rejected. Therefore, we can conclude that the
mean memory score for experts is higher than the mean memory score for novices.
A histogram is a bar graph that indicates how often a measurement falls within a particular range of
values. On the abscissa (x-axis) the range of all possible measured values is subdivided into smaller
intervals termed “bins”. The ordinate (y-axis or height of each of the bars) gives the number of
measurements in each bin. In what is termed a “normal” distribution, the mean value of a measurement
always has the greatest frequency on the histogram because it is the value most likely to come up when a
measurement is made. The further away we get from the mean, the closer the frequency gets to 0. (For
example, in the bin labeled “2-3 ft”, we would expect zero entries when measuring the heights of players on
the Celtics, while the bin labeled 7-8 ft would contain the names of many of the players.) If the bins are
made finer and finer, the bars of the bar graph blend together and smooth into a curve.
So what might be the shape of this curve? Interestingly, for most scientific measurements, the shape is
looks like a bell. This type of distribution of data is called the normal or Gaussian distribution. The curve
is often called a bell curve. When we look at a bell curve, the standard deviation of the data tells us
something about the width of the curve.
Figure 2
It turns out that the area under the bell curve centered around the mean between x +  and x –  (a width
= 2 ) is 68.3% of the total area under the curve. For the data set on page 5, and the mean is
so 68.3% of the values are between 15.24 and 15.82. We could also say that the probability, P, that a
measurement taken falls between 15.24 and 15.82 is P = 0.683. Likewise, 2and% of the
values are between 14.95 and 16.11. For 3the probability that a measurement falls between 14.66
and 16.40 is P = 0.997.
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We can talk about the confidence interval of a parameter of interest, such as the mean. The confidence
interval for the mean is a range of values within which the 'true' mean is likely to lie with a probability or
confidence level that can be specified. For example, a confidence interval of the mean with a confidence
level of 0.9 tells us that there is a 90% chance that the 'true' mean can be found within this range of values.
The confidence interval for the mean for a selected confidence level can be calculated from the histogram
,xN calculated from N sets of data points) whose standard deviation
of a series of means ( x1 ,x2 ,x3 ,...
would be
(similar to the histogram in Figure 2). The confidence level is similar to as the value of P in
Figure 2, and is also calculated as the percentage of the total area under the curve bounded by the confidence
interval, the width or range of values bounding the area under the curve within this range.
II.3
Measurements of Biological Systems
Like physical systems, measurements of biological systems are also subject to both random and
systematic errors that arise as a result of imperfections in measurement systems. When dealing with
biological systems, however, there will also be systematic variability inherent in the biological system itself.
The nature of the biological system or the conditions under which an experiment is carried out may cause
the data collected to be skewed or shifted. Statistical analysis can be used to find trends in the data that
underlie such shifts. For example, if we want to study reaction times to a stimulus in humans, there may be
certain factors affecting the distribution of the data such as age, gender or physical condition of the subject,
duration of the experiment, or variation in stimulus type and method of response. Various aspects and
differences in the physiology of the nervous system may underlie statistical variations resulting from these
factors.
III.
Purpose
In this lab, we would like to identify and model the biological variability of a stimulus- response system in
humans by subjecting participants to test conditions that are meant to probe various aspects of response
time.
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IV.
Materials and Methods
IV.1
Apparatus
Hardware




Keyboard
Line of tape
Weighted wristband
Headphones and audio
Software:



IV.2
Labview
Matlab (coded within from Labview as command lines)
Stopwatch.exe (reaction time software)
Procedure for Reaction-Time Study
Collection of Response Time Data
What to collect
In this lab, you will hit a key on the keyboard in response to a visual stimulus in the form of the appearance
of a face on the screen. Your response time to the stimulus will be recorded. Collect data as described
below under the heading “2.B. Data Collection Method”.
1) Collect two sets of control data, one set by pressing the spacebar (data set #1) and one set by
pressing the enter key (data set #2). See Table 1.
2) Collect a third set of data (data set #3) from Table 1. You may use either the enter key or spacebar
for this set, but data must come entirely from one key or the other ONLY. The relevant control set
(#1 or #2) should match the key chosen.
3) You may collect additional data sets from Table 1 (data sets #4-7 or even make up your own) for
extra credit.
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Table 1: Response Time Tests
Data collection rate = approx. 11 data points / minute
Data set
Action
Duration
Additional
Constraints
Condition Tested
#1
Press Spacebar
10 min
---
control
#2
Press Enter key
10 min
---
control
#3*
Spacebar or Enter
30 min
---
mental fatigue
#4
Spacebar or Enter
10 min
weighted wristband
physical fatigue
#5
Spacebar or Enter
10 min
audio playing
audio distraction
#6
Spacebar or Enter
10 min
use weaker hand
handedness
Spacebar or Enter
10 min
12 inch distance**
distance delay
#7
* See Data Analysis Section, Task 4. for additionally required data processing
** Place line of tape 12 inches below base of keyboard
Data Collection Method
1. Open stopwatch.exe, on the PC's desktop. Select “2) Reaction time”, then immediately place your hand
in the appropriate “starting position” (see Table 1).
Each time the face appears in the small box in the Stopwatch window hit the appropriate key (Enter or
Spacebar). The time in seconds between the stimulus and your response will be recorded. Click “1) Stop
watch” with the mouse when you are done collecting data.
2. To Practice: Select “Clear record” then “2) Reaction Time”. Practice the response method until you can
do it uniformly. Press “Clear record” again when you are ready to begin.
3. Collect data (see Table 1). Press “2) Reaction time”to start.
4. Your partner should record any information about the experiment that they feel could affect the outcome.
For example: interruptions by teacher, not ready, distractions, broken fingers, etc.
5. For each trial (data set), when you are finished with your last button press,
Press “1) Stop watch” to stop the timer.
6. Select the column of reaction times you have just generated by clicking and dragging with the mouse (or
right-clicking with the mouse and choosing Select All), until you have highlighted all the points. Copy
the data into the Windows clipboard (first part of “copy and paste”) either by using Ctrl-C (or rightclicking and choosing Copy).
7. Paste the data set to a .txt file. Save the file and name using student's initials and set number (NL1, NL2,
etc.).
8. Repeat steps 3-7 for each student in group, for all of the trials listed in Table 1.
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Data Preprocessing
If you have any outliers in your data sets (points very far from the mean), think about the cause.
Refer to the experimental notes you wrote during test time. If these points were generated due to human
error, such as not being ready to click the button, remove them from the data column in the .txt file.
IV.3
Data Processing
You have been provided with the Labview elements needed to make a data viewer (as you learned
how to build in class). Open the Labview workspace provided to find these elements and wire them
together. Read in and view your data. Then in the same workspace, use the Statistics Virtual Instrument
(VI) to calculate statistical parameters (given below) and the Histogram VI to generate a histogram for each
of your data sets. The details are explained below.
Record all data collected into text files. Later, enter all data into an excel spreadsheet similar to the
table in the "Guideline Sheet" below. Organize the data in the spreadsheet so it can be easily
analyzed. (See Data Analysis and Interpretation Section.)
View Data
1. Double click on the icon "Lab1.vi" on the desktop.
The Lab1.vi Front panel (grey screen) will open.
2. In the menu bar click on "Window" then scroll to "Show Block Diagram" and click on it.
The Lab1.vi Block Diagram (a white screen workspace) will open.
Here you will see three unwired elements:
1) "Read from spreadsheet file.vi” (a function)
2) “Transpose 2D array” (a function)
3) "Waveform Graph" (a control)
3. Wire these elements together.
4. Create a path to your data file: Go to the Front Panel and right click to view "controls".
Move the mouse to "all controls" to "string & path" to "file path control", select and drop the path icon
into the Front Panel.
5. Click the "open file" icon and browse for the data file of your choice and press "OK".
6.Click the "run" button.
You will now see a plot of the data.
7. Adjust the graph title to match the data file, y-axis to "Response Time" and x-axis to "Number of Data
Points".
Calculate Statistics
For each data set collected, determine the minimum x M IN , maximum x M A X , mean x ,
variance ^2, standard deviation , standard deviation of the mean
set. (Note: standard deviation of the mean is not directly given.)
, median and mode of the data
1. In the block diagram, right click to get the list of "functions" and find the “statistics” VI. Select the
statistical parameters desired, click "OK" and drop the icon into the block diagram.
2. Wire the elements together.
3. Right click to create numeric indicators for each of the statistical parameters.
13
4. Go to the Front Panel and click the "run" button.
You will be prompted for the data file.
5. Browse for and click on the data file of your choice and press "OK".
You will now see the parameter values.
6. Record these values in the hand-in sheet or create a spreadsheet to enter the data into.
Create Histogram for each data set
1. In the block diagram, right click get the "functions" list and find the “histogram” express VI. Click "OK"
(later, you can adjust the number of bins) and drop the icon into the block diagram.
2. Wire it up.
3. Right click to create graphic indicators for each of the statistical parameters.
4. Go to the Front Panel, click "run" and load data file.
You will see the histogram.
5. Go to the histogram icon in the block diagram and select “properties” if you need to adjust the bin size
(the resolution of the histogram) and to adjust the minimum and maximum values to get the best possible
shape. The optimum bin size is the smallest value that does not give “artificial” zeroes in the central portion
of the distribution (because there are no data points in this bin).
6. Note the final shape of the histogram.
To create a text boxes for notes and student information
1. In the front panel right click to view "controls". Move the mouse to "text controls" to "string control",
select and drop the "text box" icon into the front panel. Expand the size of the text box and enter you name,
lab partner's name, section and date.
2. Add any additional information you like to describe the data set or conditions/occurrences during the
experiment.
When you are ready to process the next data set, you can copy the entire wiring diagram.
1. In the block diagram, move the mouse to select the entire wiring diagram.
2. Copy it using Control+C on the keyboard.
3. Find a spot on the block diagram away from the wiring diagram and double click on that spot. A black
square will appear.
4. Press Control+V to paste the new wiring diagram. (Be careful that the new wiring diagram does not
overlap the old. If it does you can press control+Z and redo this step.
5. On the front panel, a new set of icons will appear offset from the original ones. Select all new icons and
move them to a convenient location in the front panel.
6.Click the "open file" icon and browse for the data file of your choice and press "OK".
7.Click the "run" button. You will now see a plot of the new data and statistics.
8. Repeat for all data sets.
IV.4
Data Analysis and Interpretation
You can make use of the excel spreadsheet to manage your data and compare different data sets. Extend the
spreadsheet to include the following analysis. (For example, to calculate the difference between the means
of two data sets, you just create a cell whose formula references the cells that hold the data in each set.)
14
SAMPLE EXCEL SPREADSHEET
1
A
B
C
D
Mean of set 1
0.233
Difference
=abs(B1-B3)
Mean of set 2
0.352
2
3
1. Calculate the difference (or shift) in mean, standard deviation, standard deviation of the mean, mode and
median between each of the following sets (remember that the standard deviation of a difference is
NOT the difference of the standard deviations!See Eq. 1 at the bottom of page 6):
set #1 and set #2
all sets using spacebar and set #1
all sets using enter key and set #2
2. Calculate the difference between data sets of the same type for different students (for example: you and
your lab partner) for sets #1, 2 and any other common sets.
3. Note any differences in shape between the histograms of the above pairs. Is all of the data you collected
normally distributed? Which sets seem to be more normally distributed?
Which of the effects (mental fatigue, physical fatigue, distraction, distance, key change on keyboard)
causes the greatest shift in each of the above calculated quantities? (mean, variance standard deviation,
standard deviation of the mean, mode and median)
4. ** For set #3: Calculate all statistical differences and differences in histogram shape between time
segments of the same set:
0-10 min segment
and. 10-20 min segment
0-10 min segment
and 20-30 min segment
What happens as progressively later segments are compared with the beginning segment? Why?
Also calculate statistics and plot histograms for data set #3 for:
0-1 min segment
0-2 min segment
0-5 min segment
0-10 min segment
0-20 min segment
0-30 min segment
What happens as the data sets get longer? How does the shape of the histogram vary between long data sets
and short data sets? What is the threshold time for obtaining a good Gaussian distribution?
4. (Optional) Calculate confidence intervals for each of the data sets at confidence levels of 0.80, 0.90, 0.95
and 0.99.
5. Based on the comparisons made in tasks 1- 4 of this section can you draw any conclusions about
physiological response? (Hint: Examine “conditions tested” in Table 1 and think about why the pairs of
trials above were selected for comparison.)
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6. Find literature to support your ideas; Here are some good references you can use:
 http://davidmlane.com/hyperstat
 http://www.sportsci.org/resource/stats/index.html
 http://www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html
 http://physics.web.cern.ch/Physics/DataAnalysis/BriefBook/
 http://www.stat.berkeley.edu/~stark/SticiGui/Text/index.htm
 http://members.aol.com/johnp71/javastat.html
V. Lab Report Guidelines
V.1 Report Format
Introduction: Motivate lab in your own words (personal assessment of topic importance); Give
justification for method used in the lab
Methods:
Description of measurements. Describe experimental protocol and care taken to perform lab
Results: Data (Objective summary of results) Focus on organization of data. This is where the Excel
spreadsheet will be useful.
Analysis: Meaning of results, how statistical analysis can help. What conclusions can or cannot be made
and why?
Include answers to questions from “Data Analysis and Interpretation” above as well as data in Excel
Spreadsheet format.
Human Error: You may want to refer to your lab notes on what happened while collecting data.
Discuss the meaning of outlying points and why you may have included some and discarded others.
Summary & Conclusions: Include only overall trends or findings from Analysis Section. There need be
no more than 3 or 4 main points.
References: (quote all sources used, including web sources, no plagiarism)
Sample Excel Spreadsheet for Results section
Name _______________ Left / Right Handed _______________
Data set
Key
#1
Spacebar
#2
Enter key
#
Ent or Spc
#
Ent or Spc
x M IN
xM A X
x
2
median
mode
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V.2 Analysis Guide Questions
The following are some more questions to help you with analysis – they are just guidelines to help you
to think critically.
Justify all answers by reasoning based on the data collected.
1. a. Based on your calculations above, are reaction times from the Spacebar and Enter keys similar or
different? What criteria and tests did you use to compare?
b. Were your and your lab partner's reactions times for “spacebar key only” and “enter key only” similar
or different? What criteria and tests did you use to compare? Hint: The two groups might have different
sample sizes; therefore make sure you apply the right test.
2. Does the data you collected contain measurement error? biological variability? How do you know?
3. What are some sources of systematic error, random error and biological variation in this experiment?
4. Which of the conditions tested is influenced most by human physiology and why? Which test conditions
are influenced by physiology the least? Why? (e.g. something that might affect response time, which has
no effect on body function)
5. How did you decide on the significance level when you performed the t-test? (Which significance level
would you use to get 95% confidence interval?)
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