Analyzing Data from an Experiment
Now that I have my experimental data, how do I begin to understand them?
Contents
I. Once I have my experimental data, how do I begin to understand them? ................................................. 2
II. What type of data do I have? .................................................................................................................... 3
Discrete data: ........................................................................................................................................... 3
Continuous data: ...................................................................................................................................... 3
Categorical data: ...................................................................................................................................... 3
Normal Data ............................................................................................................................................. 4
III. Is my data paired or unpaired? ................................................................................................................ 5
IV. Do I want to compare my values to the mean or to the standard deviation? .......................................... 6
V. Testing a Hypothesis ................................................................................................................................ 7
Why p < 0.05 ? .......................................................................................................................................... 8
VI. Using an Unpaired T-Test....................................................................................................................... 9
VII. Using a Chi-Square Test for Variance................................................................................................. 13
VIII. Using a Paired T-Test ........................................................................................................................ 15
IX. Using a Chi-Square Test for Goodness-of-Fit ...................................................................................... 17
Appendix 1: Critical Values for a T-Test.................................................................................................... 19
Appendix 2: Critical Values for a Chi-Square Test .................................................................................... 20
Created by the HHMI-FT team at the University of North Carolina at Chapel Hill, 2013.
Analyzing Data from an Experiment
I. Once I have my experimental data, how do I begin to understand them?
The point of analyzing any quantitative data that you collect in an experiment is to see if it does
or does not support your original hypothesis. One way to determine whether or not this happens
is to use a statistical test, which can tell you whether or not your data are statistically significant.
There are many different types of data that you can collect in your experiment, and each type has
its own statistical test to determine significance. However, it can sometimes be difficult to figure
out exactly what test to use, and exactly how to use that particular test.
Follow this flow chart and use the resources below to match the type of data you have to the
appropriate statistical test!
Page 2 of 22
Analyzing Data from an Experiment
II. What type of data do I have?
Data can be collected in a variety of forms.
When data are presented as numbers (known as numerical or quantitative data), instead of
words (known as categorical, descriptive, or qualitative data), they can be either continuous
or discrete.
Discrete data:
 This is in contrast with continuous data. If your data is discrete, the statistical tests on
this page are not appropriate for your calculations.
 Discrete data can be counted. It is made of numbers that cannot be split up into an infinite
amount of values.
o Some examples of variables that would be discrete are shoe sizes or the counts of
an item (number of tomatoes on a vine, number of people in a room). Something
like a size 9.452 shoe or 12.8516 tomatoes is not a meaningful piece of data.
Continuous data:
 Continuous data Continuous data is data that can be measured. It has an infinite number
of points that exist within a range of values, and any of the values have a specific
meaning.
 For example, consider 0 to 100, -17.5 to 9371, or -infinity to +infinity as possible ranges.
o In the range of 0 to 100, you can get whole numbers like 1, 2, 3, or 4, but you can
also get an unlimited amount of values between each whole number, like 1.1,
1.001, 1.0001, and so on. If you were to visualize a set of continuous data, you
would be able to draw it on a graph without picking up your pencil.
 Some examples of data sets that would be continuous:
o growth measures (height, finger length, foot width)
o things that depend on time (acceleration of a car, walking speed)
o distances (how far each person walks on an average day)
 Continuous data can be represented on a coordinate plane or scatterplot.
 Some examples of how a continuous graph looks:
On the other hand, when data is presented with words that can be described in numbers, it is
categorical data.
Categorical data:
Page 3 of 22
Analyzing Data from an Experiment



Some examples of data that would be categorical:
o types of music played on the radio and how many stations play each type
o the number of each species of animal in an area
o the number of hotels in each state of the US.
Categorical data can be represented on a bar graph, pie chart, or a dot chart (a chart that
shows the percentage of each category in the data). The data is split into different
categories (hence the name!) and the numbers describe each category.
Some examples of how a graph of categorical data looks:
Normal Data
 Many statistical tests are valid only if the population (the body from which measurements
were taken) has a normal distribution. For example, if a study was done on the heights
of kindergarteners in a class, the population would be all kindergarteners in general.
Normally distributed populations have certain measures associated with symbols that are
commonly used in statistical calculations.
 A normal distribution is data that looks like a bell curve when plotted on a density
curve (the independent axis showing the values of the data points and the dependent axis
showing the frequency of those values). The bell curve is symmetric and can describe
many bodies of data, with a majority of the subjects in the middle range and fewer and
fewer subjects in the lower and higher ranges.
 A normal distribution looks like this:
Page 4 of 22
Analyzing Data from an Experiment
III. Is my data paired or unpaired?
Paired data is data in a study that includes two sets of measurements about the same observed
subject.
Some general situations involving paired data:
 before and after (the height of a group of plants plant before applying a fertilizer and the
height of the same group of plants after applying fertilizer)
 comparing closely related things (weight gained over a span of time in sets of twins, the
number of nights experiencing insomnia in married couples)
The body of subjects in a set of paired data is called a matched population, because each
subject in the population is matched with another subject.
Independent populations, on the other hand, contain data that do not affect each other. Two or
more sets of measurements are not influenced in any way by each other. These measurements
would be unpaired.
Some situations involving unpaired data:
 comparing a sample to an expected standard (average temperature of a city for 100 years
compared to data taken over the last two years)
 comparing a treatment group to a control group (sleep time of a group treated by a
sleeping pill compared to sleep time of a group not treated by a sleeping pill)
Page 5 of 22
Analyzing Data from an Experiment
IV. Do I want to compare my values to the mean or to the standard deviation?
A population includes all possible objects of interest, whereas a sample includes only a portion
of the population. For example: in a study of United States middle school students, the
population is every middle school student in the United States and a sample would be the middle
schoolers in one school.
The mean (represented by μ ("mu") for a population and x ("x-bar") for a sample) is calculated
like any arithmetic mean – by adding all of the values and dividing it by the total number of
values.
The standard deviation (represented by σ ("sigma") for a population, and s for a sample) or
sample standard deviation (s) is a number that describes the spread of the numbers in a set of
data, or how far apart the numbers are. The larger the value of the standard deviation, the more
spread out the numbers. It is calculated by taking the square root of the average distance that
each point is from the mean.
Standard deviation is calculated using these formulas:
The degrees of freedom (df) in a statistical study is one less than the total number of subjects.
For instance, if 100 people were surveyed in a study, the degrees of freedom would be 99.
Page 6 of 22
Analyzing Data from an Experiment
V. Testing a Hypothesis
In performing statistical analysis, hypotheses must first be designated for determining statistical
significance.
The null hypothesis (represented by h0) is the statement being tested and usually assumes that
the expected effect in a study is NOT true.
The alternate hypothesis (represented by h1) is the statement of the expected effect and is
accepted if the null hypothesis can be rejected based on the statistical calculations.
For instance, in determining whether there is a significant difference in the average grades of one
class versus another class, the hypotheses would look like this:
 Null hypothesis (h0) : There is no difference in the average grades of the two classes.
 Alternate hypothesis (h1) : There is a difference in the average grades of the two classes.
An alpha value (α) is used to determine what qualifies as statistically significant. The alpha value
is the percentage likelihood that the difference found in the calculations is due to chance.
To determine the alpha value for a given statistical test, you must compare your statistic to a
table of critical values. For example, for a t-test, you would compare your t-statistic, given your
specific degrees of freedom, to find your alpha value.
 A table of critical values for T-tests can be found on page 17.
 A table of critical values for Chi-Square tests can be found on page 18.
Page 7 of 22
Analyzing Data from an Experiment
Why p < 0.05 ?
An alpha value (α) is set at a certain value to determine what qualifies as statistically significant.
The p value is the percentage likelihood that the difference found in the calculations is due to
chance.
The scientifically accepted alpha value for statistical significance is 0.05, which says that there is
a 0.05 or 5% likelihood that the data is due to chance. So if you were to find a p value lower than
0.05, your calculations would show the research is statistically significant.
In testing for the validity of an alternate hypothesis, you need to decide whether to use a one- or
two-tailed test.
 A one-tailed test determines whether the sample in question is either statistically higher
than an expected value or statistically lower than an expected value. The test only
determines whether it is ONE of these extremes and does not account for statistical
differences in the other extreme.
 A two-tailed test determines whether the sample in question is either statistically higher
OR lower than an expected value. The test determines whether it is either one of these
extremes in the same test.
If you find that your results are statistically significant, then you can reject the null hypothesis
and accept the alternate hypothesis. If you do not find that your results are statistically
significant, then you do not have a conclusive statistical analysis.
Page 8 of 22
Analyzing Data from an Experiment
VI. Using an Unpaired T-Test
One Hypothesis
Null Hypothesis: The sample mean is equal to the population mean.
How the test works:
 The test statistic is calculated using a formula that has the difference between the means
in the numerator; this makes the test statistic get larger as the means get further apart. The
denominator is the standard error of the difference in the means, which gets smaller as the
sample variances decrease or the sample sizes increase. Thus the test statistic gets larger
as the means get farther apart, the variances get smaller, or the sample sizes increase.
 The probability of getting the observed test statistic value under the null hypothesis is
calculated using the t-distribution. The shape of the t-distribution, and thus the probability
of getting a particular test statistic value, depends on the number of degrees of freedom.
The degrees of freedom for a t-test is the total number of observations in the groups
minus 2, or n1+n2-2.
Assumptions: The t-test assumes that the observations within each group are normally distributed
and the variances are equal in the two groups.
Example: In the past semesters for the last 5 years, students in the 5 p.m. section of my Biology
class had an average height of 64.6 inches. This year there were 5 students with the following
heights. Is the average heights of the previous sections significantly different from their heights?
Here is the data:
5 p.m. 68 62 67 68 69
1. Determine the mean.
Mean: (68 + 62 + 67 + 68 + 69) / 5 = 66.8
2. Determine the standard deviation.
∑(𝑥 − 𝑥̅ )2 = (68 − 66.8)2 + (62 − 66.8)2 + (65 − 66.8)2 + (68 − 66.8)2 + (69 − 66.8)2
Page 9 of 22
Analyzing Data from an Experiment
∑(𝑥 − 𝑥̅ )2 = 30.8
30.8
30.8
𝑠= √
= √
= √7.7 = 2.775
5−1
4
3. Calculate the t-statistic.
𝑡=
(𝑥̅ − 𝜇)
(66.8 − 64.6)
=
= 1.773
𝑠
2.775
5
√𝑛
4. Find the degrees of freedom.
df = n - 1 = 4
5. Use the degrees of freedom and your t-statistic to determine your p-value based on a
chart of critical values.
According to this table, 0.05 < p < 0.010, and thus the null hypothesis can NOT be
rejected.
Two Populations
Unpaired t-test
When to use: An unpaired t-test is used when you are comparing the means for two different groups or
populations. It is usually used when you are comparing the mean from an experimental group to the
mean of a control group. So the individuals in group 1 are not the same individuals in group 2.
How to do an unpaired t-test: Suppose you expose caterpillars to different temperatures (25°C and
33°C) to see if they develop into the next instar (stage) faster at the higher 33°C temperature.
25°C Caterpillars
(Control)
Days
33°C Caterpillars
(Experimental)
Days
Cat. 1
2
Cat. 4
3
Cat. 2
4
Cat. 5
2
Cat. 3
3
Cat. 6
3
Page 10 of 22
Analyzing Data from an Experiment
1. First identify null and alternative hypotheses:
Null hypothesis: The mean time for a caterpillar to develop into the next instar is the same
between the control and experimental group.
Alternative hypothesis: The mean time for a caterpillar to develop into the next instar is faster
in the experimental group than in the control group.
2. Now calculate the means for each group:
Control group: (2+4+3)/3 = 3
Experimental group: (3+2+3)/3 = 2.67
3. Now add the squares of the group mean minus each data value for both the control and
experimental group:
(control x-control mean)2 = (2-3)2 + (4-3)2 + (3-3)2 = 2
(experimental y- experimental mean)2
=
(3-2.67)2 + (2-2.67)2 + (3-2.67)2 = 0.667
4. Now Calculate variance:
So you can set it up like below (*note use your calculations from above to fill in the equation):
S2= (control x-control mean)2 + (Exp. 33°C y-Exp. 33°C mean)2
n1+ n2 – 2
S2= 2 + 0.667
3+3-2
S2 = 0.667
5. Now perform an unpaired t-test:
t=
3.0 - 2.67
= 0.495
√(0.667((1/3)+ (1/3))
6. Now look at the table of critical values of t-distributions and determine the range the p-value
must fall in (p<0.05 being significant) (degree of freedom= n1+n2 – 2).
Degree of freedom = 3+3-2= 4
Page 11 of 22
Analyzing Data from an Experiment
p>0.25
The means between the control and experimental group were NOT significantly different
therefore; you cannot reject the null hypothesis.
Page 12 of 22
Analyzing Data from an Experiment
VII. Using a Chi-Square Test for Variance
When to do this test?
 This test is useful if you have data that follows a normal distribution. This test compares
the data to the standard deviation. To use this test you need continuous data.
What is the equation?

Where:
o
o
o
n = the total number of data
s = sample variance (standard deviation)
σ = population variance (known variance)
What does it tell you?
 The chi-squared test for variance is similar to the chi squared test for goodness of fit. You
still have a null hypothesis and an alternative hypothesis. This test will tell you if the
variance in a sample matches the hypothesized variance.
 When you complete the chi-square test for variance on a data set, you will compare your
final chi-square value and your degrees of freedom to a chart of critical values. Your
critical value will then determine whether or not you can reject your null hypothesis. A
table of critical values for the chi-square test can be found on page 18.
Why is it useful?
 Imagine you work in a factory and know that the boxes get filled accurately to 50 pounds
with an average standard deviation of about 1 pound. You hire a new set of packers and
want to make sure they are as accurate as the last. You would use the chi-squared test for
variance to compare the standard deviation of the current packers to the known average
standard deviation of the factory.
 Overall, this test compares a set of data to the standard deviation and reveals the variance
in the data set compared to the expected variance in the data set.
Example:
A cereal manufacturer wishes to test the claim that the variance of sugar content of its cereals is
0.644. Sugar content is measured in grams and is assumed normally distributed. A sample of 20
cereals has a standard deviation of 1.00 gram. At α = 0.05, is there enough evidence to reject the
manufacturer’s claim?
1. State the hypotheses and identify the claim.
2. Find the critical values.
Since this test is a two-tailed test at α = 0.05, the critical values for 0.025 and 0.975 must
be found. The degrees of freedom are 19; hence, the critical values are 32.852 and 8.907,
Page 13 of 22
Analyzing Data from an Experiment
respectively. Find this using a chi squared table. A table showing critical values for a chisquare test can be found on page 17.
3. Compute the test value.
Since the standard deviation s is given in the problem, it must be squared for the formula.
4. Make the decision.
Do not reject the null hypothesis, since the test value falls between the critical values
region.
5. Summarize the results.
There is not enough evidence to reject the manufacturer’s claim that the variance of the
sugar content of the cereals is equal to 0.644.
Page 14 of 22
Analyzing Data from an Experiment
VIII. Using a Paired T-Test
Purpose: The paired t-test usually compares the mean of a population before and after a
treatment. For a paired t-test, you must take the mean of the difference between the two pairs
(before and after). This test assumes that the difference between pairs is normally distributed.
How to do a paired t-test: Suppose you take the blood pressure of three sixth graders before they
eat a piece of cake and again 30 minutes after they eat the cake and get the following:
Sixth Grader
Student 1
Student 2
Student 3
Blood Pressure Before Eating Cake
115
122
118
Blood Pressure After Eating Cake
117
123
121
You want to know if their blood pressure is significantly higher after they eat the cake than
before, so you decide to use a paired t-test.
1. First you must find the differences:
Student 1: 117 - 115 = 2
Student 2: 123 - 122 = 1
Student 3: 121 - 118 = 3
2. Now calculate the mean:
2+1+3=6
6/3=2
3. What is the null hypothesis?
The null hypothesis assumes that the mean difference between the pairs is zero.
So H0 = 0, which in the equation is subtracted from the mean difference you calculated in
step 2.
4. What is the alternative hypothesis?
The alternative hypothesis assumes that the mean difference in blood pressure will be
higher after eating the cake. Therefore, the alternative hypothesis would be H1: μ > 0.
5. Now let’s calculate standard deviation:
s = (2-2)2 + (1-2)2 + (3-2)2 = 2
s = 2/ (3-1) = 1
Page 15 of 22
Analyzing Data from an Experiment
6. Now we can do our paired t-test:
t = 1.73
7. What are our degrees of freedom?
Because df = n - 1, our degrees of freedom will be 2 for this example.
8. Find our p-value.
Now we want to use our t-statistic and our degrees of freedom to find our p-value from a
chart of critical values. A chart of critical values for the T-test can be found on page
17. Based on this chart, we find that:
0.10 < p < 0.15
9. Can our null hypothesis be rejected?
If we consider p < .05 to be statistically significant, we can NOT conclude that the mean
difference in blood pressure for these students before and after eating a piece of cake is
statistically significant. Therefore, we cannot assume that blood pressure is higher after
eating a piece of cake.
Page 16 of 22
Analyzing Data from an Experiment
IX. Using a Chi-Square Test for Goodness-of-Fit
A Chi Square Goodness of Fit test can be used when you have categorical data (ex: males and
females or heads and tails of a coin). After completing an experiment, you may want to
determine whether your observed results match your expected results (hypothesis). A Chi Square
Goodness of Fit Test will allow you to determine this.
Performing a Chi Square Goodness of Fit Test:
1. Specify a null hypothesis.
The null hypothesis states that there is no significant difference between observed and
expected. In this scenario the probability is greater than .05 (p > .05)
2. Specify an alternative hypothesis.
The alternative hypothesis states that there is a significant difference between the
observed and expected. However, this test only tells you that there is a difference. It does
not tell you what the difference is.
In this scenario, the probability is less than .05 (p < .05).
o What does this mean? Imagine you are flipping a coin 100 times. You want to
know how many times you land on heads and tails. You expect that you will land
on heads 50 times and tails 50 times. These are your expected results. However,
after performing the experiment, you land on heads 43 times and tails 57 times.
These are you observed results. You can visualize in a table like this:
Heads Tails
50
Expected 50
57
Observed 43
3. Now it is time to do the math.
X2 = [ (43-50)2 / 50] + [ (57-50)2 / 50 ]
X2 = 0.98 + 0.98 = 1.96
Now that we know the chi square value is 1.96, we will move on to the chi square table.
4. Finding degrees of freedom:
Degrees of freedom is a fancy term for number of categories minus 1. In this experiment
we have two categories (heads and tails), so df = 2 – 1.
5. Reading the chi square table:
o A table showing critical values for a chi-square test can be found on page 18.
o Degrees of Freedom are listed down the left side of the table. Find the degrees of
freedom that corresponds to your sample. In this case df = 1.
Page 17 of 22
Analyzing Data from an Experiment
o
o
Now, you go across the table until you come across you chi-square value. In this
case X2 = 1.96.
Once you have found this value, go up to the top of the table and find the alpha
value. According to the table, our alpha value is 0.15.
6. Testing Significance:
You must now compare the alpha value and the p-value of 0.05.
0.15 > 0.05
7. Conclusion:
Refer back to what the null and alternative hypotheses mean. Because 0.15 is greater than
0.5 we can say that there is no significant difference between the observed and the
expected. In other words, we fail to reject the null hypothesis. (However, you cannot state
that the null hypothesis was proven because in science it is very hard to prove something
is 100% true all the time.) Additionally, we reject the alternative hypothesis. This means
that this test shows there is not a significant difference between 57:43 (observed) and
50:50 (expected). Therefore, these results were due to chance and not an underlying
factor (such as a “trick” coin).
8. Going a step further: If the alpha value had been 0.03, would we have rejected or
failed to reject the null hypothesis?
Rejected.
Page 18 of 22
Analyzing Data from an Experiment
Appendix 1: Critical Values for a T-Test
1 tail α =
2 tails α =
df =1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
60
120
∞
0.1
0.2
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.296
1.289
1.282
0.05
0.1
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.671
1.658
1.645
0.025
0.05
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.000
1.980
1.960
0.01
0.02
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.390
2.358
2.326
0.005
0.01
63.656
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.660
2.617
2.576
*Taken from http://www.statisticsmentor.com/tables/table_t.htm
Page 19 of 22
Analyzing Data from an Experiment
Appendix 2: Critical Values for a Chi-Square Test
DF
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
0.995
0.000039
0.0100
0.0717
0.207
0.412
0.676
0.989
1.344
1.735
2.156
2.603
3.074
3.565
4.075
4.601
5.142
5.697
6.265
6.844
7.434
8.034
8.643
9.260
9.886
10.520
11.160
11.808
12.461
13.121
13.787
14.458
15.134
15.815
16.501
17.192
17.887
18.586
19.289
19.996
20.707
21.421
22.138
22.859
23.584
24.311
25.041
25.775
26.511
27.249
27.991
28.735
29.481
30.230
30.981
31.735
0.975
0.00098
0.0506
0.216
0.484
0.831
1.237
1.690
2.180
2.700
3.247
3.816
4.404
5.009
5.629
6.262
6.908
7.564
8.231
8.907
9.591
10.283
10.982
11.689
12.401
13.120
13.844
14.573
15.308
16.047
16.791
17.539
18.291
19.047
19.806
20.569
21.336
22.106
22.878
23.654
24.433
25.215
25.999
26.785
27.575
28.366
29.160
29.956
30.755
31.555
32.357
33.162
33.968
34.776
35.586
36.398
0.20
1.642
3.219
4.642
5.989
7.289
8.558
9.803
11.030
12.242
13.442
14.631
15.812
16.985
18.151
19.311
20.465
21.615
22.760
23.900
25.038
26.171
27.301
28.429
29.553
30.675
31.795
32.912
34.027
35.139
36.250
37.359
38.466
39.572
40.676
41.778
42.879
43.978
45.076
46.173
47.269
48.363
49.456
50.548
51.639
52.729
53.818
54.906
55.993
57.079
58.164
59.248
60.332
61.414
62.496
63.577
0.10
2.706
4.605
6.251
7.779
9.236
10.645
12.017
13.362
14.684
15.987
17.275
18.549
19.812
21.064
22.307
23.542
24.769
25.989
27.204
28.412
29.615
30.813
32.007
33.196
34.382
35.563
36.741
37.916
39.087
40.256
41.422
42.585
43.745
44.903
46.059
47.212
48.363
49.513
50.660
51.805
52.949
54.090
55.230
56.369
57.505
58.641
59.774
60.907
62.038
63.167
64.295
65.422
66.548
67.673
68.796
0.05
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652
38.885
40.113
41.337
42.557
43.773
44.985
46.194
47.400
48.602
49.802
50.998
52.192
53.384
54.572
55.758
56.942
58.124
59.304
60.481
61.656
62.830
64.001
65.171
66.339
67.505
68.669
69.832
70.993
72.153
73.311
0.025
5.024
7.378
9.348
11.143
12.833
14.449
16.013
17.535
19.023
20.483
21.920
23.337
24.736
26.119
27.488
28.845
30.191
31.526
32.852
34.170
35.479
36.781
38.076
39.364
40.646
41.923
43.195
44.461
45.722
46.979
48.232
49.480
50.725
51.966
53.203
54.437
55.668
56.896
58.120
59.342
60.561
61.777
62.990
64.201
65.410
66.617
67.821
69.023
70.222
71.420
72.616
73.810
75.002
76.192
77.380
0.02
5.412
7.824
9.837
11.668
13.388
15.033
16.622
18.168
19.679
21.161
22.618
24.054
25.472
26.873
28.259
29.633
30.995
32.346
33.687
35.020
36.343
37.659
38.968
40.270
41.566
42.856
44.140
45.419
46.693
47.962
49.226
50.487
51.743
52.995
54.244
55.489
56.730
57.969
59.204
60.436
61.665
62.892
64.116
65.337
66.555
67.771
68.985
70.197
71.406
72.613
73.818
75.021
76.223
77.422
78.619
0.01
6.635
9.210
11.345
13.277
15.086
16.812
18.475
20.090
21.666
23.209
24.725
26.217
27.688
29.141
30.578
32.000
33.409
34.805
36.191
37.566
38.932
40.289
41.638
42.980
44.314
45.642
46.963
48.278
49.588
50.892
52.191
53.486
54.776
56.061
57.342
58.619
59.893
61.162
62.428
63.691
64.950
66.206
67.459
68.710
69.957
71.201
72.443
73.683
74.919
76.154
77.386
78.616
79.843
81.069
82.292
0.005
7.879
10.597
12.838
14.860
16.750
18.548
20.278
21.955
23.589
25.188
26.757
28.300
29.819
31.319
32.801
34.267
35.718
37.156
38.582
39.997
41.401
42.796
44.181
45.559
46.928
48.290
49.645
50.993
52.336
53.672
55.003
56.328
57.648
58.964
60.275
61.581
62.883
64.181
65.476
66.766
68.053
69.336
70.616
71.893
73.166
74.437
75.704
76.969
78.231
79.490
80.747
82.001
83.253
84.502
85.749
0.002
9.550
12.429
14.796
16.924
18.907
20.791
22.601
24.352
26.056
27.722
29.354
30.957
32.535
34.091
35.628
37.146
38.648
40.136
41.610
43.072
44.522
45.962
47.391
48.812
50.223
51.627
53.023
54.411
55.792
57.167
58.536
59.899
61.256
62.608
63.955
65.296
66.633
67.966
69.294
70.618
71.938
73.254
74.566
75.874
77.179
78.481
79.780
81.075
82.367
83.657
84.943
86.227
87.507
88.786
90.061
0.001
10.828
13.816
16.266
18.467
20.515
22.458
24.322
26.124
27.877
29.588
31.264
32.909
34.528
36.123
37.697
39.252
40.790
42.312
43.820
45.315
46.797
48.268
49.728
51.179
52.620
54.052
55.476
56.892
58.301
59.703
61.098
62.487
63.870
65.247
66.619
67.985
69.346
70.703
72.055
73.402
74.745
76.084
77.419
78.750
80.077
81.400
82.720
84.037
85.351
86.661
87.968
89.272
90.573
91.872
93.168
Page 20 of 22
Analyzing Data from an Experiment
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
32.490
33.248
34.008
34.770
35.534
36.301
37.068
37.838
38.610
39.383
40.158
40.935
41.713
42.494
43.275
44.058
44.843
45.629
46.417
47.206
47.997
48.788
49.582
50.376
51.172
51.969
52.767
53.567
54.368
55.170
55.973
56.777
57.582
58.389
59.196
60.005
60.815
61.625
62.437
63.250
64.063
64.878
65.694
66.510
67.328
68.146
68.965
69.785
70.606
71.428
72.251
73.075
73.899
74.724
75.550
76.377
77.204
78.033
78.862
37.212
38.027
38.844
39.662
40.482
41.303
42.126
42.950
43.776
44.603
45.431
46.261
47.092
47.924
48.758
49.592
50.428
51.265
52.103
52.942
53.782
54.623
55.466
56.309
57.153
57.998
58.845
59.692
60.540
61.389
62.239
63.089
63.941
64.793
65.647
66.501
67.356
68.211
69.068
69.925
70.783
71.642
72.501
73.361
74.222
75.083
75.946
76.809
77.672
78.536
79.401
80.267
81.133
82.000
82.867
83.735
84.604
85.473
86.342
64.658
65.737
66.816
67.894
68.972
70.049
71.125
72.201
73.276
74.351
75.424
76.498
77.571
78.643
79.715
80.786
81.857
82.927
83.997
85.066
86.135
87.203
88.271
89.338
90.405
91.472
92.538
93.604
94.669
95.734
96.799
97.863
98.927
99.991
101.054
102.117
103.179
104.241
105.303
106.364
107.425
108.486
109.547
110.607
111.667
112.726
113.786
114.845
115.903
116.962
118.020
119.078
120.135
121.192
122.250
123.306
124.363
125.419
126.475
69.919
71.040
72.160
73.279
74.397
75.514
76.630
77.745
78.860
79.973
81.085
82.197
83.308
84.418
85.527
86.635
87.743
88.850
89.956
91.061
92.166
93.270
94.374
95.476
96.578
97.680
98.780
99.880
100.980
102.079
103.177
104.275
105.372
106.469
107.565
108.661
109.756
110.850
111.944
113.038
114.131
115.223
116.315
117.407
118.498
119.589
120.679
121.769
122.858
123.947
125.035
126.123
127.211
128.298
129.385
130.472
131.558
132.643
133.729
74.468
75.624
76.778
77.931
79.082
80.232
81.381
82.529
83.675
84.821
85.965
87.108
88.250
89.391
90.531
91.670
92.808
93.945
95.081
96.217
97.351
98.484
99.617
100.749
101.879
103.010
104.139
105.267
106.395
107.522
108.648
109.773
110.898
112.022
113.145
114.268
115.390
116.511
117.632
118.752
119.871
120.990
122.108
123.225
124.342
125.458
126.574
127.689
128.804
129.918
131.031
132.144
133.257
134.369
135.480
136.591
137.701
138.811
139.921
78.567
79.752
80.936
82.117
83.298
84.476
85.654
86.830
88.004
89.177
90.349
91.519
92.689
93.856
95.023
96.189
97.353
98.516
99.678
100.839
101.999
103.158
104.316
105.473
106.629
107.783
108.937
110.090
111.242
112.393
113.544
114.693
115.841
116.989
118.136
119.282
120.427
121.571
122.715
123.858
125.000
126.141
127.282
128.422
129.561
130.700
131.838
132.975
134.111
135.247
136.382
137.517
138.651
139.784
140.917
142.049
143.180
144.311
145.441
79.815
81.009
82.201
83.391
84.580
85.767
86.953
88.137
89.320
90.501
91.681
92.860
94.037
95.213
96.388
97.561
98.733
99.904
101.074
102.243
103.410
104.576
105.742
106.906
108.069
109.232
110.393
111.553
112.712
113.871
115.028
116.184
117.340
118.495
119.648
120.801
121.954
123.105
124.255
125.405
126.554
127.702
128.849
129.996
131.142
132.287
133.431
134.575
135.718
136.860
138.002
139.143
140.283
141.423
142.562
143.700
144.838
145.975
147.111
83.513
84.733
85.950
87.166
88.379
89.591
90.802
92.010
93.217
94.422
95.626
96.828
98.028
99.228
100.425
101.621
102.816
104.010
105.202
106.393
107.583
108.771
109.958
111.144
112.329
113.512
114.695
115.876
117.057
118.236
119.414
120.591
121.767
122.942
124.116
125.289
126.462
127.633
128.803
129.973
131.141
132.309
133.476
134.642
135.807
136.971
138.134
139.297
140.459
141.620
142.780
143.940
145.099
146.257
147.414
148.571
149.727
150.882
152.037
86.994
88.236
89.477
90.715
91.952
93.186
94.419
95.649
96.878
98.105
99.330
100.554
101.776
102.996
104.215
105.432
106.648
107.862
109.074
110.286
111.495
112.704
113.911
115.117
116.321
117.524
118.726
119.927
121.126
122.325
123.522
124.718
125.913
127.106
128.299
129.491
130.681
131.871
133.059
134.247
135.433
136.619
137.803
138.987
140.169
141.351
142.532
143.712
144.891
146.070
147.247
148.424
149.599
150.774
151.948
153.122
154.294
155.466
156.637
91.335
92.605
93.874
95.140
96.404
97.665
98.925
100.182
101.437
102.691
103.942
105.192
106.440
107.685
108.929
110.172
111.412
112.651
113.889
115.125
116.359
117.591
118.823
120.052
121.280
122.507
123.733
124.957
126.179
127.401
128.621
129.840
131.057
132.273
133.489
134.702
135.915
137.127
138.337
139.546
140.755
141.962
143.168
144.373
145.577
146.780
147.982
149.183
150.383
151.582
152.780
153.977
155.173
156.369
157.563
158.757
159.950
161.141
162.332
94.461
95.751
97.039
98.324
99.607
100.888
102.166
103.442
104.716
105.988
107.258
108.526
109.791
111.055
112.317
113.577
114.835
116.092
117.346
118.599
119.850
121.100
122.348
123.594
124.839
126.083
127.324
128.565
129.804
131.041
132.277
133.512
134.745
135.978
137.208
138.438
139.666
140.893
142.119
143.344
144.567
145.789
147.010
148.230
149.449
150.667
151.884
153.099
154.314
155.528
156.740
157.952
159.162
160.372
161.581
162.788
163.995
165.201
166.406
Page 21 of 22
Analyzing Data from an Experiment
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
160
170
180
190
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
79.692
80.522
81.353
82.185
83.018
83.852
84.686
85.520
86.356
87.192
88.029
88.866
89.704
90.543
91.382
92.222
93.063
93.904
94.746
95.588
96.431
97.275
98.119
98.964
99.809
100.655
101.501
102.348
103.196
104.044
104.892
105.741
106.591
107.441
108.291
109.142
117.679
126.261
134.884
143.545
152.241
196.161
240.663
285.608
330.903
376.483
422.303
468.328
514.529
560.885
607.380
653.997
700.725
747.554
794.475
841.480
888.564
87.213
88.084
88.955
89.827
90.700
91.573
92.446
93.320
94.195
95.070
95.946
96.822
97.698
98.576
99.453
100.331
101.210
102.089
102.968
103.848
104.729
105.609
106.491
107.372
108.254
109.137
110.020
110.903
111.787
112.671
113.556
114.441
115.326
116.212
117.098
117.985
126.870
135.790
144.741
153.721
162.728
208.098
253.912
300.064
346.482
393.118
439.936
486.910
534.019
581.245
628.577
676.003
723.513
771.099
818.756
866.477
914.257
127.531
128.587
129.642
130.697
131.752
132.806
133.861
134.915
135.969
137.022
138.076
139.129
140.182
141.235
142.288
143.340
144.392
145.444
146.496
147.548
148.599
149.651
150.702
151.753
152.803
153.854
154.904
155.954
157.004
158.054
159.104
160.153
161.202
162.251
163.300
164.349
174.828
185.293
195.743
206.182
216.609
268.599
320.397
372.051
423.590
475.035
526.401
577.701
628.943
680.134
731.280
782.386
833.456
884.492
935.499
986.478
1037.431
134.813
135.898
136.982
138.066
139.149
140.233
141.315
142.398
143.480
144.562
145.643
146.724
147.805
148.885
149.965
151.045
152.125
153.204
154.283
155.361
156.440
157.518
158.595
159.673
160.750
161.827
162.904
163.980
165.056
166.132
167.207
168.283
169.358
170.432
171.507
172.581
183.311
194.017
204.704
215.371
226.021
279.050
331.789
384.306
436.649
488.849
540.930
592.909
644.800
696.614
748.359
800.043
851.671
903.249
954.782
1006.272
1057.724
141.030
142.138
143.246
144.354
145.461
146.567
147.674
148.779
149.885
150.989
152.094
153.198
154.302
155.405
156.508
157.610
158.712
159.814
160.915
162.016
163.116
164.216
165.316
166.415
167.514
168.613
169.711
170.809
171.907
173.004
174.101
175.198
176.294
177.390
178.485
179.581
190.516
201.423
212.304
223.160
233.994
287.882
341.395
394.626
447.632
500.456
553.127
605.667
658.094
710.421
762.661
814.822
866.911
918.937
970.904
1022.816
1074.679
146.571
147.700
148.829
149.957
151.084
152.211
153.338
154.464
155.589
156.714
157.839
158.962
160.086
161.209
162.331
163.453
164.575
165.696
166.816
167.936
169.056
170.175
171.294
172.412
173.530
174.648
175.765
176.882
177.998
179.114
180.229
181.344
182.459
183.573
184.687
185.800
196.915
207.995
219.044
230.064
241.058
295.689
349.874
403.723
457.305
510.670
563.852
616.878
669.769
722.542
775.211
827.785
880.275
932.689
985.032
1037.311
1089.531
148.247
149.383
150.517
151.652
152.785
153.918
155.051
156.183
157.314
158.445
159.575
160.705
161.834
162.963
164.091
165.219
166.346
167.473
168.600
169.725
170.851
171.976
173.100
174.224
175.348
176.471
177.594
178.716
179.838
180.959
182.080
183.200
184.321
185.440
186.560
187.678
198.846
209.978
221.077
232.146
243.187
298.039
352.425
406.457
460.211
513.736
567.070
620.241
673.270
726.176
778.972
831.670
884.279
936.808
989.263
1041.651
1093.977
153.191
154.344
155.496
156.648
157.800
158.950
160.100
161.250
162.398
163.546
164.694
165.841
166.987
168.133
169.278
170.423
171.567
172.711
173.854
174.996
176.138
177.280
178.421
179.561
180.701
181.840
182.979
184.118
185.256
186.393
187.530
188.666
189.802
190.938
192.073
193.208
204.530
215.812
227.056
238.266
249.445
304.940
359.906
414.474
468.724
522.717
576.493
630.084
683.516
736.807
789.974
843.029
895.984
948.848
1001.630
1054.334
1106.969
157.808
158.977
160.146
161.314
162.481
163.648
164.814
165.980
167.144
168.308
169.471
170.634
171.796
172.957
174.118
175.278
176.438
177.597
178.755
179.913
181.070
182.226
183.382
184.538
185.693
186.847
188.001
189.154
190.306
191.458
192.610
193.761
194.912
196.062
197.211
198.360
209.824
221.242
232.620
243.959
255.264
311.346
366.844
421.900
476.606
531.026
585.207
639.183
692.982
746.625
800.131
853.514
906.786
959.957
1013.036
1066.031
1118.948
163.523
164.712
165.900
167.088
168.275
169.461
170.647
171.831
173.015
174.198
175.380
176.562
177.743
178.923
180.103
181.282
182.460
183.637
184.814
185.990
187.165
188.340
189.514
190.688
191.861
193.033
194.205
195.376
196.546
197.716
198.885
200.054
201.222
202.390
203.557
204.723
216.358
227.942
239.480
250.977
262.434
319.227
375.369
431.017
486.274
541.212
595.882
650.324
704.568
758.639
812.556
866.336
919.991
973.534
1026.974
1080.320
1133.579
167.610
168.813
170.016
171.217
172.418
173.617
174.816
176.014
177.212
178.408
179.604
180.799
181.993
183.186
184.379
185.571
186.762
187.953
189.142
190.331
191.520
192.707
193.894
195.080
196.266
197.451
198.635
199.819
201.002
202.184
203.366
204.547
205.727
206.907
208.086
209.265
221.019
232.719
244.370
255.976
267.541
324.832
381.425
437.488
493.132
548.432
603.446
658.215
712.771
767.141
821.347
875.404
929.329
983.133
1036.826
1090.418
1143.917
*Taken from http://www.medcalc.org/manual/chi-square-table.php.
Page 22 of 22