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Name: ___________________________________Section: _________________Date: ________________
Graded Course Assignment 4: Interpreting Data & Statistics.
Interpreting Scientific Data Using Mathematics and Statistics
Math and statistics are integral to scientific research—from forming a hypothesis, to collecting and
analyzing data, to drawing inferences beyond the data.
It is rarely practical for scientists to measure every event or individual in a population. Instead, they
typically collect data on a sample of a population and use them to draw conclusions (or make inferences)
about the entire population.
Our population is 14 day old Bean Plants.
Table 1. Bean Plant Heights at 14 days
Plant #
Height (cm)
Plant #
1
8.3
10
2
9.4
11
3
7.7
12
4
8.9
13
5
7.1
14
6
8.9
15
7
8.7
16
8
9.8
17
9
10.1
18
19
Height (cm)
6.7
8.1
9.2
10.6
9.6
8.9
7.5
8.2
9.4
9.3
One of the first steps in describing a data set is to graph the data and examine the distribution.
The first thing you notice is that the shape of
the distribution resembles the shape of a
bell. This is what’s referred to as a normal
distribution.
Figure 1. The distributions of plant heights
Measures of Average: Mean, Median, and Mode
A description of a group of observations typically includes a value for the mean, median, or mode. These
are all measures of central tendency—in other words they represent a number close to the center of the
distribution of values or observations in the data set.
Mean
The mean (also referred to as the average) is the sum of all numbers in a data set divided by the number of
values in the data set. The mean is not always the best measure of central tendency because it can be
distorted by extreme values, or outliers, which are extremely different from the rest of the sample.
Application in Science
To determine the mean of the bean plants:
I.
8.3
9.4
Find the sum of the heights:
7.7
8.7
7.1
8.9
8.7
9.8
10.1
6.7
8.1
9.2
10.6
9.6
8.7
7.5
8.2
9.4
9.3
= 166.4 cm
II.
Count the number of height measurements:
There are 19 height measurements.
III.
Divide the sum of the heights by the number of measurements to compute the mean:
Mean = 166.4 cm/19 = 8.76 cm
The mean for this sample of eight plants is 8.76 cm and may serve as an estimate for the true mean of the
population of bean plants growing under these conditions. In other words, if we collected data from
hundreds of plants and graphed the data, the center of the distribution might be around 8.76 cm.
Median
The median shows the mid-point in a distribution and is, therefore, less easily distorted by extreme values
than the mean. For this reason, it may be more useful to use the median as the main descriptive statistic
for a sample of data in which some of the measurements are extremely large or extremely small.
To determine the median of a set of values, first arrange them in numerical order from lowest to highest.
The middle value in the list is the median. If there is an even number of values in the list, then the median is
the mean of the middle two values.
Application in Science
To determine the median of our bean plants:
I.
Arrange the Height values in numerical order from lowest to highest:
6.7
7.1
7.5
7.7
8.1
8.2
8.3
8.7
8.9
1
2
3
4
5
6
7
8
9
II.
8.9
8.9
9.2
9.3
9.4
9.4
9.6
9.8
10.1
10.6
9
8
7
6
5
4
3
2
1
Find the middle value. This value is the median:
Median = 8.9cm
Note: In this case the median is 8.9cm, but the mean is 8.76cm. The mean is smaller than the median, and
the two might end up close in value, but it’s not guaranteed.
Mode
The mode is another measure that can help describe a set of data. It is the value that appears most often in
a sample of data. In the example above in Table 1, the mode is 8.9 seconds (it appears 3 times).
The mode is not typically used as a measure of central tendency in scientific research, but can be useful in
describing some distributions. For example, Figure 2 shows a distribution with two peaks, or modes—
what’s called a bimodal distribution. Describing these data with a measure of central tendency like the
mean or median would obscure this fact.
Frequencies of Body Lengths of Weaver Ants
Figure 2. Graph of body lengths of weaver ant workers.
When to Use Which?
The mean is the descriptive statistic most often used to describe the central tendency of a sample of
measurements; it is the only one of the three measurements of average that takes into account all the
information in a data set. This is why other statistical techniques, such as standard deviation, which is a
measure of variability, utilize the mean. However, as discussed above, there are occasions when taking
account of the value of every measurement in a distribution may give a distorted picture of the data; in
such cases, the median provides a more realistic description of the center of the distribution than the
mean. The mode is not used very frequently in science as a measure of central tendency but may be useful
in describing some types of distributions—for example, ones with more than one peak.
Measures of Variability: Range and Standard Deviation
Variability describes the extent to which numbers in a data set diverge from the central tendency or
average. It is a measure of how “spread out” the data are. Two common measures of variability are range
and standard deviation.
Range
The simplest measure of variability in a sample of normally distributed data is the range, which is the
difference between the largest and smallest values in a set of data.
Application in Science
Students in a biology class measured the width of eight leaves from eight different maple trees and
recorded their results in Table 2.
Table 2. Width of Maple Tree Leaves
Leaf #
Width (cm)
1
2
3
4
5
6
7
8
To determine the range of leaf widths:
7.5
10.1
8.3
9.8
5.7
10.3
9.2
8.7
I.
Identify the largest and smallest values in the data set:
Largest = 10.3 cm, Smallest = 5.7 cm
II.
To determine the range, subtract the smallest value
from the largest value:
Range = 10.3 cm – 5.7 cm = 4.6 cm
Standard Deviation
The standard deviation is the most widely applied measure of variability, there are two types.
1) Sample Standard Deviation: Used when you want to apply your results to other items not measured,
like other bean plant heights. The sample mean ( ) provides a measure of the central tendency of the
sample; the sample standard deviation (s) measures the average deviation between each
measurement in the sample and the mean ( ).
Application in Science
You are interested in knowing how tall bean plants (Phaseolus vulgaris) will grow two weeks after planting.
You plant a sample of 20 seeds in separate pots and give them equal amounts of water and light. After two
weeks, 17 of the seeds have germinated and have grown into small seedlings. Each plant is measured from
the tips of the roots to the top of the tallest stem. The measurements are recorded in Table 3.
Table 3. Plant Measurements and Steps for Calculating Standard Deviation.
Plant #
Height (mm)
Plant #
Height (mm)
1
112
10
110
2
102
11
95
3
106
12
98
4
120
13
74
5
98
14
112
6
106
15
115
7
80
16
109
8
105
17
100
9
106
Fig 3 Mean Height with +/- 1 std.dev.
Sample mean = 103
Standard Deviation (s) = 11.7
The mean height of the bean plants in this
sample is 103 mm 11.7 mm. If we use the
normal distribution table on the left. (Fig 3)
We can say that for our sample 68.2% of the
plants will fall between 114.7mm and
91.3mm.
Fig 4 Normal distribution (bell curve)
It is up to the scientist to determine how usable the data may be. Would it be prudent for a bean seed
vendor to place this data on their packages based on 17 bean plants? The easy answer is no. It’s very
possible that we would get similar data if we grew 10,000 seeds, however it’s also just as likely we may not.
2) Population Standard Deviation: Used when you have measured the entire population of items you are
interested in, like the metal cylinders below. The population mean (µ) provides a measure of the central
tendency of the sample; the population standard deviation (σ) measures the average deviation
between each measurement in the sample and the mean (µ).
TABLE 4 Densities of steel
cylinders measured by
water displacement.
Sample
Density
A
7.9
B
8.1
C
7.7
D
7.9
E
7.9
F
7.8
G
8.0
H
8.2
I
7.9
J
7.7
K
8.4
L
7.9
The data in table 4 shows the densities of 12 different steel cylinders. The volume
was measured using water displacement. This set of data is very different from
the bean plants in that the density values are expected to be close to identical.
The density of this steel shouldn’t have a normal distribution. So using a std. dev.
Can help us determine if the method(s), tools or skills used in obtaining the data
are appropriately accurate and/or precise.
For this data the mean is 7.95 g/ml, and the std dev is 0.19g/ml.
The accepted/known value for the density steel is 7.812 g/ml with a Std Dev of
0.0157g/ml. Giving us an acceptable range of 7.796g/ml to 7.828g/ml. ( +/- 1 std
dev)
While we can say that this data wasn’t that far off ( +0.138 g/ml), it doesn’t fall within one or two std
deviations. It is actually 9 std dev’s away from the known. Given the distance from this data set’s mean
from the accepted mean and the much higher std dev (0.0157 vs 0.2) it would be difficult to label this data
as worth using.
Their Formulae aren’t that different:
Sample
Population
Interpolation & Extrapolation
Sometimes scientists need to make predictions
outside the range of measured data, such as in
forecasting future climate change. This process in
which scientists make predictions between known
data points, or within the range of measured data
is called interpolation. Fig 5 shows that we could
reasonably expect to earn ~$480 if the
temperature was 21°C.
Fig 5 Ice cream sales vs. Temperature
In contrast extrapolation (fig 6) involves making
predictions outside of known data points.
Scientists can extrapolate by using a formula, data
arranged on a graph, or programmed into a
computer model. In general, extrapolation comes
with a higher degree of uncertainty.
images: www.mathisfun.com
Fig 6 Ice cream sales vs. Temperature
Correlation
When two sets of data are strongly linked together we say they have a High Correlation.
 Correlation is positive when the variables change in the same direction. For example, the length of
an iron bar will increase as the temperature increases.
 Correlation is negative when variables change in opposite directions. For example, the volume of a
gas will decrease as the pressure increases.
 If there is no relationship between the two variables such that they change independently of each
other, then there is no or zero correlation.
On the graph, written below
the equation for the line is the
Coefficient of Determination
(R2) that Excel provides. R2 is a
measure that allows us to
determine how certain one can
be in making predictions from a
certain model/graph.
R2 = 0 < +1
To find r just take the square
root of R2 and adjust the sign +
or – according to the trend line.
Figure 4. Pressure vs Temperature
Another measure of correlation is r - correlation coefficient. r measures the strength and the direction of a
linear relationship between two variables. It can have values from -1 < 0 < +1.
Fig 5 Correlation Examples
Like r, R2 values close to 1.0 are indicators of high correlation. In fig 4 the R2 = 0.9996, we can say that this
experiment represents a good display of Gay-Lussac’s Law which relates temperature and pressure, and
that the graph/line made from this data is worthy of using. However, correlation implies that the numbers
follow the same trend, and that is all. Causation is for the experimenter to determine.
In other words: “Did changing the temperature actually cause the pressure to increase?”
Statistical Analysis - The t-test
Suppose that a researcher wishes to test if a certain kind of growth hormone will produce faster growth in
mice. She injects 10 mice with the hormone and uses another 10 as a control. Three weeks later, she
weighs the mice and discovers that the mean weight of mice that have received the injections is 12.05 g
and the mean weight of control mice is 9.3 g. These values indicate that the mice receiving the hormone
are heavier. Is her value of 12.05 significantly different than 9.3? Is it possible that the hormone has no
effect; that the weight difference between the two groups is due to chance? This is like flipping a coin 10
times. You expect 5 heads and 5 tails but you might get 6 heads or 7 heads or perhaps 8 heads. Similarly, if
the hormone does not work, you expect the mean for the two groups to be similar but it may not be exactly
the same.
Table 5 – Tumor masses for 2 groups of mice.
Group 1 - Hormone - Weight (grams)
12.5
13
12
12
13
14
13
10.5
9.5
11
Mean = 12.05
Group 2 – No Hormone - Weight (grams)
12
8.5
10
8
8
13.5
9
8.5
6.5
9
Mean = 9.3
What is the chance that the two means would be as different as 12.05g and 9.3g if the hormone really did
not work? Statistical tests are used to determine whether differences in the data are real differences or
whether they are due to chance. In the example above, we test if the mean of group 1 is significantly
different than the mean of group 2. The alternative is that the difference is due to chance or random
fluctuations and the hormone did not cause additional weight gain. The test gives the probability that
difference could be due to chance. If the probability that the difference is due to chance is less than 1 out
of 20 (<0.05), then we conclude that the difference is real. If the probability is greater than 0.05, we
conclude that the difference is not significant, it could be due to chance.
There are several tests available for testing means. A commonly used test for data that are normally
distributed is the t-test.
The calculations for the test can be performed by hand but computer software can do them very quickly. To
perform the test, the weight data for the two groups of mice above are entered into a t-test program.
The software reveals that p = 0.0012. The probability that the difference between the two means (12.05
and 9.3) is due to chance (random effects) is 0.0012 (or 12 out of 10,000). Because p < 0.05, we conclude
that the two means are really different and that the difference is not due to chance. The researcher accepts
her hypothesis that the hormone produces faster growth. If p had been greater than 0.05, we would reject
her hypothesis and conclude that the two means are not significantly different; the hormone did not cause
one group to be heavier.
The word "significant" has a slightly different meaning in statistics than it does in general usage. In a
statistical test of two means, if the difference is not due to chance, we conclude that the two means are
significantly different. In the example above, the mean weight of group 1 is significantly heavier than that
of group 2.
NAME: ____________________________________ Section:
Date:______________
1. Range & Standard Deviation
Two Methods for measuring volume were examined in an experiment. The densities of twelve cylinders
were calculated using data gathered in two sets, each set using a different method for measuring the
volume of the cylinder.
Figure 1. Density Measurements Method 1
Figure 2. Density Measurements Method 2
For Method 1 the Mean is ___________g/ml;
- the std dev is 0.0346
for Method 2 the Mean is 7.95 g/ml.
- the std dev is 0.19
TABLE 1
Method 1
Sample Density
A
7.800
B
7.844
C
7.825
D
7.883
E
7.819
F
7.824
G
7.875
H
7.859
I
7.882
J
7.764
K
7.821
L
7.863
TABLE 2
Method 2
Sample
Density
A
7.9
B
8.1
C
7.7
D
7.9
E
7.9
F
7.8
G
8.0
H
8.2
I
7.9
J
7.7
K
8.4
L
7.9
The accepted value for this steel is
density of 7.812 g/ml with a
Std Dev of 0.0157g/ml.
a. Find the Mean for Method 1. ___________________
b. What are the ranges for each method? M 1 _________________ M 2 ________________
c. Which method is better? Why?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
d. Using your own judgement comparing Method 1’s results and the accepted value, is the data from
method 1 worth using? Explain using the statistical information.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. Interpolation, Extrapolation & Correlation
Figure 3. Pressure vs Temperature
Answer the following questions:
a. Does this data show good correlation? Why?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
b. What would you expect the pressure of this system to be at a temperature of 315K?
____________
c. Predict the pressure for a temperature of 335K ______________
d. With a temp range of 290K to 325K would it be prudent to extrapolate out to 1,000K?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3. T-tests
Complete the practice scenarios and determine if there is a significant difference between the means using
a Student’s t-test (alpha level= 0.05). Answer all questions and attach the t-test spreadsheet with
calculations associated with each scenario.
A researcher wishes to learn if a certain drug slows the growth of tumors. She obtains mice with tumors
and randomly divides them into two groups. She injects one group of mice with the drug and uses the
second group of mice as a control. After 2 weeks, she sacrifices the mice and weighs the tumors. The mass
of tumors for each group of mice is below. The researcher is interested in learning if the drug reduces the
growth of tumors.
Her hypothesis is: The mean weight of tumors from mice in Group A will be less than the mean weight of
tumors from mice in Group B.
Table 3 - Tumor masses for 2 groups of mice.
Tumor mass (g) of Group A mice
Treated with Drug
0.72
0.68
0.69
0.66
0.70
0.63
0.71
0.73
0.68
0.71
Tumor mass (g) of Group B mice
Control- Not Treated
0.71
0.83
0.89
0.78
0.68
0.74
0.75
0.80
0.80
0.78
Answer the following questions: (Attach the Excel t-test calculation worksheet.)
a.
b.
c.
d.
Is this a one-tailed or two-tailed t-test? Why?
Perform a t-test using the programmed Excel worksheet.
What is the calculated p-value?
Is the researcher’s hypothesis accepted or rejected? Why?
Reference:
Gregory, M. (2014, August). Statistical analysis- The t-test. Retrieved August 5, 2014 from
http://biology.clc.uc.edu/courses/bio303/statanal.htm
A group of ecology students was studying and comparing two different forest areas by taking 20 sample
plots in each area.
The researcher formulated the following hypothesis: The mean difference of sugar plus red maple trees
between Woods A and B will be different.
In each wooded area the combined number of sugar plus red maple trees in each of the 20 plots was found
to be:
Table 4
Plot # *
Woods A
Woods B
Plot Data
Plot Data
1
0
9
2
6
6
3
7
2
4
0
7
5
1
6
6
3
8
7
2
6
8
9
5
9
7
0
10
3
2
11
1
3
12
9
2
13
8
4
14
2
7
15
1
9
16
5
8
17
7
7
18
5
1
19
7
7
20
3
8
*Do not use the plot # in your calculations.
Answer the following questions: (Attach the Excel t-test calculation worksheet.)
e.
f.
g.
h.
Is this a one-tailed or two-tailed t-test? Why?
Perform a t-test using the programmed Excel worksheet.
What is the calculated p-value?
Is the researcher’s hypothesis accepted or rejected? Why?
Reference:
Carter, S.J., (2005, March 29). Statistical analysis of data. Retrieved August 4, 2014 from
http://biology.clc.uc.edu/courses/bio303/statanal.htm