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CORAL REEF RESISTANCE TO WARMING: DATA ANALYSIS ACTIVITY
STUDENT WORKSHEET
This activity goes along with the acclimatization experiment described in the annotated research article
Mechanisms of reef coral resistance to future climate change (Palumbi et al. 2014) you just worked with.
In the experiment, Dr. Stephen Palumbi and colleagues transplanted corals into both the warm pool and
the cool pool and measured chlorophyll retention after experimental heat stress (see Figure 2 of the
article). Chlorophyll retention was measured as the fraction of chlorophyll that remained in
transplanted, heat stressed coral colonies compared to control colonies. You will now analyze sample
measurements from the acclimatization experiment provided by Stephen Palumbi.
Part A: Calculating Descriptive Stats
As you complete the steps below, enter all your calculated values in Table 1.
Table 1. Amount of chlorophyll retained in corals transplanted into warm and cool pools.
COLONY
xi
AH02
AH07
AH11
AH31
AH40
AH64
AH65
AH68
AH69
AH70
AH75
Sum
Mean (𝑥̅)
SD
SE
0.87
0.89
0.65
0.67
0.71
0.96
0.57
0.57
0.66
0.70
0.75
FRACTION OF CHLOROPHYLL RETAINED
Warm (HV) pool
Cool (MV) pool
2
xi - 𝑥̅
(xi - 𝑥̅)
xi
xi - 𝑥̅
(xi - 𝑥̅)2
0.29
0.40
0.42
0.45
0.22
0.60
0.27
0.37
0.43
0.41
0.53
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
1. For the HV and MV samples in Table 1, calculate the mean amount of chlorophyll (𝑥̅) retained after
heat stress. Sum up all measurements (𝑥i) for each of the HV and MV data sets and divide by the
number of measurements (or samples) in the data set.
𝑥̅ = ∑𝑥i / 𝑛
∑𝑥i is the sum of all samples; 𝑛 is the number of samples
[College audience: Skip the formula and description.]
2. Calculate the standard deviation (s or SD) for each set of data. The standard deviation measures the
mean difference between each individual measurement (𝑥i) and the mean (𝑥̅) of the entire sample.
Standard deviation is a way to quantify how spread out a set of measurements is compared to the
mean. Use the following formula and steps and enter the result of each step into the table:
s = √(∑(𝑥𝑖− 𝑥)2 /(𝑛−1))
[High school audience: Provide the variance s2, so students only need to take the square root.]
 For each measurement (𝑥i) in the HV and MV data sets, determine the difference between the
individual measurement and the mean of the entire set (𝑥i − 𝑥̅).
 Square each result (𝑥i − 𝑥̅) 2.
 Add up (sum) all of the squared differences Σ(𝑥i − 𝑥̅)2 and enter the result in the proper column
under “Sum.”
 Divide by the sample size minus 1 (𝑛 – 1). Then take the square root of the result and round
your answer to the second decimal place. You have now calculated the standard deviation.
Enter this value into the table under “SD.”
3. Calculate the standard error of the mean (SE or SEM) for each set of data. Because Palumbi and
colleagues only collected data from a random sample of the entire coral population in the hot and
cold pools of Ofu Island, it is not possible to know for certain that the mean you have calculated for
each sample is the same as the mean of the entire population in each pool. One way to show how
close the sample mean is to the population mean is to calculate the standard error of the mean (SE
or SEM). Use the formula below to calculate the SE and enter the result into the table:
SE = s/√n
4. Calculate the 95% confidence interval (CI) for each set of data. Confidence limits serve the same
purpose as SE. The 95% CI provides a range of values within which the mean of the entire population
is likely to be found. As an approximation, use the simplified formula below to calculate the 95%
confidence interval (95% CI), which is roughly twice the SE:
95% CI = 2 x SE
Upper limit: 𝑥 + (2 x SE); Lower limit: 𝑥 - (2 x SE)
Part B: Graphing the Data
In the graphing space below (or on your computer), construct a bar graph that compares the mean
amount of retained chlorophyll in HV and MV corals after heat stress. Label both axes and show either
the SE or 95% CI as error bars depending on your instructor’s directions.
TIP: If you use SE as errors bars, the top of the bar (or upper limit) equals 𝑥 + SE, and the bottom of the
bar (or lower limit) equals 𝑥 - SE. The upper and lower limits for 95% CIs as error bars are listed in
question 4 above.
[High school audience: You could show an example of a properly drawn and labeled bar graph here.]
5. Describe the differences between HV and MV corals you observe in the graph.
Part C: Performing a t-Test
By just looking at the bar graph, you can sometimes make a guess whether or not the apparent
difference between two sets of measurements is statistically significant. For example, the means may
seem clearly different and the error bars may not overlap, suggesting that the difference between the
two means could be significant. But a statistical test is required to confirm that the difference is in fact
significant.
The appropriate statistical test for comparing two means is the Student’s t-Test for independent
samples (the t-Test). The t-Test determines the probability (p) that any observed differences between
the means of two samples (e.g., HV and MV corals) occurred simply by chance. You will calculate the t
statistic called “observed t” (tobs) and then compare it to the critical t statistic (tcrit). This critical t-value is
a cutoff value that determines whether you can reject the null hypothesis that the mean of the one
particular sample is equal to the mean of the other sample, or 𝑥̅ 1 = 𝑥̅ 2. If your observed t value (tobs) is
less than the critical value (tcrit), then you cannot reject the null hypothesis. If the calculated statistic is
larger than the critical value, then you reject the null hypothesis and accept the alternative hypothesis
that the means are significantly different, or 𝑥̅ 1 ≠ 𝑥̅ 2.
The tcrit for your sample size of 11 per data set is 2.086.
[With proper instruction, both high school and college students could be asked to determine the
degrees of freedom themselves and find the critical using a table for alpha = 0.05.]
6. Now determine tobs by entering your calculated values in Table 1 into the formula below.
tobs= |𝑥MV - 𝑥HV|/(SEMV + SEHV) = ______________
7. Compare your calculated tobs with the critical t-value (tcrit) of 2.086. Based on your bar graph, the
associated error bars, and the results of your t statistic calculations, make a claim about the
differences you observe between HV and MV corals. Support your claim with evidence from the
graph and statistical test.