1. Statistics in Biology: Standard Error of the Mean & Error Bars
Essential Question: How are statistics used to interpret data and determine the accuracy of experimental results?

2. Sampling and Capturing the Actual Mean
Let’s suppose that 3 students are assigned to determine the mean of black-eyed peas from a 1 pound bag of black-eyes peas.
It would take hours to mass each and every pea. So the students decide to mass a random sample of the peas and determine the mean of the sample.

3. Standard Error or Standard Error of the Mean
It was agreed that each group would mass 100 black-eyed peas. The mean and standard deviation was determined for each of the three random samples.
All three means are different but certainly close. What is the probability that each group captured the actual mean?
Sample
Mean (g) S 1
0.220
0.039 2
0.215
0.043 3
0.208
0.042

4. Standard Error or Standard Error of the Mean
In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error or the standard error of the mean (SEM).
To compute the SE or SEM just simply divide the standard deviation by the square root of the number of data points.

5. Mean
Bar graph of the three sample means. What is the probability of capturing the actual mean?

6. Use the SE equation to calculate the standard error for the sample data

7. Mean Bar graph with +/-1 SE bars inserted
To the 68% confidence level, there is a statistically significant difference in the mean mass of black-eyed peas in sample 1 and sample 3.

8. So what conclusions can I draw from using +/- 1 SE bars?
Error bars that DO NOT overlap, suggest there is a significant difference in the data of samples 1 & 3
Difference in means is due to something OTHER THAN random sampling
68 % confidence level

9. How do I report this conclusion? TWO Conclusions…
At a 68% confidence level, the difference in means of samples 1 and 3, is statistically significant because the error bars do not overlap, using +/- 1 Standard Error of the mean. At a 68% confidence level, the population mean of a one pound bag of black-eyed peas is between g and g using data from sample 1.

10. Calculate the 2x the standard error for the sample data

11. Sample
Mean (g) SD SE
2 SE 1
0.220
0.039
0.0039
0.0078 2
0.215
0.043
0.0043
0.0086 3
0.208
0.042
0.0042
0.0084
Mean
How does the confidence interval and conclusions change with +/- 2 SE bars?
To the 95% confidence level, there is not a statistically significant difference between the mean mass of black-eyed peas in set 1, 2, or 3.

12. So what conclusions can I draw from using +/- 2 SE bars?
Error bars that overlap suggest there is NO significant difference in the data of the 3 samples
Difference in means IS due to random sampling
95 % confidence level

13. How do I report this conclusion? Your Turn!!
At a 95% confidence level, the difference in means of samples 1, 2 and 3, is not statistically significant because the error bars overlap, using +/- 2 Standard Error of the mean.

14. How do I report this conclusion? Your Turn!!
At a 95% confidence level, the population mean of a one pound bag of black-eyed peas is between g and g using data from sample 1.

15. Video on Calculating Standard Error of the Mean & Error Bars
YouTube: Bozeman Standard Error

17. Calculate +/- 2 SEM for problem 1 and add the error bars to YOUR graph
Using +/- 2 SEM write a conclusion based on problem 1 data & graph
Point out what happens to the SE bars. With a larger sample size, you increase the probability of capturing the actual mean.

18. Notice what happens to the error bar as the sample size increases
Point out what happens to the SE bars. With a larger sample size, you increase the probability of capturing the actual mean.

19. The larger the sample size, the smaller the standard error tends to be.
The greater the probability of capturing the actual mean of the population
Point out what happens to the SE bars. With a larger sample size, you increase the probability of capturing the actual mean.

20. Comparing the Mass of Pinto Beans and Black-eyed Peas
A student was interested in determining if the mass of pinto beans was significantly larger than the mass of black-eyed peas.
It is obvious that the mass can vary.

21. Comparing the Mass of Pinto Beans and Black-eyed Peas
Upon observation, it appears that pinto beans are larger but it needs to be quantified. The mass of 300 random seeds are measured and recorded and histograms are made.

22. Mean s SE
2 SE
Black-eyed Peas
0.21
.042
Pinto Beans
0.37
.074
Point out what happens to the SE bars. With a larger sample size, you increase the probability of capturing the actual mean.
Which sample of seeds will have a larger standard deviation, the pinto beans or black-eyed peas?

23. The mean for black-eyed peas is 0
The mean for black-eyed peas is 0.21 g and the mean for pinto beans is 0.37 g. Remember this is only a sample of a larger population.
Point out what happens to the SE bars. With a larger sample size, you increase the probability of capturing the actual mean.

24. Is the difference in means due to sampling?
Is the difference in the means of these two samples statistically significant?
Is the difference in means due to sampling?
Point out what happens to the SE bars. With a larger sample size, you increase the probability of capturing the actual mean.

25. Mean SD SE
2 SE
Black-eyed Peas
0.21
.074
.0043
.0086
Pinto Beans
0.37
.041
.0024
.0048
The 2 SE bars do not overlap, so it is most likely that the difference between the mass means is statistically significant.
With 95% confidence, there is a statistically significant difference between the mean mass of black-eyed peas and pinto beans.

26. Should I report Standard Deviation or Standard Error of the Means Bars????
If you are trying to support or reject a hypothesis -- in other words, when you are reporting on the results of an experiment -- you will most likely be using standard error of the means (usually 2 SE for a 95% confidence level for your error bars) but if you want to illustrate the variation in the population, you will use standard deviation bars.
With the 95% confidence level, we can say that there is only a 5% chance that the range between the error bars of 2SEMS excludes the mean of the actual data.