1. Descriptive Statistics Used in Biology

2. 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 these data to draw conclusions (or make inferences) about the entire population. Statistics is a mathematical discipline that relates to this type of analysis.

3. One of your first steps in analyzing a small data set is to graph the data and examine the distribution. Here are two graphs of beak measurements taken from two samples of medium ground finches that lived on the island of Daphne Major, one of the Galápagos Islands, during a major drought in 1977.

4. The measurements tend to be more or less symmetrically distributed across a range, with most measurements around the center of the distribution. This is a characteristic of a normal distribution.

5. Measures of Average: Mean, Median, and Mode A description of a group of observations can include 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.

6. The Mean You calculate the mean (also referred to as the average or arithmetic mean) by summing all the data points in a data set (ΣX) and then dividing this number by the total number of data points (N): What scientists want to understand is the mean of the entire population, which is represented by µ. They use the sample mean, represented by ̅, as an estimate of µ.

7. Calculate the mean for two subsets of the Grants’ Data Nonsurvivors Survivors 5-bird sample15-bird sample5-bird sample15-bird sample Bird ID # Beak Depth (mm) Bird ID # Beak Depth (mm) Bird ID # Beak Depth (mm) Bird ID # Beak Depth (mm) 127.5228311.209439.103169.85 3479.312889.1016438.806238.80 4138.2029410.5018849.1567310.10 5228.393158.80224411.016789.70 60910.503218.48819110.868918.00 3527.70 101911.21 4138.20 147710.10 4689.02 15288.55 5039.10 17979.31 5078.85 185010.40 56110.20 18849.15 6109.00 22429.45 6199.25 23789.86 6217.60 224910.68 6769.70 29398.31 Mean

8. Note that the mean values are different for the five and fifteen-samples. Which is a better estimate of the true mean, µ?

9. Median When the data are ordered from the largest to the smallest, the median is the midpoint of the data. It is not distorted by extreme values, or even when the distribution is not normal. For this reason, it may be more useful for you 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. Find the median for each of your four sets of finch data.

10. 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. You can use the range for data that are not normally distributed. For any data, a larger range value indicates a greater spread of the data—in other words, the larger the range, the greater the variability. An extremely large or small value in the data set will make the variability appear high. Calculate the range for your four samples of the Grants’ data. The standard deviation provides a more reliable measure of the “true” spread of the data.

11. Definitions for Median and Range on BRT

12. Standard Deviation and Variance The standard deviation is the most widely used measure of variability. The sample standard deviation (s) is essentially the average of the deviation between each measurement in the sample and the sample mean (). The sample standard deviation is an estimate of the standard deviation in the larger population.

13. The formula for calculating the sample standard deviation follows:

14. What does standard deviation indicate? If a population has a normal distribution, 68% of the sample should be within one standard deviation of the mean. Approximately 2 standard deviations should account for 95% of all samples.

16. Variance and Standard Deviation Note that the number calculated at this step provides a statistic called variance (s 2 ). Variance is an important measure of variability that is used in certain statistical methods. It is the square of the standard variation. 6. Take the square root to calculate the standard deviation (s) for the sample. Calculate the standard deviation for the two five-bird samples (survivors and non-survivors)

17. Results Note that the standard deviation is smaller for the larger samples. This is often, but not always the case.. Now let’s look at a larger data set:

18. Data for 50 Finches Four variables, including beak depth Calculate the mean for each variable Variance is given; you can calculate the standard deviation easily Enter in table

19. Fill in everything except 95% confidence interval

20. Measures of Confidence: Standard Error of the Mean and 95% Confidence Interval The standard deviation provides a measure of the spread of the data from the mean. A different type of statistic reveals the uncertainty in the calculation of the mean. The sample mean is not necessarily identical to the mean of the entire population. Every time you take a sample and calculate a sample mean, you would expect a slightly different value. In other words, the sample means themselves have variability.

21. Standard Error of the Mean This variability can be expressed by calculating the standard error of the mean (abbreviated as̅ SEM) or the 95% confidence interval (95% CI).

22. Why are SEM and 95% CL useful? The standard error of the mean represents the standard deviation of a distribution around the mean and estimates how close the sample mean is to the population mean. The greater the sample size (i.e., 50 rather than 15 or 5 finches), the more closely the sample mean will estimate the population mean, and therefore the standard error of the mean becomes smaller. The 95% confidence interval (95% CI) is equivalent to 1.96 (typically rounded to 2) standard errors of the mean. Because the sample means are assumed to be normally distributed, 95% of all sample means should fall between 2 standard deviations above and below the population mean, estimated by 95% CI.

23. Calculate the 95% Confidence Interval for each variable mean. Both SEM̅ and 95% CI can be illustrated as error bars in a bar graph of the means of two or more samples that are being compared. Depicting SEM or the 95% CI as error bars in a bar graph provides a clear visual clue to the uncertainty of the calculations of the sample means.

25. Make a Bar Graph with Error Bars On a sheet of graph paper, construct four bar graphs that compare the means of non-survivors and survivors for each physical characteristic (wing length, body mass, tarsus length, and beak size). Label both axes of each graph and show the 95% CI as error bars. Once you complete your four bar graphs, describe any differences between non-survivors and survivors you observe in each graph.

27. What have we learned? Measurements often cluster around the mean is a form known as a normal distribution. The average variability about the mean is known as the standard deviation of the mean. Since it usually not possible to sample every individual in a population, there is also variability in the measurement of the mean. If the variation of the mean follows a normal distribution, we can estimate how close our sample mean is to the actual mean. 95% confidence intervals allow us to compare sample means of different groups and infer statistically significant differences.

28. Observation: seeds of weeds seem to need light as well as water to germinate You will investigate this observation using garden seeds of various species You have the following: lots of seeds, petri dishes, filter paper, water Design an experiment to address the observation. Each petri dish should have exactly 30 seeds; each group should use at least 10 petri dishes We will collect germination data and analyze it next week. A written report, in scientific format, will be completed for homework in your lab notebook…

29. Observation: seeds of weeds seem to need light as well as water to germinate You will investigate this observation using dill weed (Anethum graveolens) You have the following: lots of dill seeds, petri dishes, filter paper, water Design an experiment to address the observation. Each petri dish should have exactly 30 seeds; each group should use at least 6 petri dishes We will collect germination data and analyze it next week. A written report, in scientific format, will be completed for homework…