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.
One of the first steps in describing a data set is to graph the data and examine the distribution. Figure 1 shows 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, before a major drought in 1977. The graph on top shows beak measurements of finches that died during the drought. The graph on the bottom shows beak measurements of finches that survived the drought.
Beak Depths of 50 Medium Ground Finches That Did Not Survive the Drought
Beak Depths of 50 Medium Ground Finches That Survived the Drought
Figure 1. The distributions of beak depth measurements in two groups of medium ground finches.
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The first thing you notice is that the shape of both distributions resembles the shape of a bell. This is what’s referred to as a normal distribution.
Another thing you will notice by looking at the graphs in Figure 1 is that the center and spread of each distribution is different. Descriptive statistics allow you to describe and quantify these differences.
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
Students in a biology class planted eight bean seeds in separate plastic cups and placed them under a bank of fluorescent lights. Fourteen days later, the students measured the height of the bean plants that grew from those seeds and recorded their results in Table 1.
Table 1. Bean Plant Heights
Plant # 1 2 3 4 5 6 7 8
Height
(cm)
7.5 10.1
To determine the mean of the bean plants:
8.3 9.8 5.7
I.
Find the sum of the heights:
7.5 + 10.1 + 8.3 + 9.8 + 5.7 + 10.3 + 9.2 + 8.7 = 69.6 cm
10.3
II.
Count the number of height measurements:
There are eight height measurements.
9.2 8.7
III.
Divide the sum of the heights by the number of measurements to compute the mean:
Mean = 69.6 cm/8 = 8.7 cm
The mean for this sample of eight plants is 8.7 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 8.7 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.
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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
A researcher studying mouse behavior has recorded the time (in seconds) it took for 13 different mice to locate food in a maze in Table 2.
Table 2. Length of Time for Mice to Locate Food in a Maze
Mouse # 1 2 3 4 5 6 7 8 9 10 11 12 13
Time
(seconds)
31 33 81 33 28 29 33 27 27 34 35 28 32
To determine the median time that the mice spent searching for food:
I.
Arrange the Time values in numerical order from lowest to highest:
27, 27, 28, 28, 29, 31, 32, 33, 33, 33, 34, 35, 81
II.
Find the middle value. This value is the median:
Median = 32 seconds
Note: In this case the median is 32 seconds, but the mean is 35 seconds. The mean is larger than the median because a very slow mouse distorts the mean in the direction of that large value.
Mode
The mode is another measure of the average. It is the value that appears most often in a sample of data. In the example shown in Table 2, the mode is 33 seconds.
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.
Figure 2. Graph of body lengths of weaver ant workers.
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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.
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 3.
Table 3. Width of Maple Tree Leaves
8 Plant # 1 2 3 4 5 6 7
Width
(cm)
7.5 10.1 8.3 9.8 5.7
To determine the range of leaf widths:
I.
Identify the largest and smallest values in the data set:
10.3 9.2
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
8.7
Standard Deviation
The standard deviation is the most widely applied measure of variability. The sample mean ( 𝑥 ) 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 ( 𝑥 ).
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 4.
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Table 4. Plant Measurements and Steps for Calculating Standard Deviation.
Plant # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Height
(mm)
112 102 106 120 98 106 80 105 106 110 95 98 74 112 115 109 100
Sample mean = 103
Standard Deviation (s) = 11.7
Summary:
The mean height of the bean plants in this sample is 103 mm ± 11.7 mm. This means that this sample of 17 observations is drawn from a population with a standard deviation of 11.7 mm. A low standard deviation indicates that the data points tend to be very close to the mean ; a high standard deviation indicates that the data points are spread out over a large range of values.
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.
Group 1 - Hormone - Weight (grams) Group 2 – No Hormone - Weight (grams)
12.5
12
13
12
12
8.5
10
8
13
14
13
10.5
9.5
11
Mean = 12.05
8
9
9
13.5
8.5
6.5
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.
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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.
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 outside of known data points is known as extrapolation. Scientists can extrapolate by using a formula, data arranged on a graph, or programmed into a computer model. In contrast, interpolation involves making predictions between known data points, or within the range of measured data. In general, extrapolation comes with a higher degree of uncertainty.
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.
Correlation can have a value, which shows how good the correlation is, and if it is positive or negative.

1 is a perfect positive correlation

0 is no correlation (the values don't seem linked at all)

-1 is a perfect negative correlation
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