Descriptive Statistics and Graphing

Measures of Central Tendency

Measures of central tendency are used a "typical" number in a data set. There are different ways to find a typical number and there may be advantages and disadvantages of each, depending on the data. The mean, median, and mode are described below.

Mean

The mean is the average. The mean of a group of numbers is obtained by adding the numbers and then dividing the sum by the total number of data. For example, suppose that a population biologist obtains the following population estimates for waterfowl in six lakes:

Data: 400, 200, 220, 210, 340, 250

The mean of the six numbers below is equal to the sum (1620) divided by the number of data points (6) = 270.

Median

The median is the middle observation. It is obtained by arranging the data in increasing (or decreasing) order. The number that is in the middle is the median. 

Example

Data: 27, 40, 3, 51, 34

Data arranged in ascending order: 3, 27, 34, 40, 51

The median for this data set is 34.

If there is an even number of data points, there will be two middle numbers. In this case, the value that is half-way between the two middle numbers is the median. This is equivalent to taking the mean of the two middle numbers.

Example

Data: 400, 200, 220, 210, 340, 250

Data arranged in ascending order: 200, 210, 220, 250, 340, 400

The numbers 220 and 250 are both the middle numbers. The median is therefore the mean of 220 and 250. It is 235.

Mode

The mode is the most common value.

Example

Data: 6, 7, 6, 1, 3, 3, 4, 9, 0, 6, 8, 1

The mode for the data set above is 6 because it occurs three times.

Which to use? Some Disadvantages of Each:

Mode – The mode is good if you want to know the most common occurrence but it may not be representative of the data, particularly with small data sets. For example, in the following data the mode is 2, but 2 is not a central measurement: 2, 2, 12, 15, 17, 19, 20. Another problem is that there may be more than one mode.

Median – The median is useful if the data distribution is skewed (does not follow a bell-shaped curve). With a small number of data points, the median may not be a central number. In the following data set, the median is 0, which is not a representative number: 52, 50, 48, 0, 0, 0, 0.

Mean – The mean gives a good representation of the data if the data are normally distributed (a bell-shaped curve). The mean is influenced by outliers, particularly if the number of data points is small. If there are outliers, the median may be a better because it is more immune to this sort of bias. For example, suppose that a population estimates for waterfowl in seven lakes are:

400

200

220

210

340

250

44,000

The last number in this data set does not seem to belong with the rest; it is an outlier. The typical lake seems to have approximately 200 to 400 birds but the mean for this data set is 6,517. The median (250) may be a better measurement of central tendency.

Measures of Dispersion

Dispersion refers to the spread of the data. Two measures of dispersion- the range and the standard deviation- are discussed below.

Range

The range is the difference between the largest and smallest data points.

Example

Each of the following data sets have a mean of 50. Notice that data set 1 is more dispersed; the range is 100. The range of data set two is 4.

Data set 1: 100, 75, 50, 25, 0

Data set 2: 52, 51, 50, 49, 48

Variance and Standard Deviation

Variance and standard deviation refer to the the difference between a typical data point and the mean.

Calculation

The calculations below are for the following data set:

25, 26, 26, 31, 35, 36, 38

1. Calculate the mean. The mean for the data set above is 31.

Data
25
26
26
31
35
36
38
Sum = 217
Mean = 217/7 = 31

2. Calculate the deviation from the mean.

Data	Deviation
25	31 - 25 = 6
26	31 - 26 = 5
26	31 - 26 = 5
31	31 - 31 = 0
35	31 - 35 = -4
36	31 - 36 = -5
38	31 - 38 = -7
Sum = 217
Mean = 217/7 = 31

3. Square the deviations.

Data	Deviation	Deviation squared
25	31 - 25 = 6	36
26	31 - 26 = 5	25
26	31 - 26 = 5	25
31	31 - 31 = 0	0
35	31 - 35 = -4	16
36	31 - 36 = -5	25
38	31 - 38 = -7	49
Sum = 217
Mean = 217/7 = 31

4. Sum the squared deviations.

Data	Deviation	Deviation squared
25	31 - 25 = 6	36
26	31 - 26 = 5	25
26	31 - 26 = 5	25
31	31 - 31 = 0	0
35	31 - 35 = -4	16
36	31 - 36 = -5	25
38	31 - 38 = -7	49
Sum = 217
Mean = 217/7 = 31
		Sum = 176

5. Divide the sum of squared deviations by the number of data points minus 1 (also called n-1). This is the variance.

Data	Deviation	Deviation squared
25	31 - 25 = 6	36
26	31 - 26 = 5	25
26	31 - 26 = 5	25
31	31 - 31 = 0	0
35	31 - 35 = -4	16
36	31 - 36 = -5	25
38	31 - 38 = -7	49
Sum = 217
Mean = 217/7 = 31
		Sum = 176
		Variance = Sum/(n-1) = 29.33

6. The standard deviation is the square root of the variance.

Data	Deviation	Deviation Squared
25	31 - 25 = 6	36
26	31 - 26 = 5	25
26	31 - 26 = 5	25
31	31 - 31 = 0	0
35	31 - 35 = -4	16
36	31 - 36 = -5	25
38	31 - 38 = -7	49
Sum = 217
Mean = 217/7 = 31
		Sum = 176
		Variance = Sum/(n-1) = 29.33
		Standard Deviation = Square root of 29.33 = 5.42

Normal Distribution

A normal distribution is a bell-shaped curve. For example, suppose that 10,000 runners finish a marathon race (26.2 miles). A few of the finishers will be among the fastest runners and a few will be the slowest. Most of the finishers will be closer to the mean. The plot below represents the type of data that are expected.

Many kinds of biological data have a normal distribution as shown above. For example, we could plot the weights or height of people in a population or their scores on exams. A large sample size will likely show that the data are normally distributed. One reason is because many biological characteristics are due to more than one gene and the genes often act additively.

The Central Limit Theorem states that when numbers are summed, the distribution of the numbers is normal. For example, we could roll 5 dice and add their numbers. The smallest number that we could get is 5 and the largest number that we could get is 30. We could then roll the dice again and sum all of the numbers for a second data point. If we repeated this a large number of times, the results would be normally distributed. A Java applet is available at the University of South Carolina website to illustrate this principle. Go to this web page (http://www.stat.sc.edu/~west/javahtml/CLT.html) and try rolling 5 dice 10,000 times and observe the distribution. Notice that most of the numbers are in the range of 16 to 19 with very rolls producing 5 or 30. Repeat the procedure with 2 dice and notice that most rolls are 7 with relatively few 2s and 12s. Try rolling 1 die 10,000 times. Summing is not used and the data are not normally distributed.

The normal distribution is useful in statistical analysis. For example, approximately 68% of the data points fall within one standard deviation of the mean (mean plus or minus 1 S.D.).

Suppose that we take a sample of marathon runners and calculate their mean finishing time and standard deviation. The mean is 4.3 hours (260 minutes) and the standard deviation is 41 minutes. Using this information, we can calculate that 68% of the runners in a race are expected to finish within 260 + or - 47 minutes. 

Mean = 260 minutes
Standard Deviation = 41 minutes

260 minutes + 41 minutes = 301 minutes (or 5 hrs. 1 min.)

260 minutes - 41 minutes = 216 minutes (or 3 hrs. 36 min.)

Therefore, based on our sample of runners, approximately 68% of the runners in a race are expected to finish between 216 minutes and 301 minutes.

Approximately 95% of the data points fall within two standard deviations of the mean.

260 + (2 * 41) = 342 minutes ( or 5 hrs. 42 minutes)

260 – (2 * 41) = 178 minutes (or 2 hrs. 58 minutes)

Therefore, based on our sample of runners, approximately 95% of the runners in a race are expected to finish between 178 minutes and 342 minutes.

Approximately 99% of the data points fall within three standard deviations of the mean.

Presenting Data

Tables and graphs are convenient for presenting data. They present the data in an organized format, enabling the reader to find information quickly.

Tables

Tables should be numbered sequentially beginning with “Table 1.” Include a descriptive title. The title should enable the reader to understand the table without reading the rest of the document. When creating tables, be sure to state the units of all measurements. Some examples are minutes, milliliters, parts per million, etc.

Example

Table 1.  Number of bird species observed in 10 different woodlots in Clinton County, NY on January 18, 2006. Each bird count was done by two observers over a 1-hour period beginning at 8:00 AM.

Size of Woodlot (ha)	# Bird Species
3.3	11
8	13
3.6	13
13	14
1.1	7
11.0	14
7.4	12
6.6	14
8	12

Graphs

Graphs, drawings, and other illustrations are called “Figures.” They should be numbered and titled just like tables.

The scale used on each axis should cover the values in the data set. For example, if you are graphing 52 to 87, start at 45 or 50 and go to 90 or 95.

Variables

The data in the graph below was obtained by measuring the heart rate of a runner while running at different speeds. As can be seen, when the runner ran faster, heart rate increased.

Heart rate and speed are both referred to as variables. Heart rate depends on speed, so it is called a dependent variable. Speed does not depend on heart rate; it is an independent variable.

Dependent variables should be plotted on the Y-axis (vertical axis). Independent variables should be plotted on the X-axis (horizontal axis).

Bar Graphs

Bar graphs are best when the data are in groups or categories.

The data in the example below come from population counts of several different kinds of mammals in a woodlot in Clinton County, NY in July 2006.

Grey squirrel – 8

Red squirrels – 4

Chipmunks - 17

White-footed mice – 26

White-tailed deer – 2

A bar graph is best for these data because they are categories; it is not possible to have a data point that is between grey squirrels and red squirrels.

Line graphs

Line graphs are used when the data are continuous.

The data in the example below are pH measurements of an a pond in Clinton County, NY on 5/11/05. Samples of water were obtained every 2 hours beginning at 1:00 AM and ending at 11:00 PM.

  1:00 AM – 5.2

  3:00 AM – 5.1

  5:00 AM – 5.1

  7:00 AM – 6.0

  9:00 AM – 6.6

11:00 AM – 6.9

  1:00 PM – 7.0

  3:00 PM – 7.0

  5:00 PM – 6.6

  7:00 PM – 5.9

  9:00 PM – 5.3

11:00 PM – 5.2

A line graph is appropriate because the data are continuous. Although the researcher in this experiment did not measure the pH at 2:00 AM, it is possible that she could have done that. A line graph shows pH values for time periods that are between the times when samples were taken.

Scatter Plots

Scatter plots are useful to see if there is a relationship between two variables.

Suppose that a researcher wants to learn if there is a relationship between the size of the forest and the number of bird species that live in the forest. She collects the following data from different woodlots around Clinton County , NY in January 2006.

The points on the graph indicate that larger woodlots generally have more bird species.

Suppose that the data collected produced the graph below instead of the one shown above. The graph below shows a negative relationship; larger woodlots have fewer bird species.

Suppose that the data collected produced the graph below instead of the ones shown in the previous two examples. The graph below shows that size of woodlot does not seem to be related to number of bird species.

Rounding and Significant Digits

It is often desirable to round numbers. For most purposes in this laboratory course, numbers should be rounded to 3 significant digits. Some examples below illustrate this concept.

The number 35,832,487 can be rounded to 35,800,000. We use the three digits that are furthest to the left; the rest become zeros. The number 35,852,487 becomes 35,900,000. If the number to the right of the 3rd digit is 5 or greater, the 3rd digit is rounded up. If it is less than 5, it is rounded down.

The number 2.4815 becomes 2.48. The number 2.4855 becomes 2.49.

Example

The table below shows the number 12753122 rounded to several different significant digits.

Number of significant digits	Number
1	10000000
2	13000000
3	12800000
4	12750000

The table below shows the number 0.4382251 rounded to several different significant digits.

Number of significant digits	Rounded Number
1	0.4
2	0.44
3	0.438
4	0.4382