A population is a group of individuals of a single species within a circumscribed area. Populations are the fundamental unit for paleobiological studies since their evaluation permits examination of group attributes, such as patterns of growth and mortality. Two fundamental properties of populations can be examined in fossils; population size and population age structure.

Population Size
        Population size is the more difficult of the two properties to estimate due to problems of delineating a true population. Most fossil "populations" are actually  time-averaged samples that represent the accumulation of specimens over time period of few years to a few centuries. Consequently, they represent a sampling over successive populations of a species in the same area. If conditions are relatively constant this temporal sampling does not drastically alter population age structure, but population size estimates are almost impossible to obtain because the total time period of accumulation is rarely known.

        Population size estimates are usually only possible for fossil assemblages resulting from catastrophic death where a population is decimated by an environmental disaster. Catastrophic assemblages allow for the estimation of population size for single point in time. Several different methodologies are available for estimating population size although the most widely used is the minimum number of individuals (MNI = number of skeletal element X in the assemblage/number of skeletal element X in an intact skeleton) that can account for the assemblage. For example, a sample of 294 bison bones has 26 lumbar vertebrae. Since an intact buffalo skeleton has seven lumbar vertebrae, MNI = 26/7 = 3.7. In other words, there had to be at least four buffalo in the assemblage. In general, it is preferable to calculate MNI's for several different skeletal elements to produce separate estimates. This helps to eliminate problems arising from durability differences between different skeletal elements.

Population Age Structure
        The population age structure indicates the number of individuals in different size classes. The age structure records responses of the population to the environment and arises from interaction of two age-specific rates, growth and survivorship.

        Age-specific growth is the mean body size at different age classes. Age-specific growth for each population can assume one of three basic growth curves: linear growth (= growth is constant over time), slow exponential growth (= moderate elevation of juvenile growth rates and moderate reduction of adult growth rate), and rapid exponential growth (= pronounced elevation of juvenile growth rates and pronounced reduction of adult growth rate).

        Age-specific survivorship is the percentage of one age class surviving into the next age class. There are three basic survivorship curves: linear survivorship (= mortality constant over time), convex survivorship (= highest mortality among oldest adults), and concave survivorship (= highest mortality among juveniles).

        An understanding of growth and survivorship curves allows paleobiologists to make reconstructions of life histories of extinct species. Life history strategies for organisms are based on the nature of the environment and how it is perceived by the organism. In general, there are three common life history strategies.

        Opportunistic species are adapted to life in short-lived, variable environments. Their populations are characterized by rapid exponential growth and a short life span. In these species, relatively little energy is devoted to antipredator and anticompetitor devices. Consequently, they are readily outcompeted by many species. But because they are semelparous (= reproduce only once) they are adapted for quickly invading new habitats, growing and then reproducing rapidly before other species arrive.

        Stress-tolerant species are adapted to life in physiologically stressful environments, where many other species can not survive and reproduce. They have slow exponential growth, a long life span and are iteroparous (= reproduce repeatedly). Like opportunistic species, they devote little energy to antipredator and anticompetitor devices, relying instead on the inability of other species to survive in the same habitat.

        Finally, biotically competent species are adapted to life in stable, physiologically favorable environments. Like stress-tolerant species they have slow exponential growth, a long life span, and are iteroparous. But unlike either opportunistic or stress-tolerant species, biotically competent species divert large amounts of energy to antipredator and anticompetitor devices.

        Not all species neatly fit into one of these categories. Life history strategies that are intermediate in form are known. Also, some species can alter their life history strategy based on the environment where they occur.

        As an example, a sample of 180 specimens of the thick-shelled, extant bivalve (Protothaca staminea) were collected from a shell flat in Mugu Lagoon, California (data from R. R. Schmidt & J. E. Warme. 1969. Population characteristics of Protothaca staminea (Conrad) from Mugu Lagoon, California. Veliger 12: 193-199).

Size-frequency curve (Figure 1)

Age-frequency curve (Figure 2)

Growth curve (Figure 3)

Survivorship curve (Figure 4)

The size-frequency distribution is of relatively little value in reconstructing growth and survivorship curves. But the shells of bivalves record annual growth breaks that allow us to determine the age in years for each individual and produce an age-frequency distribution. From this data, a growth curve and a survivorship curve can be constructed. Growth in P. staminea is of the slow exponential form, while survivorship is convex. Unlike idealized curves, the survivorship curve for P. staminea is not purely convex, but rather weakly sigmoidal. Survivorship is lowest for older adults between ages five and six, but a few adults survive until age seven. This pattern is actually fairly typical of many invertebrates, where a few "lucky" individuals survive longer than expected.

        Based on the available data, P. staminea appears to be a biotically competent species. For a comparatively small bivalve (maximum size in study of 42 mm), P. staminea lives a relatively long time (up to seven years). Slow exponential growth and convex survivorship are consistent with a biotically competent life history strategy. Further, the robust shell of P. staminea provides a strong defense against a variety of potential predators. There is no direct evidence of reproductive strategy used by P. staminea, but the small body size and long life are compatible with iteroparity.

        To help you prepare for your term project, over the next two weeks we will reconstruct the population attributes for a sample of the extinct bivalve, Anadara staminea, from the Church Formation (Upper Miocene) of Maryland. This species, like most bivalves, has distinct annual growth lines that permit individuals to be easily assigned to specific age classes. In this exercise, we will determine the size-frequency distribution, age-frequency distribution, age-specific mortality curve, and age-specific survivorship curve for this species.

        In the next exercise, we will examine mortality in Anadara in more detail. Since annual growth lines are present in Anadara, it is possible to reconstruct seasonal differences in mortality within the sampled population. Seasonal aspects of mortality provides additional insights into the life style of Anadara.

        The most tedious part of analyzing this data is the grouping of individual measurements into classes. Fortunately, Excel has a sorting feature that will perform this task for you. Consider the following shell length data (in centimeters): 3.2, 0.8, 6.2, 1.2, 1.4, 0.4, 1.0, 1.1, 4.3 and 2.2. These data could easily be put into two centimeter size classes (2 cm, 4 cm, etc.) by inspection. But for a large data set, this procedure is prohibitively slow. This same data can be quickly sorted into size classes by Excel. Construct a dedicated spreadsheet to sort this data, using the following procedure:

1. Enter the heading Length into cell A1 and the heading Bins into cell C1. Enter the shell lengths from the data set into the spreadsheet (cells A2 through A11). Bins are the upper limits for each of the size classes we specify. We would like to sort the data in categories that are multiples of 2 cm. Since our largest shell length is 6.2, we will set the largest bin value at 8. Beginning in cell C2, enter the upper limit of the size classes in the individual cells (2 in C2, 4 in C3, etc.). It is important to list the bins sequentially in ascending order!

2. Click on Tools and then on Data Analysis. Scroll through the options and click on Histogram and OK. Note: if you do not have Data Analysis as a menu item in the Tools menu, click  [here].

3. Click in the register for Input Range: and then drag the pointer from cell A1 through cell A11. The register should now contain the range $A$1:$A$11. Repeat for the Bin Range (C1 through C5). Check the labels box, since the labels are included in the data and bins ranges. Place the upper lefthand corner of the Output Range at any convenient cell (e.g., E1). Check the Chart Output box and then on OK. The counts of shells within each bin and a graph of this data appear within the spreadsheet.

4. Edit the graph properties as in the previous exercise to make the graph easier to read.

To be really useful, a spreadsheet also should be able to perform more comprehensive analyses on the displayed data. Spreadsheets have a large number of built in functions that can be used to quickly perform almost any test. These functions are in the form of resident formulas that can be activated with the appropriate commands.

Mathematical Functions
Addition +
Subtraction -
Multiplication *
Division /
Summation  =SUM(cellX:cellZ)
Absolute Value =ABS(cellX-cellZ)
Square Root =SQRT(cellX)

Logical Functions
Comparison =IF(cellX>cellY, a, b) -- if value in cell X is greater than value in cell Y, use value a, otherwise use value b
Nested Comparison =IF(cellX>cellY, a, IF(cellX=cellY, b, c))-- if value in cell X is greater than value in cell Y, use value a, or if equal, use value b; otherwise use value c

Logical functions can be used with any of the following operators: = (equal); >(greater than); < (less than); >= (greater than or equal to); <= (less than or equal to) and <> (not equal). Also; constants can be used in place of cell addresses; e.g., =IF(cellX>constant, a, b).

Data Manipulation Functions
Count =COUNT(cellX:cellZ) -- counts the number of cells in thespecified range containing numerical values
Conditional Count =COUNTIF(cellX:cellZ, criterion) -- counts the number of cells in the specified range matching the criterion value
Table Definition =OFFSET(ref_cell, row, col, height, width) --indictes the number of rows and columns from a reference cell where a table begins and the height and width of the table


Descriptive Statistics
Mean =AVERAGE(array)
Median =MEDIAN(array)
Mode =MODE(array)
Standard Deviation =STDEV(array)
Variance =VAR(array)


        Excel also contains numerous additional resident functions to help with the customizing spreadsheets. Go to the menu bar and click on Help to access information on additional functions. The easiest means of accessing this information in Help is to click on the Contents tab and then browse through the available subjects. Find the topic category 'Creating Formulas and Auditing Workbooks'. Click on the + symbol to the left of this topic to open it. Browse through the subtopics and click on the + symbol to the left of Worksheet Function Reference. Three different areas provide a wealth of information about built-in functions: logical functions, math and trigonometry functions and statistical functions. Almost every function you would ever need to build a customized data analysis spreadsheet is already present.

        One obvious problem with these functions is that they each have very specific syntaxes. A simple typing error, such as a period where there should be a comma, can prevent the function from functioning. While the syntaxes have similar formats, memorizing the often subtle differences is tedious, at best. To overcome this problem, Excel has two features that greatly help with functions. First, since many of the functions used in statistics rely on whole ranges of data values, a shorthand means of calling whole data sets, or arrays, would be very useful. The standard format for calling a range of cells is to list the first and last cells of the array separated by a colon. For example, we might find it useful to know the mean length of the shells. The basic formula for this would be: =AVERAGE(A2:A11). When typed anywhere in the worksheet containing the data, the cell containing this formula would display the calculated mean value of 2.2. This method of specifying an array is adequate for simple functions, but gets bothersome with large complex data sets and complicated formulas. A simpler method is to give descriptive titles to arrays. Return to the shell data and use the mouse to highlight cells F2 through F5. Now click on the Insert menu, then on Name, then on Define. A dialog box appears asking you to name this array. Since this column of data has the heading Length, Excel will suggest this as the array name. Click on [OK]. We can now write the equation as: =AVERAGE(Length), which is much easier to remember and type without error.

        A second, even more powerful means of handling formulas and functions is the Excel a Function Wizard, indicated by the symbol; fx in the toolbar. The Function Wizard is invoked by clicking on the cell where the function is to be placed and then on the; fx button. Select the appropriate function from the menu, and then provide the information requested (e.g., data fields, sample sizes, etc.) and Excel inserts the function into the cell in the proper syntax!

Growth and Survivorship Curves

        Each research group should obtain one of the Anadara subsamples available in the laboratory for study:

1. Measure and record the length of each specimen (in millimeters).

2. Determine the age (in years) for each specimen by counting growth lines on the shell. Record this data.

3. Pool the length and age data from all research groups. This pooled data should be used to determine: (1) the size-frequency distribution for your specimens, (2) the age-frequency distribution, (3) the age-specific growth curve, and (4) the age-specific survivorship curve.

The equation for age-specific survivorship is:

where: %Sx = number surviving to age X, NT = total number of individuals in the sample, Nx = number of individuals in age 1 through age X.

        Open the lifehist spreadsheet to analyze this data. Data fields are set up for specimen identification number, length and age. Bins have been inserted for sorting the size and age data. Once you have entered and sorted the raw data, the spreadsheet will automatically calculate the mean length and % survivorship for each age class. With this information, you should construct the following four graphs:

  • size-frequency distribution (graph of number of individuals in each size class)
  • age-frequency distribution (graph of number of individuals in each age class)
  • age-specific growth curve (graph of mean length in each age class)
  • age-specific survivorship curve (graph of percentage of individuals surviving to a specific age class)
You should print copies of each of these graphs, as well as, a copy of the entire data table for your records.