Applying General Mathematics

Basic Arithmetic and Algebraic Calculations

• Arithmetic focuses on computations with specific numbers using operations like addition, subtraction, multiplication, and division.  
• Algebra is an abstract form of arithmetic that uses variables (represented by letters) to stand for unknown or unspecified quantities.  

Calculations Involving Proportions

• Decimals, fractions, percentages, and ratios all represent relationships between quantities, often parts of a whole, and can be converted into one another.
  • ​Decimals: Use place value to show parts of a whole in base-ten (e.g., 0.25). Most biological calculations use decimals, for example, calculating the size of a bacterial cell.
  • Fractions: Explicitly show a part divided by a whole (e.g., 1/4).  For example, use of Punnett grids to predict results genetic crosses. 
  • Percentages: Relate a number to a whole of 100, indicating a part out of 100 (e.g., 25%).  For example, calculations of percentage change and percentage difference.
  • Ratios: Compare quantities in a part-to-part or part-to-whole relationship (e.g., 1:3).  For example, calculation of the surface-area-to-volume ratio.
• Density is the proportion of the number of individual items divided by a defined area or volume. For example, population density in ecology or stomatal density in plant biology.
• Relative frequency of a data value is the frequency divided by the total number of recorded values. It indicates the proportion of results which take that value.​  Commonly used in understanding change in allele frequency in a population.
• Reciprocals are the multiplicative inverse of a number, such as 1/x.  Reciprocals are often calculated to estimate the rate of a reaction, using 1/T where T is time.

Measures of Central Tendency
Measures of central tendency are used to find a "typical" or "central" value in a dataset. These descriptive statistical provide a concise summary of complex data distributions.

• Mean: the arithmetic average, calculated by summing all data points and dividing by the total number of points
• Median: the middle value when the data is arranged in order
• Mode: the value that appears most frequently in the dataset

Measures of Dispersion
Measures of dispersion are measures of data variability.

• Range:  the difference between the highest and lowest values in a dataset.  Highly susceptible to outliers and is generally not recommended for datasets with more than five trials. 
• ​Standard Deviation (SD):  a descriptive statistics that calculates the typical amount by which individual data points in a sample deviate from the mean of the data. Used when the data follows a normal (bell-shaped) distribution and there are at least five trials (repeats) of measurement. When data points are tightly clustered around the mean, the SD is low; when data points are spread out, the SD is high. 
• Interquartile range (IQR): a descriptive statistics that represents the range of the middle half of a dataset, calculated as the difference between the third quartile (75th percentile) and the first quartile (25th percentile).  Used when outliers are present or the data is not normally distributed. 
• Standard Error (SE): an inferential statistics that quantifies the precision of a sample mean (average of measurements in a sample of a population) as an estimate of the population mean (the actual mean if a measurement was made of every individual in the population). A smaller standard error suggests a more precise estimate, implying the sample mean is likely closer to the population mean. ​ ​

Scientific Notation
Many people doing scientific work deal with very large or very small numbers. To avoid having to write and count lots of zeros, they use scientific notation to write numbers. Scientific notation (or standard form) involves writing a given number as a number between 1 and 10, multiplied by a power of 10.

Example:  9,448,800,000 

Example:  0.000 000 053 04

Calculators often display numbers using scientific notation.  However, on an exam it is not acceptable to write an answer as a calculator display.  For example:  870,000 X 95,000,000 = 8.265E13 would be written as = 8.265 x 10^13 

Approximation and Estimations
Both methods used to obtain values that are close to the true or accurate values.

• Approximation: a mathematical technique that finds a value close to the true value, simplifying a complex problem. Approximations are not necessarily be very precise or accurate. Itis often used when an exact calculation is challenging ortime-consuming and a reasonably close value is sufficient.
• Estimation:  making an informed prediction (or educated guess) about a biological quantity, often using available data and experience.  Itis used when the true value of a quantity is unknown or cannot be directly measured.  For example, when estimating a sample size in a microscopic field of view without a ruler. 

Scales of Magnification
Scales of magnification refer to how much a specimen is enlarged for viewing.  A labeled scale bar can be measured with a ruler and the magnification of an image is calculated using the formula:

Rates of Change
The rate of change measures how quickly a biological variable, such as population size, enzyme reaction rate, or genetic makeup, changes over a specific period.

• To calculate the rate of change from data, calculate the difference between the final and initial values and then divide by the elapsed time for the change to occur.
  
• To determine the rate of change from graph, calculate the slope by picking two points (x₁, y₁) and (x₂, y₂) and using the formula Slope = (y₂ - y₁) / (x₂ - x₁), where 'y' represents the dependent (responding) variable and 'x' the independent (manipulated) variable. The steeper the slope, the greater the rate of change; a positive slope indicates an increase, and a negative slope indicates a decrease. 

Describing Relationships between Variables
Proportionate means one variable is always a fixed, constant multiple of another, indicating a constant relationship.

Directly Proportionate:  two variables change in the same direction, consistently increasing or decreasing together. 

​Inversely Proportionate:  two variables consistently change in opposite directions: one increases while the other decreases.  

Correlated means two variables tend to change together in some way, but not necessarily at a constant ratio or with a fixed relationship. See this page for more information about correlations. 

Positively Correlated:  two variables move in the same direction, meaning as one increases, the other tends to increase or as one decreases, the other tends to decrease.

Negatively Correlated:  two variables move in opposite directions, meaning as one increases, the other tends to decrease or as one decreases, the other tends to increase.

A direct proportion is a specific type of positive correlation, but correlation does not imply proportionality, as variables can be correlated without being proportional.

Percent Change and Percent Difference

Percent change indicates how much a value has increased or decreased relative to its original value over time, calculated by dividing the difference between the new and old values by the original value. Used to track changes over time or when one value is clearly an "initial" or "original" value and the other is a "final" or "new" value. 

Percent difference is used to compare two values to determine how much they differ from each other as a percentage. It is used when comparing two values of the same kind that do not have a distinct start or end point, such as comparing two different measurements. Percent difference can be calculated in two ways:

1.  If provided a baseline or number to compare to, divide the difference by the reference value. ​

2.  When there is no baseline or provided number to compare to, divide the difference by the average of the two numbers.  

Continuous and Discrete Variables
​Quantitative observations can be discrete or continuous

Discrete:  Involve data that can only take specific, separate values, typically whole numbers.  Arise from counting or categorizing phenomena.

Discrete data is often visualized on a histogram or bar chart

Continuous: Involve data that can take any value within a given range, including fractions or decimals.  Arise from measuring phenomena. 

Continuous data is often visualized on a line chart or scatterplot

Biological Index
An index calculation converts raw numbers for different variables into a single, relative value.  Index numbers make it easier to understand trends and differences in samples.  For example:

• The mitotic index is used to compare rates of cell division in different tissue samples. See D2.1.17

• The Simpson’s reciprocal index is used to measure the relative biodiversity of a given community It accounts for both the number of species present (richness) and the number of individuals per species (evenness).  ​

• The Lincoln index is used to estimate the population size of a species using a mark-and-recapture technique.  See C4.1.4

Chi-Square Test
​The Chi-square (χ²) test is an inferential statistical tool used to analyze categorical data. It is used to assess phenomena such as Mendelian inheritance patterns in offspring (see D3.2.21*), community ecology questions about species associations (see C4.1.15), or the distribution of organisms in an environment. The test compares actual (observed) data with predicted (expected) data to see if the difference is likely due to random chance or a real underlying factor.  See these pages for more information about the chi-square test of independence or the chi-square goodness-of-fit test. 

The t-test
The t-test is an inferential statistical method used to determine if the means of two sampled groups are significantly different from each other. The t-test helps researchers conclude if observed differences between a control group and an experimental group (or between two conditions) are likely due to chance.  See this page for more information about the t-test.