Applying General Mathematics in Biology (HL IB Biology)

• Biology often requires the use of calculations, which can include
  • Decimals
    • Most biological calculations use decimals, e.g. calculating the size of a bacterial cell
  • Fractions
    
    • Most scientific calculators will initially give answers as fractions
      • Make sure you know where the S⇔D button is so that you convert the fraction into a decimal 
  • Percentages
    
    • There are many percentage calculations, including percentage change and percentage difference
  • Ratios
    • The most common ratio requiring understanding is that of surface area to volume ratio
  • Proportions
    • Proportionality can be used to understand quantity and scale and is important in biology in topics such as cell biology when creating biological drawings of cells and tissues from a microscope image or micrograph
  • Frequencies
    
    • This is most commonly used in understanding change in allele frequency
  • Densities
    • We often look at and examine population density in ecology or stomatal density in plant biology
  • Approximations 
    • This is used to obtain an approximate value for example when using the magnification formula
  • Reciprocals
    • We frequently used reciprocals (1/n) when dealing with concentration versus rate graphs, using 1/T where T is time

Measures of central tendency

•  Measures of central tendency involve calculations of mean, median and mode which you should be able to apply to a range of scenarios and contexts
  • Mean
    • The mean is an average of a group of numbers calculated by totaling all values and dividing by the number of values
    • Mean is used to summarise a dataset with a single number which represents the data's typical value
  • Median
    • This is the middle number which can be found by ordering all values and picking out the one in the middle
    • It helps us to understand that 50% of values have are smaller or equal to the median and 50% of values are higher or equal to the median
  • Mode
    • This is the most frequent value in a dataset
    • It can be useful to understand the most common value in categorical data when the mean and median can't be used

Measures of dispersion

• Measures of dispersion involve applying calculations of standard deviation (SD), standard error (SE) and interquartile range (IQR) to a range of contexts
• These ideas are also considered here with reference to the use of error bars on graph 
  • Standard Deviation
    • The mean is a more informative statistic when it is provided alongside standard deviation
    • Standard deviation measures the spread of data around the mean value
      • It is very useful when comparing consistency between different data sets
    • The mean must be calculated before working out the standard deviation

• • Standard Error
    • Standard error of the mean measures how far the mean of the data is likely to be from the true mean
    • It measures the accuracy with which a sample represents a population
    • The SE is always smaller than the SD
  • Interquartile Range
    • This is another method of analysing dispersion of data
    • It is the difference between the 75th and 25th percentiles of the data
      • Quartiles are the values that divide the whole series into four equal parts

Scientific notation

• Scientific notation is also known as standard form
• It is a system of writing and working with very large or very small numbers
• Numbers in scientific notation are written as:

a × 10ⁿ

• They follow these rules:
  • a is a number above 1 and below 10
  • For large numbers, n is an integer that is greater than 0 
    • i.e It shows how many times a is multiplied by 10
  • For small numbers, n is an integer that is less than 0 
    • i.e It shows how many times a is divided by 10
  • n < 0 for small numbers i.e how many times a is divided by 10

Approximation and estimation

• Approximation and estimation are both methods used to obtain values that are close to the true or accurate values
  • While they share some similarities, they have distinct characteristics and are used in different contexts

Approximation

• Approximation involves finding a value that is close to the actual value of a quantity
  • It may not necessarily be very precise or accurate
• It is often used when an exact calculation is challenging or time-consuming and a reasonably close value is sufficient 

Estimation

• Estimation involves making an educated guess or assessment based on available information or data
• It is used when the true value of a quantity is unknown or cannot be directly measured
  • For example biologists estimate dates of the first living cells and the last universal common ancestor or the method of estimating times by use of the “molecular clock”

Scales of magnification

• Magnification is an important skill used widely in biology and frequently assessed in examinations
• For more information and worked examples see our revision note on microscope skills

Rates of change

• The rate of change tells us how something changes over time
  • For example oxygen consumption in germinating seeds over a period of days
• To determine rates of change from tabulated data, you can use the average rate of change or gradient, if the data has been plotted as a graph
• The average rate of change between two points on a graph or in a table is:

Rate of change = 

Proportionality and correlations

• There are a number of terms that are commonly applied to trends, particularly in graphs
  
  • Direct and inverse proportionality
    • Direct proportionality applies to a trend that has a clearly linear relationship which means the relationship can be described as "when one variable increases, the other increases" or "if x doubles, then y doubles"
    • Inverse proportionality means that the relationship can be described as "when one variable increases, the other decreases" or "if x doubles, then y halves" 
  • Positive and negative correlations
    
    • Positive correlations show when the gradient of the graph is positive / slopes or curves upwards and describes a relationship where as x increases, y also increases
    • Negative correlations is when the gradient of the graph is negative / slopes or curves downwards; this describes a relationship where as x increases, y decreases

Percentage change and percentage difference

• Percentage change and percentage difference are commonly used to express the relative change between two values
  • They are useful for comparing experimental results, determining reaction yields and analysing other chemical data

Percentage change

• Percentage change is used to express the relative change between an initial value and a final value
• It is calculated using the following formula:

Percentage Change = 

Percentage difference

• Percentage difference is used to compare two values to determine how much they differ from each other as a percentage
• This can be done in two different ways:
• Method 1: as a measure of relative change
  • This method is the most commonly seen in exams, it calculates how much one value has increased or decreased relative to another.
  • The formula you need to use depends on which number you use as the devisor.
  • If you are calculating the difference as a percentage of number A, you should use the following formula:

Percentage Difference = 

• Method 2: a measure of symmetrical percentage difference
  • This method is used when there is no baseline, or starting number to compare to.
  • You should use the following formula:

Continuous and discrete data

• Discrete data is quantitative
  • It consists of separate, distinct and countable values
  • For example:
    • Number of an organism in a sample
• Continuous data is also quantitative
  • It is based on measurements and can include decimal numbers or fractions
  • This allows for an infinite number of values 
  • For example:
    • The temperature of an enzyme reaction as time progresses
    • The volume of oxygen gas produced during a photosynthesis reaction

Statistical tests

• Statistical tests can be used to analyse a range of different data sets
• The type of test used will depend on a number of factors such as 
  • The size of the sample
  • They type of data, i.e. is it discrete or continuous
  • The nature of the question being investigated

Simpson's reciprocal index

• The Simpson’s reciprocal index can be 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)
• A higher index value is indicative of a greater degree of biodiversity within the community

The Lincoln index.

• This calculation allows an estimate of population sizes of individual animal species
• You can read more about the Lincoln Index here

Chi-squared test

• A chi-square test is a statistical test that is used to compare observed and expected results
• Our revision notes here cover this in detail

The t-test

• The t-test can be used to compare the means of two sets of data and determine whether they are significantly different or not
• The sets of data must follow a rough normal distribution, be continuous and the standard deviations should be approximately equal