St Aloysius College. A-level Biology MATHS SKILLS BOOKLET. Name: Teacher:

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1 St Aloysius College A-level Biology MATHS SKILLS BOOKLET Name: Teacher:

2 10% of the marks in your Biology exams will require the use of mathematical skills The following tables indicate where these mathematical skills could be assessed. Those shown in bold will only be tested in the full A Level course. These skills could be assessed in other areas of specification content but these are a guide as to where you may have encountered these skills before in a Biological context.

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6 Equations you need to learn: Density Density= Mass/Volume Hardy Weinberg Equation p + q = 1 p = frequency of dominant allele p 2 + 2pq + q 2 = 1 q = frequency of recessive allele

7 Equations you need to be able to use but do not have to learn. NB. You will not be expected to calculate a statistical test in your examinations.

8 Indices: Indices are a shorthand for multiplying a term by itself any number of times. They are also called powers or exponents. You may look at this when comparing volumes of liquid, rates of reaction, or converting units. e.g. This allows us to express = Addition: for any term a: a m a n = a m+n 2. Subtraction: for any term a : (3 3 3) ( ) = = = a m an = am n = = = = Multiplication: for any term a: (a m ) n = a m n (y y) (y y) (y y) (y y) (y y) = y 2 y 2 y 2 y 2 y 2 = (y 2 ) 5 However, from the original line, we can see this is equal to y 10 so (y 2 ) 5 = y Division: for any term a: a 1 n n = a From the first rule, we know that a 1 2 a 1 2 = a 1. We also know that a a = a Therefore, a 1 2 = a. There is a similar argument to show that a = a. Axioms: There are three more rules of indices which are important to remember and come directly from the definition of indices. a 1 = a Example: 2 1 = 2 a 0 = 1 Example: 8 0 = 1 a n = 1 a n Example: 2 3 = = 1 8 It is important to note that a common misconception is that there is a law for a m + a n. In general, there is no way to simplify this expression.

9 Example Qs on indices:

10 Standard Form: We use standard form to easily manage very large or very small numbers. Decimal Standard Form For example, the number may be written as In this form, is the product of two numbers: 8.7 is the digit number, and is the exponential number. A number is in standard form when it is written as a 10 n, where 1 a < 10 In standard form, the power of 10 shows the number of places the decimal point must be shifted to give the number in decimal form. A positive power will shift to the right, and a negative power will shift to the left. In standard form, the digit number also contains the number of significant figures in the number. The exponential number positions the decimal point. To convert to standard form, shift the decimal until there is one non-zero digit left of the decimal point, and count the number of places the decimal point has moved (this will be negative if your initial number was less than one). This number is the power of 10. Example Q 1: moles of a particular substance were dissolved in 2.5 dm 3 of water. What is the concentration of this substance? Give your answer in standard form To type a number in standard form on your calculator, - Input the digit number followed by the multiplication sign. - Locate the 10 x symbol, and use this to insert the exponent. - Check your equation for any needed brackets. To check, multiply and The answer should be Example Q 2: A cross section of an artery contains m 3 of blood. If this blood weighs g, calculate the density of the blood. Leave your answer in standard form.

11 Rounding and Significant Figures There are some simple rules to use when working out significant figures. Rule 1: All non-zero digits are significant. For example, 78 has 2 significant figures, has four significant figures and 340 has two significant figures. Rule 2: Intermediate zeros are significant. For example, 706 has 3 significant figures, and has six significant figures. Rule 3: Any leading zeroes are not significant. For example, has 3 significant figures (5, 6 and 7; ignore the leading zeroes). Rule 4: Zeroes at the ends of numbers containing decimal places are significant. For example, has 4 significant figures and has 5 significant figures. Significant Figures and Rounding: In rounding, when the next number is 5 or more round up, while if it is 4 or less don t round up. Measurement expressed by rounding to decimal places Number of decimal places Measurements expressed by rounding to significant figures Number of significant figures Measured to the nearest thousandth Ten thousandth Thousandth Hundredth Tenth Whole number If you aren t sure how to round your answer, you can work out the number of significant figures that you should round to by looking at the measurements you re using in the calculation. Just count the number of significant figures for each measurement and use the lowest number of significant figures for your answer. E.g = has 2 sf, 1.85 has 3 sf. So round your answer to 2 sf = 0.65 Significant figures and standard form: In standard form only the significant figures are written as digits, for example x 10 3 has four significant figures. If this were written as a straight number it would be This looks like it has only two significant figures but the significant figures are defined as the ones that contribute to its precision. Writing the number as 5600 implies precision only to the nearest whole hundred (could be or 5633). Using standard form allows precision to remain clearly as part of the stated number because all significant figures are written. Example Qs: 1. The growth rate of a plant is cm hour -1. What is the rate to: a) 3 decimal places? b) 3 significant figures? 2. A student is calculating the average growth rate of a tray of seedlings by dividing the average change in seedling height by the incubation time. The average change in seedling height is 17.5cm and the incubation time is 60 days. What is the average growth rate (in cm day -1 )? Give your answer to an appropriate number of significant figures.

12 Units and prefixes: One of the reasons we use the international system of units is because it makes the conversion of units (especially those with different prefixes) mathematically simple. We use prefixes as shorthand for standard form when using commonly occurring very large or very small numbers. For example, the length m may be written as m 2.3 is the digit number and is kept is known as the exponential number and can be replaced with the prefix n pronounced as nano. Hence: m = m = 2.3 nm Standard Units (SI Units) You should always clearly write units in your calculations: Base Units: Metre (m) for length, height distance Kilogram (kg) for mass Second (s) for time Mole (mol) for the amount of a substance Derived Units: Square metres (m 2 ) for area Cubic metre (m 3 ) for volume Cubic centimetre (cm 3 ) or ml for volume Degrees Celcius ( ) for temperature Mole per litre (mol dm -3 ) for concentration Joule (J) for energy Pascal (Pa) for pressure Volt (V) for electrical potential You may also encounter these non-si units: Litre (cubic decimetre) (L, dm 3 ) for volume Minute (min) for time Hour (h) for time Svedberg (S) (for sedimentation rate), used for ribosome particle size

13 To accommodate the huge range of dimensions in our measurements, units may be further modified using appropriate prefixes. Here is a table to show the range of dimensions in our measurements. In Biology we tend to use the shaded rows the most. Division Factor or Prefix Length units Mass units Volume units Time units exponential number One billion 10 9 giga times One million 10 6 mega times One 10 3 kilo kilometre km kilogram kg thousand times Whole unit 10 metre m gram g litre L or second S dm 3 Tenth 10-1 deci One 10-2 centi centimetre cm hundredth One 10-3 milli millimetre mm milligram mg millilitre ml millisecond ms thousandth or cm 3 One 10-6 micro micrometre µm microgram µg microlitre µl microsecond µs millionth One thousand millionth or one billionth 10-9 nano nanometre nm nanogram ng nanolitre nl nanosecond ns Example 1 The length of a DNA nucleotide is 0.6 nm. a) Convert this number into standard form. b) If a strand of DNA is 1.6 m long, how many nucleotides is it made up of? Converting between units: If you are converting a smaller unit to a larger one you divide. If you are converting a larger unit to a smaller one you multiply. Divide by 1000 for each step to convert in this direction nano micro milli Whole unit Kilo e.g. nm e.g. µm e.g. mm e.g. m e.g. km Multiply by 1000 for each step to convert in this direction

14 Example Qs: 1. Convert 1m to mm 2. Convert 1m to µm 3. Convert 20,000 µm to mm: Converting between square or cube units: One m 2 = 1000 x 1000 = mm 2 so your conversion factor becomes x or by One m 3 = 1000 x 1000 x 1000 = mm 3 so your conversion factor now becomes x or by Example Qs: 1. Convert 20m 2 to km 2 : 2. Convert 1m 2 to mm 2 : 3. Convert mm 3 to m 3 : 4. Convert m 3 to mm 3 : Manipulating Units: A number and a unit (like 3 m) is a magnitude (3) multiplied by a unit (metre). The rules of algebra apply not only to the numbers you are manipulating, but also to the units attached to them. For example: 3 m 3 m = 3 3 m m = 9 m m = 9 m 2 = 9 m 2 Units can be multiplied and divided just like regular number. For example: 6 m 3 2 m 2 = 3 m3 m 2 = 3 m m m m m = 3 m m m m m = 3 m At A-level, rather than write m/s to mean metres per second, we will write ms 1. This makes it easier to combine units via the following rules (remember the indices section!) unit a unit b = unit a+b 1 Nb This means that: = kg or more generally 1 kg 1 a n = an unit a unit b = unita b Example Qs: 36 cm3 a) Calculate the following: 12 cm 2 b) Calculate the following: 36 kg cm 3 64 cm 2

15 Ratios Understanding ratio allows us to easily compare separate quantities. We can then examine patterns, comment on the relationship, or use ratios to help us solve equations. For example: - Use 3 parts red paint to 1 part white paint. - Use 1 teabag to 250 ml of water. The order of the ratio is very important. We can use ratios to scale measurements, drawings, and calculations up and down. We can write a ratio as a fraction by scaling the ratio so that it is divided by the total number of parts. Example: To make mortar, we need 1 part cement, and 2 parts sand. The total number of parts for one batch of mortar is = 3. Thus the ratio for creating mortar is 1:2 which can also now be expressed as 1 3 : 2 3 From this form, it is easy to see how much of the total mixture is sand ( 2 3 ) and how much is concrete (1 3 ). Example Qs:

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17 Plotting 2 Variable Equations Plotting equations gives us a sense of what a relationship looks like. We can then easily see properties of the equation as well as compare it to other equations. To draw a representation of an equation, we first need coordinates to plot. We can achieve this by creating a table of values. These are just the x and y values that are true for the given line. To create a table of values, choose a set of x values, and substitute them into the equation and evaluate to get the y values. For example for the equation: y = 2x + 2 see the table of values to the right. Once you have completing your table of values, draw neatly and label a pair of axis. Once you have done this, plot the coordinates present in your table of values onto the graph. If they form a straight line, use a ruler to connect the coordinates. Otherwise, draw a curve freehand.

18 Averages Mean The mean is the total of the numbers divided by how many numbers there are. To work out the mean: Add up all the numbers = 72 Divide the answer by how many numbers there are. There are 9 numbers = 8 So the mean value is 8. Median is the middle value. To work out the median: Put the numbers in order: The number in the middle of the list is the median. So the median value is 7. If there are two middle values, the median is halfway between them. Work out the median for this set of numbers: There are two middle values, 7 and 8. The median is halfway between 7 and 8, so the median is 7.5. Mode The mode is the value that appears the most. To work out the mode: Put the numbers in order: Look for the number that appears the most. 6 appears more than any other number. So the mode value is 6. Percentages A percentage is simply expressing a fraction as a decimal. Percentage as proportion and fraction: e.g. percentage of pin eyed plants is 323/790 = 0.41 x 100 = 41% Percentage as chance e.g. in genetics ¼ x 100 = 25% of cystic fibrosis child from two parents. Percentage change. E.g. in osmosis experiments. A sample weighed 18.50g at the start and at the end it weighed 11.72g. Percentage change = mass change starting mass x 100 = Mass change = 6.78g x 100 = -36.7% (Remember it is negative because it was a loss in mass). Substituting into equations There are several equations that you will need to be able to use in A Level Biology. E.g. Simpson s Index of Biodiversity N means the total number of all individual organisms in a survey n means the total number of each different species. Σ = total of Brackets indicate sub-calculations must be done. Plant species A 84 B 2 C 6 D 8 TOTALS N(N-1) Number at site 1 (n) N n(n-1)

19 Number of cells Log 10 number of cells Rearranging equations The individual parts of terms in equations are all related, but sometimes you might know all the values of the terms except one. The equation can be re-written so that the unknown term can be calculated changing the subject of an equation. E.g. magnification = size of image size of real object Can be rearranged to: Size of image = magnification x real size and Real size = image size magnification Using Log in Biology Time The use of logarithms (log10) allows very large ranges of numbers to be easily accommodated on a simple axis by reducing the numbers to a scale between 0 and 10. Rather than trying to scale an axis using divisions at 1, 10, 100 and 1000 you can use the logs of these numbers, which are 0, 1, 2 and 3 respectively. The logs express numbers in powers of 10. Logarithmic data is often obtained from growth experiments with microorganisms. You calculate the log10 by typing in the number and clicking log10 on your calculator. To convert values you read off a log10 graph back use base 10 (10 x ). So for example you might want to determine the population at 150 minutes, read the y-value off the graph where time = 150. This is Converting this value in log10 back to base 10 gives 178 (3sf). You click 10 x and then the number on your calculator. Population as number of cells Log10 number of cells Number of cells against time for a culture of E. Coli Time Example Q: The table shows population counts of Vibrio bacteria after an initial inoculum at time = Log 10 number of cells against time for a culture of E. Coli Time a) Plot your data using a logarithmic scale for the population. Use the graph paper to the right but remember to add axes labels and a title. b) Use your graph to estimate the likely population of cells after 27 minutes. c) Use extrapolation to work out the population expected after 120 minutes. Time Populations/cells per cm

20 Using Statistics in Biology There are three statistical tests you need to be able to understand for A-level Biology as well as standard deviation: 1) the chi-squared test to test the significance of the difference between observed and expected results 2) the Student s t-test to compare the difference between two means 3) the correlation coefficient, e.g. Spearman s Rank Correlation Coefficient to look for an association between two sets of data. 4) the use of standard deviation What could you be asked to do in your exam: In preparing for written examinations, it will be important for students to understand how to select a statistical test that is appropriate for given data and to be able interpret the results of such a statistical test. Students could also be asked to justify their choices and interpretation as well as the following: Formulate a null hypothesis for the experiments they perform during their class work or when given appropriate information, for experiments carried out by others. Evaluate the null hypothesis of another investigator. Devise and justify an appropriate table in which to record their raw data. Devise and justify an appropriate way to represent their processed data graphically Evaluate the way in which another investigator has represented processed data Select and justify the selection of an appropriate statistical test for data they will subsequently collect themselves or data that might be collected by others. The statistical tests are restricted to: o chi-squared test when the data are categoric o the Student s t test when comparing the mean values of two sets of data o a correlation coefficient when examining an association between two sets of data. Evaluate the choice of a statistical test made by another investigator. Interpret a given probability value in terms of the probability of the difference between observed data and expected data (chi-squared test), the difference between the means of two samples (Student s t test) or a correlation between two variables (correlation coefficient) being due to chance. Interpret a given probability value in terms of acceptance or rejection of a null hypothesis, using 0.05 as the critical probability value. Evaluate the conclusions from the same data made by another commentator. Show an understanding of degrees of freedom so that, when given appropriate information, a student can use a given result of a statistical test to find the correct probability value from an abridged table of values. What will you not be asked to do in your exam: In written examinations, students might be asked to perform simple calculations such as finding a mean value. Students will not be asked to perform a calculation using a statistical test (or to calculate the standard deviation of a mean).

21 Writing a null hypothesis A null hypothesis is a hypothesis that says there is no statistical significance between the two variables. It is usually the hypothesis a researcher or experimenter will try to disprove or discredit. Example: An investigation to determine whether ph affects the rate of an enzyme controlled reaction Null hypothesis: There is no significant difference between the rate at which the enzyme works at different phs. Investigation 50 Grids were placed on the north side of 50 trees and the percentage cover of lichens was counted. This was repeated on the south side of the trees and the average percentage cover was compared. Null Hypothesis A group of students were classified in terms of personality (introvert or extrovert) and in terms of colour preference (red, yellow, green or blue) with the purpose of seeing whether there is an association (relationship) between personality and colour preference. A laboratory experiment was carried out in which the rate of digestion of egg white was determined at 12 different concentrations of protease enzyme. Criminologists have long debated whether there is a relationship between weather conditions and the incidence of violent crime. The article Is There a Season for Homicide? classified 1361 homicides according to season; spring, summer, autumn or winter. Measurements of the diameter were taken of 100 xylem vessels and 100 phloem cells to determine whether there was a difference in size of these cells. Volunteers who suffer from migraines were asked to trial a new drug, Pain-Go. Some were given the established market leader, Soreaway, and others were given Pain-Go. Each volunteer was asked to record the time taken to experience relief. On average, Pain-Go worked 38 minutes faster than Soreaway. As light intensity increases, so does the rate of photosynthesis (as long as CO2 and temperature are not limiting) Design an investigation to determine whether this statement is true.

22 Choosing a statistical test Learn this flow chart for how to decide which of the three statistical tests to use. Use the exact wording in the boxes in your exam answer. Examples: Q: Was the experimenter right to choose the Chi Squared test? A: Yes, because they were finding the number of individuals in particular categories Q: Which statistical test should be used for this data? Explain your answer A: T-test. Because the investigation involves taking measurements and looking for differences between mean values.

23 Which statistical test would you choose for these investigations? Investigation 50 Grids were placed on the north side of 50 trees and the percentage cover of lichens was counted. This was repeated on the south side of the trees and the average percentage cover was compared. Choice of Statistical Test A group of students were classified in terms of personality (introvert or extrovert) and in terms of colour preference (red, yellow, green or blue) with the purpose of seeing whether there is an association (relationship) between personality and colour preference. A laboratory experiment was carried out in which the rate of digestion of egg white was determined at 12 different concentrations of protease enzyme. Criminologists have long debated whether there is a relationship between weather conditions and the incidence of violent crime. The article Is There a Season for Homicide? classified 1361 homicides according to season; spring, summer, autumn or winter. Measurements of the diameter were taken of 100 xylem vessels and 100 phloem cells to determine whether there was a difference in size of these cells. Volunteers who suffer from migraines were asked to trial a new drug, Pain-Go. Some were given the established market leader, Soreaway, and others were given Pain-Go. Each volunteer was asked to record the time taken to experience relief. On average, Pain-Go worked 38 minutes faster than Soreaway. As light intensity increases, so does the rate of photosynthesis (as long as CO2 and temperature are not limiting) Design an investigation to determine whether this statement is true. Deciding if your results are significant For each statistical test calculated (and given to you in an exam) you will end up with an answer this is called the test statistic. Your next job is to work out the probability that you could get this test statistic by chance (fluke) or does it mean your results are significant. You need to decide if you test statistic is significant or not and come up with a conclusion as to whether to accept or reject the null hypothesis. You need to compare your test statistic with something called a critical value. If you test statistic is greater than the critical value your data is significant and you can reject your null hypothesis. If your test statistic is less than your critical value than your data is not significantly different that you would expect to happen by chance, and so you accept your null hypothesis. Example: if you were doing an experiment to find out which colour environment maggots preferred and you found 20,000,000 maggots went towards green light and 0 maggots went towards red light then you could be certain that maggots preferred green light. You could be 100% confident in rejecting your null hypothesis there is a 0% probability that these results occurred by chance. Statistical tests produce a test statistic that may be found in a table of probability. These tables (see next page) show the probability that the data you observed are different due to chance alone. In Biology we generally accept any probability greater than 5% as likely to be just chance or fluke, but probabilities of 5% or below show us that the data do differ significantly and there must be a cause influencing the outcome. Chi squared degrees of freedom is the number of categories minus 1 T-test useful for small sample sizes a few repeats. Degrees of freedom = total number of repeats/tests for both sets of data recorded, minus 2 (one for each category) Correlation coefficient n is the number of pairs of data investigated

24 Wording in your exam: Statistical test Correlation coefficient Statement if calculated value is greater than critical value at 0.05/ 5% significance level Calculated value is greater than the critical value so reject null hypothesis A probability of less than 0.05 or 5% that the correlation in results is due to chance OR Our calculated value of Spearman s rank correlation coefficient, rs is closer to +1 (if positive) or -1 (if negative) than the critical value. E.g. calculated value of closer to +1 than critical value of There is less than 5% probability that the positive correlation between the length and mass of the whale is due to chance. We reject our null hypothesis. Statement if calculate value is lower than the critical value at 0.05/ 5% significance level Calculated value is less than the critical value so accept the null hypothesis A probability of more than 0.05 or 5% that the correlation in results occurred due to chance OR Our calculated value of Spearman s rank correlation coefficient, rs is not closer to +1 (if positive) or -1 (if negative) than the critical value. E.g. critical value of is closer to +1 than calculated value of There is more than 5% probability that the positive correlation between the length and mass of the whale is due to chance. We accept our null hypothesis. Chi squared T-Test Our calculated value of Chi-squared is greater than the critical value of Chisquared. There is less than 5% probability that the differences between the observed and expected data are due to chance. We reject our null hypothesis. Our calculated value of t is greater than the critical value of t. There is less than 5% probability that the differences in the means (mean mass of bacterium A and mean mass of bacterium B) are due to chance. We reject our null hypothesis. Our calculated value of Chi-squared is less than the critical value of Chisquared. There is more than 5% probability that the differences between the observed and expected data are due to chance. We accept our null hypothesis. Our calculated value of t is less than the critical value of t. There is more than 5% probability that the differences in the means (mean mass of bacterium A and mean mass of bacterium B) are due to chance. We accept our null hypothesis.

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26 Statistics Exam Questions & Mark-Scheme Answers:

27 1) Species richness and an index of diversity can be used to measure biodiversity within a community. (a) What is the difference between these two measures of biodiversity? (1) Scientists investigated the biodiversity of butterflies in a rainforest. Their investigation lasted several months. The scientists set one canopy trap and one understorey trap at five sites. The canopy traps were set among the leaves of the trees m above ground level. The understorey traps were set under trees at m above ground level. The scientists recorded the number of each species of butterfly caught in the traps. The table below summarises their results. Species of butterfly Mean number of butterflies P value In canopy In understorey Prepona laertes 15 0 < Archaeoprepona demophon < Zaretis itys > 0.05 Memphis arachne < Memphis offa 21 3 < Memphis xenocles 32 8 < (b) The traps in the canopy were set at m above ground level. Suggest why there was such great variation in the height of the traps (1)

28 (c) By how many times is the species diversity in the canopy greater than in the understorey? Show your working. Use the following formula to calculate species diversity. d = where N is the total number of organisms of all species and n is the total number of organisms of each species. Answer =... (3) (d) The scientists carried out a statistical test to see if the difference in the distribution of each species between the canopy and understorey was due to chance. The P values obtained are shown in the table. Explain what the results of these statistical tests show (Extra space) (3) (Total 8 marks)

29 (a) Species richness measures only number of (different) species / does not measure number of individuals. 1 (b) Trees vary in height. 1 (c) 1. Index for canopy is 3.73; 2. Index for understorey is 3.30; 3. Index in canopy is 1.13 times bigger; If either or both indices incorrect, allow correct calculation from student s values. 3 Question (d) 1. For Zaretis itys, difference in distribution is probably due to chance / probability of being due to chance is more than 5%; 2. For all species other than Zaretis itys, difference in distribution is (highly) unlikely to be due to chance; 3. Because P < which is highly significant / is much lower than 5%. 3 Answer [8] A statistical test was carried out on the results to see if the results were significant. Both the probability values they obtained were p<0.01 Explain what this means A statistical test was carried out on the results to see if the results were significant. Both the probability values they obtained were p>0.05 Explain what this means 1. Differences significant; 2. Probability of difference being due to chance less than 1% 1. Differences insignificant 2. Probability of the differences being due to chance are greater than 5% What information does standard deviation tell you? Shows how spread out all the measurements are around the mean Gives an idea of how reliable measurements are/ mean is higher SD = less reliable If standard deviation bars overlap there is no significant difference between the two bars and any difference seen is due to chance What is a running mean used for? Look for number of samples where mean does not change/changes little/mean shows less fluctuation; Were two results enough to calculate a mean? No: Yes: Cannot recognise which is anomalous Need at least 3 to calculate representative mean Both results were concordant

30 Further Practice A Level Calculations Example Question: June 2013 Q 5 (b) Calculate the percentage increase in the mean rate of uptake of imatinib when the temperature is increased from 4 C to 37 C at a concentration of imatinib outside the cells of 1.0 µmol dm -3. Give your answer to one decimal place. 4 C 37 C Example Question: Jan 2013 Q4(b)(ii)

31 Question: June Q4 (a)

32 Question: Jan 2012 Q5 (b)

33 Question: June 2011 Q4 (b) HINT! Units: the clue to what you need to divide by what dm 3 s -1 = dm 3 /s 1 Question: June 2011 Q6 (c)

34 Question: Jan 2011 Q1 (d) Question: Jan 2011 Q3 (d)

35 Question: Jan 2011 Q5 (b)