Mathematics and Statistics for AP Biology

AP Biology Testing You will be given six ‘grid in’ calculation problems on the national AP exam, as

well as on the class final. There may also be calculation and graphing components to both the multiple choice and the free response sections. To answer those questions a simple calculator is needed. Buy a cheap five function (+, -, x, , ). You will not be allowed to use a graphing calculator on the test.

You will be provided with a formula sheet on the day of the test. Do not bring any formulas or written notes with you on the testing day. Mr. Ballog will provide you with the most recent official formula sheet for class tests as well as for completing this set of exercises.

Practice ExercisesThis set of exercises includes sample problems that are, for the purpose of

instruction, fairly simple. The problems on your exams will not involve any more complex mathematics but the concepts they are associated with may be of a more complex nature.

PART 1 – Measures of Central Tendency

Identifying what is happening in the middle of any data set often offers researchers a lot of information. Many sets of biologically derived data fit a normal curve. A normal curve shows the distribution of the range of data. If you were to measure everyone’s shoe size in the class there would be a few students who wear very small shoes and a few who wear very large shoes, with most students wearing shoes somewhere near the middle of the range. The mean, median, and mode are the most widely used measures to describe how collected data clusters in the middle of a normal distribution (central tendency). As a general rule use;Mean when the data closely fits a normal curve,Median when data are skewed to one end of the distribution or the other or when there are extreme outliers in the data,Mode is not often used in Biological research but is valuable to identify data patterns that are bimodal.

Mean (average) = average

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APAP BiologyBiology

SmallShoes

Figure 1 Normal Distribution Curve

LargeShoes

N = total number of individuals in the entire populationn = total number of individuals in a samplei = the number of measurementsxi = any given single measurement = sum ofWhat this formula says is; add up all instances of the data and divide by the number of data points – but you already knew that!

Below is a table of data recorded during a behavioral study of fruit flies. The data was taken over a 10 minute period by counting the number of flies found in two different chambers. The left (treatment) chamber had a cotton ball saturated with a substance selected by the students. Ask Mr. Ballog to show you the set-up of the experiment.

Time (minutes)

Number of Drosophila in right

chamber

Number of Drosophila in left (treated)

chamber0 5 5

0.5 9 11.0 8 21.5 8 22.0 9 12.5 10 03.0 9 13.5 7 34.0 9 14.5 9 15.0 8 25.5 7 36.0 9 16.5 7 37.0 7 37.5 7 38.0 8 28.5 9 19.0 7 39.5 8 210.0 9 1

1) What is the value for N for this experiment? _______2) What is the value for i for this experiment? _______3) Calculate the mean of the data for both chambers over the course of the experiment.

right chamber = _________ left chamber = _________

Median The median is the data value that lies in the very middle of a set of data. Half of the data will be below the median while the other half will lie above the median. Unlike the

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mean, whose value may not even be represented in the data, the median is one of the data values – well, usually. In a data set with an even number of data points the median will be the average (mean) of the two central data points. The median is used when there are a few extreme values in the data set that might give an erroneous view of the central value of the data set. It has the advantage of showing what value the data set ‘revolves’ around.To find the median you arrange the data points in ascending numerical order. The middle data point in this arrangement is the median.

4) What is the median of the data sets collected in the drosophila (fruit fly) experiment?

Medianright = _______________ Medianleft = ________________

ModeThe mode is the data value that occurs most frequently in a set of data. At times it may be useful to describe a data set as being bimodal. This occurs in populations that exhibit disruptive selective pressures. Neither the mean or median would show this tendency in a data set.

5) What is the mode of the data sets collected in the drosophila experiment?

Moderight = _______________ Modeleft = ________________

PART 2 – Measures of Variability

While the measures of central tendency show how the collected data clusters, measures of variability describe how data spreads out. These measures give an idea of the shape of the normal distribution and how much variation individual data points exhibit. Range, standard deviation and variance are the most widely used measures of variability.

RangeThe range in a data set simply shows how far apart the smallest and largest data points are. These data values populate the two extreme tails of the full data set. To determine the range identify the smallest data value and subtract it from the largest data value.6) What is the range of the two data sets collected on drosophila behavior?

Rangeright = _______________ Rangeleft = ________________

Standard Deviation and Variance

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Figure 2 Bimodal distribution pattern

Smallestvalue

Largestvalue

Figure 3 Data range

Variance(s2 or σ2) and standard deviation(s or σ) are two closely related measures of variability. In order to calculate the standard deviation of a data set you must first calculate the variance of the same data set. Standard deviation basically tells us how far data points deviate from the mean. You measure how far a data point is from the mean and then find the average of all of the calculated distances from the mean. The formula sheet provides the following algebraic definition:

Where;s = the standard deviations2 = variance = averagen = total number of individuals in a sample

n-1 = the degrees of freedomi = the number of measurementsxi = any given single measurement = sum of

7) Go ahead and calculate the variance and then standard deviation of the drosophila data. By this time you should notice that the two sets are in essence just inverses of each other (If a fly is not in one chamber it is in the other) so if you determine the standard deviation for sides data set it will be the same as for the second sides set.

Standard Deviation (s) = ________________

What does the standard deviation tells us about the distribution of the data?In a normal curve the distribution of the data is determined by the standard deviation as shown at right. 68% of the data is within one standard deviation of the mean; 95% is within 2 standard deviations of the mean; while 97% of the collected data will fall within 3 standard deviations of the mean. As the standard deviation becomes smaller the data clusters more closely to the mean. The distribution curve is therefore more closely centered about the mean (below).

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Figure 4 Normal distribution w/standard deviationFigure 5 Normal distribution w/ changing standard deviation

8) Calculate statistics for the Galápagos Finch data.

Died in Drought

Survived Drought

MeanMedianModeStandard Deviation

9) What trend does the data show? ________________________________________________________________________

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To gain further practice and to make this a bit less repetitive we will introduce a different data set. This is the data collected by researchers Peter and Rosemary Grant on Daphne Major in the Galápagos Islands. The data shows the change in beak depth of a population of finches following a draught year (1977)

Figure 6 Daphne Major

Died in Drought Survived DroughtBand Beak Depth

(mm)Band Beak Depth

(mm)283 11.2 1019 11.21278 10.6 1919 11.2294 10.5 2244 11.01609 10.5 8191 10.86674 10.5 1659 10.78422 10.3 1861 10.7428 10.2 1599 10.7561 10.2 2249 10.68605 10.2 1426 10.61461 9.8 2206 10.5611 9.8 1850 10.4343 9.7 1418 10.38420 9.7 572 10.3676 9.7 3642 10.28456 9.6 673 10.1458 9.6 1477 10.1293 9.5 2887 10.1512 9.4 2210 10.08347 9.3 1592 10564 9.3 710 10619 9.3 618 9.9311 9.2 2378 9.86506 9.2 316 9.85509 9.2 309 9.8288 9.1 354 9.8503 9.1 1452 9.8468 9 2211 9.7610 9 678 9.7452 8.9 616 9.6315 8.8 1372 9.48462 8.8 2242 9.45507 8.8 1797 9.31511 8.8 1587 9.3307 8.6 8190 9.28687 8.6 1884 9.15321 8.5 1635 9.1356 8.5 943 9.1298 8.4 8136 8.9345 8.4 2226 8.9522 8.4 931 8.99 8.3 1643 8.8519 8.3 623 8.8413 8.2 2940 8.78276 8 1528 8.55342 8 1248 8.5346 7.9 560 8.5457 7.85 1527 8.38352 7.7 2939 8.31621 7.6 685 8.212 7.5 891 8

10) Propose an explanation for the data. ________________________________________________________________________________________________________________________________________________

11) Using the standard deviation calculations sketch the population standard curve for each set of data. (Use the information from the bottom of page four to help you plot the curve) Label graph and axis.

PART 3 – Measures of Confidence

When you sample a population it is just that, a sample, and may not give accurate information concerning the entire population (here a population also refers to a set of any recorded data).The measurements taken of the finches just represents a sample of the entire population of finches. Statistics provides a way to communicate how much error may have been in collected data due to sampling error. The more closely the sample size approaches the entire population the smaller the sampling error until the point where the entire population is sampled and no error is present. Two measures of confidence will be presented here; The Standard Error of the Mean and the 95% Confidence Interval.

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The standard error of the mean utilizes the standard deviation of the sample and the sample size to estimate how closely the sample data approximates the data that would be collected if the entire population were measured. The formula for the standard error of the mean is;

Where;s = the standard deviation n = total number of individuals in a sample

The standard error of the mean tells you that 68.3% of the sample means are within ±1standard error of the entire population mean. This can be expanded and refined to show a 95% confidence interval using the 95% Confidence Interval.

95% Confidence IntervalMost research is aimed at having sample populations model the entire population with 95% confidence. The 95% confidence interval shows the range of data that may be represented in the population within 2 SD of the sample mean. To adjust for sample size a table of values based on the degrees of freedom is referenced. The following equation is used for determining the 95% confidence interval and placement of error bars on data graphs.

95 %CL=SE x tP (n−1)

Where;SE = Standard Error of the MeantP (n−1 ) = value from t table at the .05 level for n-1 degrees of freedom.For large sample sizes (≥ 30) the t value approximates 2 so the following equation may be used without calculating the standard

error. 95 %CL=2.0 s√n

This puts your level of sampling error within 95% 0f the entire population. (Note: The AP exam will only address the simpler Standard Error of the Mean calculation)The Standard Error of the Mean and the 95% Confidence Interval is used to provide error bars for graphs showing the mean values of data sets. The error bars show the range 1 standard error above and one below the mean value.The graph below shows data means graphed with error bars for a calculated SE = 5. Bars are drawn 5 units above and below the sample means. The AP exam may have you include error bars in graphs.

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Figure 7 t table at the .05 (95%) level

12) Calculate the standard error of the mean and 95% confidence Interval for the Finch data.

Died in Drought Survived DroughtsSEM95% CL

13) Graph the data as a bar chart of the means showing error bars for both SEM and 95% CL.

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