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276225123825AP Biology00AP Biology 4429125-415925-180975565785Mathematics and Statistics for AP BiologyAP 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 Tendency3752850878840LargeShoesFigure SEQ Figure \* ARABIC 1 Normal Distribution Curve SmallShoesLargeShoesFigure SEQ Figure \* ARABIC 1 Normal Distribution Curve SmallShoesIdentifying 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.11525251841500Mean (average) ? = averageN = 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 chamberNumber of Drosophila in left (treated) chamber0550.5911.0821.5822.0912.51003.0913.5734.0914.5915.0825.5736.0916.5737.0737.5738.0828.5919.0739.58210.0911) 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 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 = ________________Mode38195258255Figure SEQ Figure \* ARABIC 2 Bimodal distribution patternFigure SEQ Figure \* ARABIC 2 Bimodal distribution patternThe 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 VariabilityWhile 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.4086225183515Smallestvalue00Smallestvalue374332598425561022554610Largestvalue00LargestvalueRangeThe 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.453390012065Figure SEQ Figure \* ARABIC 3 Data range00Figure SEQ Figure \* ARABIC 3 Data range6) What is the range of the two data sets collected on drosophila behavior?Rangeright = _______________Rangeleft = ________________Standard Deviation and Variance2200275104584500Variance(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 samplen-1 = the degrees of freedomi = the number of measurementsxi = any given single measurement = sum of7) 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?33147001098550Figure SEQ Figure \* ARABIC 4 Normal distribution w/standard deviationFigure SEQ Figure \* ARABIC 4 Normal distribution w/standard deviationIn 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).-86677558420Figure SEQ Figure \* ARABIC 5 Normal distribution w/ changing standard deviationFigure SEQ Figure \* ARABIC 5 Normal distribution w/ changing standard deviationDied in DroughtSurvived DroughtBandBeak Depth (mm)BandBeak Depth (mm)28311.2101911.2127810.6191911.229410.5224411.0160910.5819110.8667410.5165910.7842210.3186110.742810.2159910.756110.2224910.6860510.2142610.614619.8220610.56119.8185010.43439.7141810.384209.757210.36769.7364210.284569.667310.14589.6147710.12939.5288710.15129.4221010.083479.31592105649.3710106199.36189.93119.223789.865069.23169.855099.23099.82889.13549.85039.114529.8468922119.761096789.74528.96169.63158.813729.484628.822429.455078.817979.315118.815879.33078.681909.286878.618849.153218.516359.13568.59439.12988.481368.93458.422268.95228.49318.998.316438.85198.36238.84138.229408.78276815288.55342812488.53467.95608.54577.8515278.383527.729398.316217.66858.2127.58918-91440381635To 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)4000020000To 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)-403860283337000-5746751945640Figure SEQ Figure \* ARABIC 6 Daphne MajorFigure SEQ Figure \* ARABIC 6 Daphne Major8) Calculate statistics for the Galápagos Finch data.Died in DroughtSurvived DroughtMeanMedianModeStandard Deviation9) What trend does the data show? ________________________________________________________________________10) Propose an explanation for the data. ________________________________________________________________________________________________________________________________________________4191005797550011) 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 ConfidenceWhen 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.Standard Error of the Mean 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 deviationn = total number of individuals in a sample4324350523875Figure SEQ Figure \* ARABIC 7 t table at the .05 (95%) level00Figure SEQ Figure \* ARABIC 7 t table at the .05 (95%) levelThe 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=2.0snThis 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.14859001009650012) Calculate the standard error of the mean and 95% confidence Interval for the Finch data. Died in DroughtSurvived Drought?sSEM95% CL13) Graph the data as a bar chart of the means showing error bars for both SEM and 95% CL.13049252667000 ................
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