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Statistical analyses are necessary when observed differences are smallcompared to experimental imprecision and biological variability. Whenyou work with experimental systems with no biological variability and littleexperimental error, heed these aphorisms:Introduction to statisticalcomparisonsIf you need statistics to analyze your experiment, then you've done thewrong experiment.If your data speak for themselves, don't interrupt!But in many fields, scientists can't avoid large amounts of variability, yetcare about relatively small differences. Statistical methods are necessary todraw valid conclusions from such data.The key concept: Sampling from a populationGarbage in, garbage outComputers are wonderful tools for analyzing data. But like any tool, dataanalysis programs can be misused. If you enter incorrect data or pick aninappropriate analysis, the results won't be helpful. Heed the first rule ofcomputers: Garbage in, garbage out.Sampling from a populationWhile this volume provides far more background information than mostprogram manuals, it cannot entirely replace the need for statistics texts andconsultants. If you pick an inappropriate analysis, the results won't beuseful.To do this, statisticians have developed methods based on a simple model:Assume that all your data are randomly sampled from an infinitely largepopulation. Analyze this sample, and use the results to make inferencesabout the population.GraphPad provides free technical support when you encounterproblems with the program. However, we can only providelimited free help with choosing statistics tests or interpreting theresults (consulting can sometimes be arranged for a fee).When do you need statistical calculations?When analyzing data, your goal is simple: You wish to make the strongestpossible conclusion from limited amounts of data. To do this, you need toovercome two problems: Important differences can be obscured by biological variability andexperimental imprecision. This makes it difficult to distinguish realdifferences from random variability. The human brain excels at finding patterns, even from random data.Our natural inclination (especially with our own data) is to concludethat differences are real, and to minimize the contribution of randomvariability. Statistical rigor prevents you from making this mistake.Introduction to statistical comparisons3www.graphpad.comThe basic idea of statistics is simple: you want to extrapolate from the datayou have collected to make general conclusions about the largerpopulation from which the data sample was derived.This model is an accurate description of some situations. For example,quality control samples really are randomly selected from a largepopulation. Clinical trials do not enroll a randomly selected sample ofpatients, but it is usually reasonable to extrapolate from the sample youstudied to the larger population of similar patients.In a typical experiment, you don't really sample from a population. But youdo want to extrapolate from your data to a more general conclusion. Theconcepts of sample and population can still be used if you define thesample to be the data you collected, and the population to be the data youwould have collected if you had repeated the experiment an infinitenumber of times.Note that the term sample has a specific definition in statisticsthat is very different than its usual definition. Learning newmeanings for commonly used words is part of the challenge oflearning statistics.Analyzing Data with GraphPad Prism4Copyright (c) 1999 GraphPad Software Inc.

The need for independent samplesIt is not enough that your data are sampled from a population. Statisticaltests are also based on the assumption that each subject (or eachexperimental unit) was sampled independently of the rest. The concept ofindependence can be difficult to grasp. Consider the following threesituations. You are measuring blood pressure in animals. You have five animalsin each group, and measure the blood pressure three times in eachanimal. You do not have 15 independent measurements, because thetriplicate measurements in one animal are likely to be closer to eachother than to measurements from the other animals. You shouldaverage the three measurements in each animal. Now you have fivemean values that are independent of each other. You have done a biochemical experiment three times, each time intriplicate. You do not have nine independent values, as an error inpreparing the reagents for one experiment could affect all threetriplicates. If you average the triplicates, you do have threeindependent mean values. You are doing a clinical study, and recruit ten patients from an innercity hospital and ten more patients from a suburban clinic. You havenot independently sampled 20 subjects from one population. Thedata from the ten inner-city patients may be closer to each other thanto the data from the suburban patients. You have sampled from twopopulations, and need to account for this in your analysis.Data are independent when any random factor that causes a value to betoo high or too low affects only that one value. If a random factor (that youdidn’t account for in the analysis of the data) can affect more than onevalue, but not all of the values, then the data are not independent.How you can use statistics to extrapolate from sampleto populationStatisticians have devised three basic approaches to make conclusionsabout populations from samples of data:The first method is to assume that the populations follow a specialdistribution, known as the Gaussian (bell shaped) distribution. Once youassume that a population is distributed in that manner, statistical tests letyou make inferences about the mean (and other properties) of thepopulation. Most commonly used statistical tests assume that thepopulation is Gaussian.The second method is to rank all values from low to high, and thencompare the distribution of ranks. This is the principle behind mostIntroduction to statistical comparisons5www.graphpad.comcommonly used nonparametric tests, which are used to analyze data fromnon-Gaussian distributions.The third method is known as resampling. With this method, you create apopulation of sorts, by repeatedly sampling values from your sample.This isbest understood by an example. Assume you have a single sample of fivevalues, and want to know how close that sample mean is likely to be fromthe true population mean. Write each value on a card and place the cardsin a hat. Create many pseudo samples by drawing a card from the hat, andreturning it. Generate many samples of N 5 this way. Since you can drawthe same value more than once, the samples won’t all be the same. Whenrandomly selecting cards gets tedious, use a computer program instead.The distribution of the means of these computer-generated samples givesyou information about how accurately you know the mean of the entirepopulation. The idea of resampling can be difficult to grasp. To learn aboutthis approach to statistics, read the instructional material available atwww.resample.com. Prism does not perform any tests based onresampling. Resampling methods are closely linked to bootstrappingmethods.Limitations of statisticsThe statistical model is simple: Extrapolate from the sample you collectedto a more general situation, assuming that your sample was randomlyselected from a large population. The problem is that the statisticalinferences can only apply to the population from which your samples wereobtained, but you often want to make conclusions that extrapolate evenbeyond that large population. For example, you perform an experiment inthe lab three times. All the experiments used the same cell preparation, thesame buffers, and the same equipment. Statistical inferences let you makeconclusions about what would happen if you repeated the experimentmany more times with that same cell preparation, those same buffers, andthe same equipment. You probably want to extrapolate further to whatwould happen if someone else repeated the experiment with a differentsource of cells, freshly made buffer and different instruments.Unfortunately, statistical calculations can't help with this furtherextrapolation. You must use scientific judgment and common sense tomake inferences that go beyond the limitations of statistics. Thus, statisticallogic is only part of data interpretation.The Gaussian distributionWhen many independent random factors act in an additive manner tocreate variability, data will follow a bell-shaped distribution called theGaussian distribution, illustrated in the figure below. The left panel showsthe distribution of a large sample of data. Each value is shown as a dot,Analyzing Data with GraphPad Prism6Copyright (c) 1999 GraphPad Software Inc.

with the points moved horizontally to avoid too much overlap. This iscalled a column scatter graph. The frequency distribution, or histogram, ofthe values is shown in the middle panel. It shows the exact distribution ofvalues in this particular sample. The right panel shows an ideal Gaussiandistribution.ColumnScatter GraphFrequency Distribution orHistogram250Probability Density501005000102030405060Bin Center70010203040506070ValueThe Gaussian distribution has some special mathematical properties thatform the basis of many statistical tests. Although no data follow thatmathematical ideal exactly, many kinds of data follow a distribution that isapproximately Gaussian.The Gaussian distribution is also called a Normal distribution.Don’t confuse this use of the word “normal” with its usualmeaning.The Gaussian distribution plays a central role in statistics because of amathematical relationship known as the Central Limit Theorem. Tounderstand this theorem, follow this imaginary experiment:1. Create a population with a known distribution (which does not haveto be Gaussian).2. Randomly pick many samples from that population. Tabulate themeans of these samples.3. Draw a histogram of the frequency distribu

analysis programs can be misused. If you enter incorrect data or pick an inappropriate analysis, the results won't be helpful. Heed the first rule of computers: Garbage in, garbage out. While this volume provides far more background information than most program manuals, it cannot entirely replace the need for statistics texts and consultants.