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Statistics for Biology Descriptive Statistics mean number of times each value occurs Repeated measurements in biology are rarely identical, due to random errors and natural variation. If enough measurements are repeated they can be plotted on a histogram, like the one on the right. This usually shows a normal distribution, with most of the repeats close to some central value. Many biological phenomena follow this pattern: eg. peoples' heights, number of peas in a pod, the breathing rate of insects, etc. 95% CI 95% CI normal distribution curve values The central value of the normal distribution curve is the mean (also known as the arithmetic mean or average). But how reliable is this mean? If the data are all close together, then the mean is probably good, but if they are scattered widely, then the calculated mean may not be very reliable. The width of the normal distribution curve is given by the standard deviation (SD), and the larger the SD, the less reliable the data. For comparing different sets of data, a better measure is the 95% confidence interval (CI). This is derived from the SD. The purpose of taking a random sample from a lot or population and computing a statistic, such as the mean from the data, is to approximate the true mean of the population. A 95% confidence interval provides a range of values which is likely to contain the population parameter of interest (in this case, the average) in approximately 95% of the cases. You can be pretty confident that the real mean lies somewhere in this range. Whenever you calculate a mean you should also calculate a confidence limit to indicate the quality of your data. small confidence limit, low variability, data close together, mean is reliable large confidence limit, high variability, data scattered, mean is unreliable In Excel the mean is calculated using the formula =AVERAGE (range) , the SD is calculated using and the 95% CI is calculated using =CONFIDENCE (0.05, STDEV(range), COUNT(range)) . =STDEV (range) , This spreadsheet shows two sets of data with the same mean. In group A the confidence interval is small compared to the mean, so the data are reliable and you can be confident that the real mean is close to your calculated mean. But in group B the confidence interval is large compared to the mean, so the data are unreliable, as the real mean could be quite far away from your calculated mean. Note that Excel will always return the results of a calculation to about 8 decimal places of precision. This is meaningless, and cells with calculated results should always be formatted to a more sensible precision, usually 2 decimal places (Format menu > Cells > Number tab > Number). see also: https://explorable.com/statistics-tutorial Plotting Data Once you have collected data you will want to plot a graph or chart to show trends or relationships clearly. With a little effort, Excel produces very nice charts. First enter the data you want to plot into two columns (or rows) and select them. Drawing the Graph. Click on the chart wizard . This has four steps: 1. Graph Type. For a bar graph choose Column and for a scatter graph (also known as a line graph) choose XY(Scatter) then press Next. Do not choose Line. 2. Source Data. If the sample graph looks OK, just hit Next. If it looks wrong you can correct it by clicking on the Series tab, then the red arrow in the X Values box, then highlight the cells containing the X data on the spreadsheet. Repeat for the Y Values box. 3. Chart Options. You can do these now or change them later, but you should at least enter suitable titles for the graph and the axes and probably turn off the gridlines and legend. 4. Graph Location. Just hit Finish. This puts the chart beside the data so you can see both. Changing the Graph. Once you have drawn the graph, you can now change any aspect of it by double-clicking (or sometimes right-clicking) on the part you want to change. For example you can: move and re-shape the graph change the background colour (white is usually best!) change the shape and size of the markers (dots) change the axes scales and tick marks add a trend line or error bars (see below) Lines. To draw a straight "line of best fit" right click on a point, select Add Trendline, and choose linear. In the option tab you can force it to go through the origin if you think it should, and you can even have it print the line equation if you are interested in the slope or intercept of the trend line. If instead you want to "join the dots" (and you don't often) double-click on a point and set line to automatic. Error bars. These are used to show the confidence intervals on the graph. You must already have entered the 95% confidence limits on the spreadsheet beside the X and Y data columns. Then double-click on the points on the graph to get the Format Data Series dialog box and choose the Y Error Bars tab. Click on the red arrow in the Custom + box, and highlight the range of cells containing your confidence limits. Repeat for the Custom box. The standard error of the mean (SEM) is calculated by dividing the standard deviation by the square root of number of measurements that make up the mean (often represented by N). In this case, 5 measurements were made (N = 5) so the standard deviation is divided by the square root of 5. By dividing the standard deviation by the square root of N, the standard error grows smaller as the number of measurements (N) grows larger. This reflects the greater confidence you have in your mean value as you make more measurements. You can make use of the of the square root function, SQRT, in calculating this value. 2 X SEM = 95% CI The standard deviation is a measure of the fluctuation of a set of data about the sample mean. The SEM is an estimate of the fluctuation of a sample mean about the "true" population mean. The error bars are attempting to give a range of plausible values for the population mean (at that point in time), given the fact that sample means will fluctuate from sample to sample. "The standard deviation is a measure of the variability within a sample population, and that SEM is a measure of how well that sample population represents the whole population." Problems 1. Here are the results of an investigation into the rate of photosynthesis in the pond weed Elodea. The number of bubbles given off in one minute was counted under different light intensities, and each measurement was repeated 5 times. Use Excel to calculate the means, standard deviation and 95% confidence limits of these results, then plot a graph of the mean results with error bars and a line of best fit. light intensity (Lux) 0 500 1000 2000 3500 5000 repeat 1 repeat 2 repeat 3 repeat 4 repeat 5 5 12 7 42 45 65 2 4 20 25 40 54 0 5 18 31 36 72 2 8 14 14 50 58 1 7 24 38 28 36 There is a bewildering variety of statistical tests available, and it is important to choose the right one. This flow chart will help you to decide which statistical test to use, and the tests are described in detail on the following pages. normal data Testing for a correlation non-normal Spearman correlation coefficient data =CORREL (range 1, range 2) on ranks of data 0=no correlation/ 1=perfect correlation Plot scatter graph Finding how one factor affects another Testing for a relation between 2 sets Calculate mean and 95% CI from replicates Measurements start here What kind of test? Testing for a difference between sets Pearson correlation coefficient =CORREL (range 1, range 2) 0 = no correlation 1 = perfect correlation Linear regression Add Trendline to graph and Display Equation. Gives slope and intercept of line same individuals Paired t-test =TTEST(range1, range2, 2, 1) If P<5% then significant difference If P>5% then no significant difference different individuals Unpaired t-test =TTEST(range1, range2, 2, 2) If P<5% then significant difference If P>5% then no significant difference 2 sets Plot bar graph What kind of data? >2 sets Frequencies (counts) Comparing observed counts to a theory What kind of test? Testing for a difference between counts Testing for an association between groups of counts ANOVA Tools menu > Data analysis > Anova If P<5% then significant difference If P>5% then no significant difference 2 test =CHITEST(obs range, exp range) If P<5% then disagree with theory If P>5% then agree with theory 2 test =CHITEST(obs range, exp range) If P<5% then significant difference If P>5% then no significant difference 2 test for association =CHITEST(obs range, exp range) If P<5% then significant association If P>5% then no significant association Statistics to Test for a Correlation Correlation statistics are used to investigate an association between two factors such as age and height; weight and blood pressure; or smoking and lung cancer. After collecting as many pairs of measurements as possible of the two factors, plot a scatter graph of one against the other. If both factors increase together then there is a positive correlation, or if one factor decreases when the other increases then there is a negative correlation. If the scatter graph has apparently random points then there is no correlation. variable 1 No Correlation variable 2 Negative Correlation variable 2 variable 2 Positive Correlation variable 1 variable 1 There are two statistical tests to quantify a correlation: the Pearson correlation coefficient (r), and Spearman's rank-order correlation coefficient (rs). These both vary from +1 (perfect correlation) through 0 (no correlation) to –1 (perfect negative correlation). If your data are continuous and normally-distributed use Pearson, otherwise use Spearman. In both cases the larger the absolute value (positive or negative), the stronger, or more significant, the correlation. Values grater than 0.8 are very significant, values between 0.5 and 0.8 are probably significant, and values less than 0.5 are probably insignificant. In Excel the Pearson coefficient r is calculated using the formula: =CORREL (X range, Y range) . To calculate the Spearman coefficient rs, first make two new columns showing the ranks (or order) of the two sets of data, and then calculate the Pearson correlation on the rank data. The highest value is given a rank of 1, the next highest a rank of 2 and so on. Equal values are given the same rank, but the next rank should allow for this (e.g. if there are two values ranked 3, then the next value is ranked 5). In this example the size of breeding pairs of penguins was measured to see if there was correlation between the sizes of the two sexes. The scatter graph and both correlation coefficients clearly indicate a strong positive correlation. In other words large females do pair with large males. Of course this doesn't say why, but it shows there is a correlation to investigate further. Linear Regression to Investigate a Causal Relationship. If you know that one variable causes the changes in the other variable, then there is a causal relationship. In this case you can use linear regression to investigate the relation in more detail. Regression fits a straight line to the data, and gives the values of the slope and intercept of that line (m and c in the equation y = mx + c). The simplest way to do this in Excel is to plot a scatter graph of the data and use the trendline feature of the graph. Rightclick on a data point on the graph, select Add Trendline, and choose Linear. Click on the Options tab, and select Display equation on chart. You can also choose to set the intercept to be zero (or some other value). The full equation with the slope and intercept values are now shown on the chart. **You can do this in Logger Pro -Analyze: Linear Fit, and look at the Correlation value. In this example the absorption of a yeast cell suspension is plotted against its cell concentration from a cell counter. The trendline intercept was fixed at zero (because 0 cells have 0 absorbance), and the equation on the graph shows the slope of the regression line. The regression line can be used to make quantitative predictions. For example, using the graph above, we could predict that a cell concentration of 9 x 107 cells per cm3 would have an absorbance of 1.37 (9 x 0.152). The larger the absolute value (positive or negative), the stronger, or more significant, the correlation. Values grater than 0.8 are very significant, values between 0.5 and 0.8 are probably significant, and values less than 0.5 are probably insignificant Correlation Coefficient, r The quantity r, called the linear correlation coefficient, measures the strength and the direction of a linear relationship between two variables. The linear correlation coefficient is sometimes referred to as the Pearson product correlation coefficient The larger the absolute value (positive or negative), the stronger, or more significant, the correlation. Values grater than 0.8 are very significant, values between 0.5 and 0.8 are probably significant, and values less than 0.5 are probably insignificant Coefficient of Determination, r 2 or R2 The coefficient of determination is such that 0 < r 2 < 1, and denotes the strength of the linear association between x and y. The coefficient of determination represents the percent of the data that is the closest to the line of best fit. For example, if r = 0.922, then r 2 = 0.850, which means that 85% of the total variation in y can be explained by the linear relationship between x and y (as described by the regression equation). The other 15% of the total variation in y remains unexplained. The coefficient of determination is a measure of how well the regression line represents the data. If the regression line passes exactly through every point on the scatter plot, it would be able to explain all of the variation. The further the line is away from the points, the less it is able to explain. http://mathbits.com/MathBits/TISection/Statistics2/correlation.htm T-Test to Compare Two Sets of Data Another common form of data analysis is to compare two sets of measurements to see if they are the same or different. For example are plants treated with fertiliser taller than those without? If the means of the two sets are very different, then it is easy to decide, but often the means are quite close and it is difficult to judge whether the two sets are the same or are significantly different. To compare two sets of data use the t-test, which tells you the probability (P) that there is no difference between the two sets. This is called the null hypothesis. P varies from 0 (impossible) to 1 (certain). The higher the probability, the more likely it is that the two sets are the same, and that any differences are just due to random chance. The lower the probability, the more likely it is that that the two sets are significantly different, and that any differences are real. Where do you draw the line between these two conclusions? In biology the critical probability is usually taken as 0.05 (or 5%). This may seem very low, but it reflects the facts that biology experiments are expected to produce quite varied results. So if P > 5% then the two sets are the same (i.e. fail to reject the null hypothesis), and if P < 5% then the two sets are different (i.e. reject the null hypothesis). For the t test to work, the number of repeats should be at least 5. In Excel the t-test is performed using the formula: =TTEST (range1, range2, tails, type) . For the examples you'll use in biology, tails is either 1 or 2 (for a "one-tailed" or "two-tailed" test). You need to decide which of the following types of effect you expect to find: The first mean to be larger than the second The first mean to be smaller than the second The first mean to be different from the second in either direction The tail is the extreme end of the distribution of the data and your experiment can be one of two types: One tailed tests expect the effect to be in a certain direction, so the first two points above are examples of 1 tailed experiments Two tailed tests are used when you have no idea which sample will be larger than the other, but you are looking for any difference. The third point above is such a case. If you have stated your experimental hypothesis with care, it will tell you which type of effect you are looking for. For example, the hypothesis that "Coffee improves memory" is one tailed because you expect an improvement. Testing whether the pH of a stream changed from the previous year suggests a two tailed test as no direction is implied. So remember, don't be vague with your hypothesis if you are looking for a specific effect. If unsure, use a 2-tailed test so you don't miss something! Type can be either 1 for a paired test (where the two sets of data are from the same individuals), or 2 for an unpaired test (where the sets are from different individuals). The cell with the t test P should be formatted as a percentage (Format menu > cell > number tab > percentage). This automatically multiplies the value by 100 and adds the % sign. This can make P values easier to read and understand. It’s also a good idea to plot the means as a bar chart with error bars to show the difference graphically. In the first example the yield of potatoes in 10 plots treated with one fertiliser was compared to that in 10 plots treated with another fertiliser. Fertiliser B delivers a larger mean yield, but the unpaired t-test P shows that there is a 8% probability that this difference is just due to chance. Since this is >5% we fail to reject the null hypothesis that there is no significant difference between the two fertilizers. In the second example the pulse rate of 8 individuals was measured before and after eating a large meal. The mean pulse rate is certainly higher after eating, and the paired t-test P shows that there is only a tiny 0.005% probability that this difference is due to chance, so the pulse rate is significantly higher after a meal and we reject the null hypothesis. http://www.gla.ac.uk/sums/users/jdbmcdonald/PrePost_TTest/chooset1.html The outcome of a significance test is a probability, referred to as a P value. First, let’s be clear what the P value means. It will be simpler to do that in the context of a particular example. Suppose we wish to know whether treatment A is better (or worse) than treatment B (A might be a new drug, and B a placebo). We’d take a group of people and allocate each person to take either A or B and the choice would be random. Each person would have an equal chance of getting A or B. We’d observe the responses and then take the average (mean) response for those who had received A and the average for those who had received B. If the treatment (A) was no better than placebo (B) (that would be our null hypothesis), the difference between means should be zero on average. But the variability of the responses means that the observed difference will never be exactly zero. So how big does it have to be before you discount the possibility that random chance is all you were seeing. You do the test and get a P-value. The P value is the probability that you would find a difference as big as that observed, or a still bigger value, if in fact A and B were identical. If this probability is low enough, the conclusion would be that it’s unlikely that the observed difference (or a still bigger one) would have occurred if A and B were identical, so we conclude that they are not identical, i.e. that there is a genuine difference between treatment and placebo. (you do not accept the null hypothesis and offer an alternative hypothesis that the drug caused the difference). http://www.dcscience.net/?p=6518 ANOVA to Compare >2 sets of Data The t test is limited to comparing two sets of data, so to compare many groups at once you need analysis of variance (ANOVA). From the Excel Tools menu select Data Analysis then ANOVA Single Factor. This brings up the ANOVA dialogue box, shown here. Enter the Input Range by clicking in the box then selecting the range of cells containing the data, including the headings. Check that the columns/rows choice is correct (this example is in three columns), and click in Labels in First Row if you have included these. The column headings will appear in the results table. Leave Alpha at 0.05 (for the usual 5% significance level). Click in the Output Range box and click on a free cell on the worksheet, which will become the top left cell of the 8 x 15-cell results table. Finally press OK. The output is a large data table, and you may need to adjust the column widths to read it all. At this point you should plot a bar graph using the averages column for the bars and the variance column for the error bars. The most important cell in the table is the Pvalue, which as usual is the probability that the null hypothesis (that there is no difference between any of the data sets) is true. This is the same as a t-test probability, and in fact if you try ANOVA with just two data sets, it returns the same P as a t test. If P > 5% then there is no significant difference between any of the data sets (i.e. the null hypothesis is true), but if P < 5% then at least one of the groups is significantly different from the others. In the example on this page, which concerns the grain yield from three different varieties of wheat, P is 0.14%, so is less than 5%, so there is a significant difference somewhere. The problem now is to identify where the difference lies. This is done by examining the variance column in the summary table. In this example, varieties 2 and 3 are very similar, but variety 1 is obviously the different one. So the conclusion would be that variety 1 has a significantly lower yield than varieties 2 and 3. Download:http://www.analystsoft.com/en/products/statplusmac/ Chi-squared Test for Frequency Data Sometimes the data from an experiment are not measurements but counts (or frequencies) of things, such as counts of different phenotypes in a genetics cross, or counts of species in different habitats. With frequency data you can’t usually calculate averages or do a t test, but instead you do a chi-squared (2) test. This compares observed counts with some expected counts and tells you the probability (P) that there is no difference between them. In Excel the 2 test is performed using the formula: =CHITEST (observed range, expected range) . There are three different uses of the test depending on how the expected data are calculated. Sometimes the expected data can be calculated from a quantitative theory, in which case you are testing whether your observed data agree with the theory. If P < 5% then the data do not agree with the theory, and if P > 5% then the data do agree with the theory. A good example is a genetic cross, where Mendel’s laws can be used to predict frequencies of different phenotypes. In this example Excel formulae are used to calculate the expected values using a 3:1 ratio of the total number of observations. The 2 P is 53%, which is much greater than 5%, so the results do indeed support Mendel’s law. Incidentally a very high P (>80%) is suspicious, as it means that the results are just too good to be true. Other times the expected data are calculated by assuming that the counts in all the categories should be the same, in which case you are testing whether there is a difference between the counts. If P < 5% then the counts are significantly different from each other, and if P > 5% then there is no significant difference between the counts. In the example above the sex of children born in a hospital over a period of time is compared. The expected values are calculated by assuming there should be equal numbers of boys and girls, and the 2 P of 6.4% is greater than 5%, so there is no significant difference between the sexes. 1. 2. If the count data are for categories in two groups, then the expected data can be calculated by assuming that the two groups are independent. If P < 5% then there is a significant association between the two groups, and if P > 5% then the two groups are independent. Each group can have counts in two or more categories, and the observed frequency data are set out in a table, called a contingency table. A copy of this table is then made for the expected data, which are calculated for each cell from the corresponding totals of the observed data, using the formula E = column total x row total / grand total . In this example the flow rate of a stream (the two categories fast / slow) is compared to the type of stream bed (the four categories weedchoked / some weeds / shingle / silt) at 50 different sites to see if there is an association between them. The 2 P of 1.1% is less than 5%, so there is an association between flow rate and stream bed. 3. Problems 1. In a test of two drugs 8 patients were given one drug and 8 patients another drug. The number of hours of relief from symptoms was measured with the following results: Drug A 3.2 1.6 5.7 2.8 5.5 1.2 6.1 2.9 Drug B 3.8 1.0 8.4 3.6 5.0 3.5 7.3 4.8 Find out which drug is better by calculating the mean, SD and 95% confidence limit for each drug, then use an appropriate statistical test to find if it is significantly better than the other drug. 2. In one of Mendel's dihybrid crosses, the following types and numbers of pea plants were recorded in the F2 generation: Yellow round seeds Yellow wrinkled seeds Green round seeds Green wrinkled seeds 289 122 96 39 According to theory these should be in the ratio of 9:3:3:1. Do these observed results agree with the expected ratio? 3. The areas of moss growing on the north and south sides of a group of trees were compared. North side 20 43 53 86 70 of tree South side 63 11 21 54 9 of tree Is there a significant difference between the north and south sides? 54 74 4. Five mammal traps were placed in randomly-selected positions in a deciduous wood. The numbers of field mice captured in each trap in one day were recorded. The results were: Trap A B C D E no. of mice 22 26 21 8 23 Trap D caught far fewer mice than the others. Did this happen by chance or is the result significant? 5. In an investigation into pollution in a stream, the concentration of nitrates was measured at six different sites, and a diversity index was calculated for the species present. Site 1 2 3 4 5 6 413.3 439.7 726 850 567.3 766.7 Conductivity (S) Diversity index 7.51 5.17 4.49 3.82 5.88 3.74 Is there a correlation between conductivity and diversity, and how strong is it? (The diversity index is calculated from biotic data, so is not normally distributed.) 6. The blood groups of 400 individuals, from 4 different ethnic groups were recorded with the following results: Ethnic group Blood Group O Blood Group A Blood Group B Blood Group AB 1 46 40 7 3 2 48 39 12 2 3 53 33 12 4 4 55 30 13 3 Is there as association between blood group and ethnic group? 7. The effect of enzyme concentration on rate of a reaction was investigated with the following results. Enzyme concentration (mM) 0 0.1 0.2 0.5 0.8 1.0 Rate (arbitrary units) 0 0.8 1.1 3.2 6.6 7.2 Plot a graph of these results, fit a straight line to the data, and find the slope of this line. Use the slope to predict the rate at an enzyme concentration of 0.7mM. More Practice: 1. Go http://mathbench.umd.edu/modules/prob-stat_normal-distribution/page01.htm Complete the activity: Normal Distributions and the Scientific Method 2. Go to: http://mathbench.umd.edu/modules/prob-stat_bargraph/page01.htm Complete the activity: Bar Graphs and Standard Error 3. Go to: http://mathbench.umd.edu/homepage/statistical_tests.htm Complete the t-test activity (pages 1-5, 13) and the chi-square activity. Let’s Talk About Stats: Understanding the Lingo by Laura Fulford on 26th of February, 2014 in Lab Statistics & Math http://bitesizebio.com/19291/a-basic-guide-to-stats-understanding-the-lingo/ The type of data you have, the number of measurements, the range of your data values and how your data cluster are all described using statistical terms. To determine which type of statistical test is the best fit for analyzing your data, you first need to learn some statistics lingo. Variables Variables are anything that can be measured; they are your data points, and the type you have affects the statistical test you use. Measurement or numerical variables are the main type of variables that are obtained in biological research, so I’ll focus on these. Measurement variables can either be continuous, which means they can be any value between two points, (for example half and quarter measurements) while discrete variables are whole numbers (such as ranking 1-5). As I will show you later, some tests can only be used with continuous variables, while others can accommodate discrete values. Sample size (n) Sample size refers to the number of data points in your set of data. In general, the larger your sample size, the better. However, factors such as time, cost and practicality limit the sample size you use. As an absolute minimum you need an n of 3 to perform a statistical test, but look at publications with similar experiments to determine what is considered acceptable. The size of your sample will affect the variance of your data (see below). Data Spread How spread out your data is gives you an idea of how reliable it is – data with low variance is more reliable than data with high variance. It is therefore useful to know how variable your data is, and there are several simple measures for determining this. Variance Variance is the simplest measure of the spread of the data and is the average of the squared differences of the mean. It tells us how spread out the data is from the mean; the larger the number the more spread out the data (higher variance). To calculate variance, first find the mean of the data points. Next, find the difference between each sample and the mean and then square the result. Finally, average the results of the squared differences. Note: if you are calculating sample variance (which you most likely are, since this means you are measuring just a sample of a population rather than the entire population) then you divide by n-1 when finding the average of the squared differences rather than n. This is to correct for the fact that you are only estimating the variance (since you are not measuring the entire population) rather than accurately computing it. Standard Deviation The standard deviation (SD) is the most widely used method for measuring the spread of the data. SD is simply the square root of the variance and similarly tells us how much the samples deviate from the mean. The standard deviation is often preferred to the variance as it is produces figures in which the majority of the data is on the same scale, making the results easier to display. Standard Error of the mean While SD tells you about the variability of your data, SE provides information on the precision of the sample mean. SE is calculated by dividing the SD by the square root of the sample size. When you take a sample of observations from a population, the mean of the sample is an estimate of the mean of all of the observations in the population. If your sample size is small, your estimate of the mean won't be as good as an estimate based on a larger sample size. You'd often like to give some indication of how close your sample mean is likely to be to the parametric mean. One way to do this is with the standard error of the mean. If you take many random samples from a population, the standard error of the mean is the standard deviation of the different sample means. About two-thirds (68.3%) of the sample means would be within one standard error of the parametric mean, 95.4% would be within two standard errors, and almost all (99.7%) would be within three standard errors. As you increase your sample size, sample standard deviation will fluctuate, but it will not consistently increase or decrease. It will become a more accurate estimate of the parametric standard deviation of the population. In contrast, the standard error of the means will become smaller as the sample size increases. With bigger sample sizes, the sample mean becomes a more accurate estimate of the parametric mean, so the standard error of the mean becomes smaller. Confidence intervals Confidence limits and standard error of the mean serve the same purpose, to express the reliability of an estimate of the mean. In some publications, vertical error bars on data points represent the standard error of the mean, while in other publications they represent 95% confidence intervals. I prefer 95% confidence intervals. When I see a graph with a bunch of points and vertical bars representing means and confidence intervals, I know that most (95%) of the vertical bars include the parametric means. When the vertical bars are standard errors of the mean, only about two-thirds of the bars are expected to include the parametric means; I have to mentally double the bars to get the approximate size of the 95% confidence interval. Distribution Distribution, as the name implies, describes how your data is distributed. There are many ways your data can be distributed and this can affect the statistical test you use. The most well-known distribution is of course the normal distribution, which has a bell shape. A normal distribution means the data is symmetrical, with values higher and lower than the mean equally likely, but the frequency of values drops off quickly the further away from the mean. Non-normal distributions are skewed; the mean is usually not in the middle. Most statistical tests assume that the distribution is normal, but beware – many common statistical tests are not valid for highly skewed data. p value The p value is what you are searching for – the number that will tell you whether you have achieved the holy grail of science: statistical significance! It is generally considered that a result with a p value less than 0.05 is unlikely to have occurred by random chance and is therefore statistically significant. In contrast, results with a p value greater than 0.05 are not considered significant, as it cannot be ruled out that they did not occur by random chance. The p value is affected by sample size and if your sample size is too small you will not obtain a significant result even if the observed effect is real. Therefore you need to ensure you have a suitable sample size. Paired or unpaired data One factor that will be important in determining which type of test to use is whether or not your data is paired. Paired data is derived from equivalent and matched populations. For example, if you are comparing two drugs and you give drug A to 10 people of a certain age and population one day and 10 people of the same age and population drug B another day, your data is matched and you can use a paired test. If 10 people are given drug A but 15 people given drug B, then your data is unpaired. Parametric vs non-parametric test A parametric test is used when the data is assumed to be of normal distribution and equal variance. In contrast, non-parametric tests make no assumptions about distribution or variance. In general, non-parametric tests are less powerful, but more conservative. Any significance you find with the test is probably more real. Type 1 and Type 2 errors A statistical test can give a false result – often when the wrong test is used or a test is used incorrectly. Two types of errors can be encountered. A Type 1 error is a false positive. It is when you conclude that a result is statistically significant when in fact it isn’t. A Type 2 error is a false negative, it occurs when actual significance is missed. Resources: General Overview of Statistical Terms: **http://click4biology.info/c4b/1/stat1.htm **http://www.drexel.edu/dvsf/statistics_help.htm **http://www.statsoft.com/textbook/stathome.html **http://udel.edu/~mcdonald/statintro.html Overview of which test to use: http://www.le.ac.uk/bl/gat/virtualfc/Stats/introst.html http://www.graphpad.com/guides/prism/6/statistics/ http://onlinestatbook.com/rvls.html http://www.psychstat.missouristate.edu/introbook/sbk00.htm