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APPENDIX B Programmed Text on Data Analysis (Membrane Potential) STATISTICS When data are collected, it is advisable to make more than one measurement of a variable x. The mean x of the n measurements can then be calculated by the formula_________ Sx n This mean is a measure of central tendency; it provides a single summary figure that best describes the central location of the entire set of observations. It is also useful to have a measure of the spread or variability in the observations, i.e. the extent to which they are clustered around the mean or scattered more widely. The variance can be calculated by the formula ______________ (x x) 2 n where the numerator is the sum of the squares of the deviation from the mean. From this is calculated the standard deviation, given by___________________ variance (x x) 2 n When the variable falls on a normal distribution, 95% of within_____________with the remaining 5% lying outside these limits. the values ±2 standard deviations of the mean, If a series of samples, each of size n, are drawn from the population, these sample means will also be normally distributed about the population mean. The standard deviation of these means is ____________________, and 95% of the sample means will again lie within_____________________________. the standard error of the mean, ± 2 standard deviations of the mean. Standard error of the mean is obtained by calculating ___________________. standard deviation n When two sets of data are being compared with a view to deciding whether they are significantly different, it is usual to assume that the variations between them are due to measurement (random sampling) error. This assumption is called ________________. lie the Null Hypothesis. If one calculates the probability of these variations arising purely by chance, and finds it to be very low (usually less than 1 in 20 or 5%), one can then reject the null hypothesis and accept that the differences did NOT arise randomly, but are in fact real. Examples of such data sets are the sample means mentioned earlier. By performing Student’s t-test (using the difference between the means and the standard error of this difference), and taking into account the number of degrees of freedom, one obtains a number which indicates whether or not there is a real difference between the two samples. GRAPHS When considering experimental observations it often appears that two variables being measured in a population are somehow inter-related. For example, the MEMPOT program is based on an equation, which gives one variable, membrane voltage, in terms of a second variable, electrolyte concentration. However, as this equation is fairly complex, it is useful to start with a relationship, which can be expressed by a simple linear equation. What is the general form of a linear relationship? __________________. y = mx + b. When does the straight line represented by this equation pass through the origin? ___________. when b = 0. What does m represent?_____________________________ the slope of the line. When is this equal to the ratio of y to x i.e. when is y proportional to x?__________________ only when b = 0. Considering a simplified version of the equation in the MEMPOT program, when only one cation is assumed to be contributing to the membrane voltage, one obtains the following relationship_______________________________ [K]o RT Vm = ln F [K]i Describe in words the variable on the x-axis when Vm is plotted on the yaxis?________________________________________________________________________________ __. x is the natural logarithm of the ratio of the outside to the inside concentration of K+ ions. What are the units of x? _____________________________. it is a ratio, i.e. a number without any units. If linear graph paper have?_________________ is used for this plot, what shape will the curve exponential. What is the mathematical explanation for this? ___________________________________________ The equation can be rewritten with voltage as a power of e. To produce a straight line, rather than a curve, we use semi-log graph paper. What is another advantage of plotting data on a logarithmic scale? ___________________________________________________________________________________ ___________________________________________________________________________________ To compress data which would otherwise be spread out over a very wide range of values. There is a very important question one should ask about the relationship between two variables: how well do the data points on the graph “hug” the straight line expressing the relationship between them? To ascertain this, one calculates the________________________ correlation coefficient, r which, if the “fit” is poor, will approach a value of _____________, zero, but if the “fit” is excellent, will approach close to _____________. one. Note that a high degree of correlation does not imply anything about a causal relationship between x and y, but what it does do is enable one to predict values of y from x with a high degree of probability that they will be correct. Of course, some variables will be positively correlated, with r having a value between 0 and +1, whereas others will be negatively correlated, with r falling between 0 and -1; see if you can think of examples of these differences from everyday life. In the MEMPOT program you are given the opportunity to test the goodness of fit of different lines to a certain set of data you have generated, by varying one of the “constants” in the relevant equation. The mathematical method of doing this is to calculate the sum of the squares of the deviations of the experimental points from the predicted curve; the smaller this sum, the better the fit, since the calculation of r (which involves this term being subtracted from 1) will result in a higher value.