Populations, Investigations and Power 19 January 2021 17:22 Session 2 - Datancefrom mean [s) Standard normal distribution: · Mean, mode and median are identical · 'Bell' shaped curve · Equal number of people each side of mean The Normal Distribution -40 -20 A = Probability Densityof x . 12 - 25 40 Normal distributions with the same mean, but different spread (variance, measured as standard deviation s.d.) Normal distributions with different means and different standard deviations Most biological measurements will follow a normal distribution - height, weight, BP, drug response etc The Normal Distribution Figure 6: Normal Distribution Curves for Mean of 0 and Variances of 25, 100, and 400 0.08 0.07 0.06 0.05 $ 0.04 0.03 0.02 0.01 0 -40 A -20 × Consider drawing samples from a population - how many samples would we need to draw before we could be certain we had got a good approximation of the whole population? The wider the variance in our sample, the less chance we have of drawing someone close to the mean, so the larger the sample you have to draw to be sure you are representative. When performing any study, it is much easier to analyse that study if we can assume our data is normally distributed, and this will have important implications for study design. Having a good idea of what you are looking for before you start will help you plan an experiment, study or trial appropriately. -25 -100 -400 40 You will typically be looking for a difference between two groups, or before or after an intervention (treatment). Standard Deviation 68-95-99.7 Rule 0.40 0.35 0.30 0.25 Probability Density 0.20 0.15 68.27% 95.45% 0.10 99.73% 0.05 0.00 ?-3? ?-2? uto H+ 20 u+ 30
Standard deviation is used to determine the 'spread' of a population - the smaller the value, the narrower the bell curve 68% of a population will be within 1 standard deviation of the mean 95% of a population will be within 2 standard deviations of the mean 99.7% of a population will be within 3 standard deviations of the mean Knowing your standard deviation can therefore tell you how close a random individual drawn from a population is to be to the mean of that population. Drawing more people allows better estimates of the mean and the distribution Confidence Intervals 0.40 0.35 0.30 Probability Density 0.20 0.25 0.15 68-95-99.7 Rule 68.27% F 95.45% 1 I 0.10 1 0.05 99.73% 0.00 1-30 1-20 p+ 20 #+30 These measurement are actually confidence intervals - if you draw an individual from a normally distributed population, you can be 95% confident that individual will lie with 2 standard deviations of the mean for that population You will often see mean and 95% confidence interval plotted in graphs, with the mean as the spot and the confidence interval as the bar If two error bars overlap it is unlikely there will be a significant difference Sample Testing When performing an experiment, the default position is always that