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Statistics The usual course of events for conducting scientific work “The Scientific Method” Observation Develop a Working Hypothesis Conduct an experiment or a series of controlled systematic observations Appropriate statistical tests Reformulate or extend hypothesis Confirm or reject hypothesis The usual course of events for conducting scientific work “The Scientific Method” In a group of crickets, Observation small ones seem to avoid large ones There willabe movement away from Develop Working Hypothesis large cricket by small ones Record the of Conduct an number experiment that of small crickets ortimes a series controlled move awayobservations from small systematic and large crickets. Chi squarestatistical test Appropriate tests Reformulate Avoidance may dependor on extend hypothesis previous experience Confirm or There is a significant reject hypothesis difference in the number of times small crickets move away from large vs. small ones Imagine that you are collecting samples (i.e. a number of individuals) from a population of little ball creatures - Critterus sphericales Little ball creatures come in 3 sizes: Small = Medium = Large = You end up with a total of five samples -sample 1 -sample 2 -sample 3 -sample 4 -sample 5 The real population (all the little ball creatures that exist) Your samples Each sample is a representation of the population BUT No single sample can be expected to accurately represent the whole population To be statistically valid, each sample must be: 1) Random: Thrown quadrat?? Guppies netted from an aquarium? Assign numbers from a random number table 13 4 6 1 3 7 5 2 9 14 10 11 8 12 15 To be statistically valid, each sample must be: 2) Replicated: But not - ‘Pseudoreplication’ Not pseudoreplication 10 samples from 10 different trees Sample size = 10 Pseudoreplication 10 samples from the same tree Sample size = 1 TYPES OF DATA RATIO DATA - constant size interval - a zero point with some reality e.g. Heights, rates, time, volumes, weights INTERVAL DATA - constant size interval - no true zero point zero point depends on the scale used e.g. Temperature Ordinal Scale - ranked data -grades, preference surveys Nominal Scale Team numbers Drosophila eye colour The kind of data you are dealing with is one determining factor in the kind of statistical test you will use. Statistics and Parameters Measures of: Central tendency - mean, median, mode Dispersion - range, mean deviation, variance, standard deviation, coefficient of variation Central tendency - Mean The real population (all the little ball creatures that exist) The real population (all the little ball creatures that exist) Your samples The real population (all the little ball creatures that exist) Central Tendency 1) Arithmetic mean At Population level Measuring the diameters of all the little ball creatures that exist m = SXi N m - population mean Xi - every measurement in the population N - population size Your samples X = SXi n X = SXi n X = SXi n X = SXi n X = SXi n Sample mean X = SXi n Sum of all measurements in the sample Sample size If you have sampled in an unbiased fashion X = SXi n X = SXi n X = SXi n X = SXi n X = SXi n Each roughly equals m Central tendency - Median Median - middle value of a population or sample e.g. Lengths of Mayfly (Ephemeroptera) nymphs 5th value (middle of 9) 1 2 3 4 5 6 7 8 9 Odd number of values Even number of values Median value Median value Median = middle value Median = + 2 Or - to put it more formally Odd number of values (i.e. n is odd) Median = X(n+1) 2 Even number of values Median = X(n/2) + X(n/2) + 1 2 Central tendency - Mode c. Mode - the most frequently occurring measurement Mode Frequency (= number of times each measurement appears in the population Values (= measurements taken) Measures of Dispersion Why worry about this?? -because not all populations are created equal Distribution of values in the populations are clearly different BUT means and medians are the same Mean & median Measures of Dispersion 1. Range - difference between the highest and lowest values Remember little ball creatures and the five samples Range = - Range - crude measure of dispersion Note - three samples do not include the highest value and - two samples do not include the lowest Measures of Dispersion 2. Mean Deviation X is a measure of central tendency Take difference between each measure and the mean Xi - X BUT SXi - X = 0 So this is not useful as it stands Measures of Dispersion 2. Mean Deviation (cont’d) But if you take the absolute value -get a measure of disperson S |Xi - X| and S |Xi - X| n = mean deviation Measures of Dispersion 3. Variance -eliminate the sign from deviation from mean Square the difference (Xi - X)2 And if you add up the squared differences - get the “sum of squares” S(Xi - X)2 (hint: you’ll be seeing this a lot!) Measures of Dispersion 3. Variance (cont’d) Sum of squares can be considered at both the population and sample level Population SS = S(Xi - Sample m)2 ss = S(Xi - X)2 Measures of Dispersion 3. Variance (cont’d) If you divide by the population or sample size - get the mean squared deviation or VARIANCE Population s2 = S(Xi N Sample m)2 Population variance s2 = S(Xi - X)2 n-1 Sample variance Measures of Dispersion 3. Variance (cont’d) Note something about the sample variance s2 = S(Xi - X)2 n-1 Degrees of freedom or df or n Measures of Dispersion 4. Standard Deviation - just the square root of the variance Population s = S(Xi N Sample m)2 s = S(Xi - X)2 n-1 Standard Deviation - very useful Most data in any population are within one standard deviation of the mean NORMAL DISTRIBUTION From previous slide show Type of data Discrete 2 categories & Bernoulli process Use a Binomial model to calculate expected frequencies Continuous Other distributions > 2 categories Use a Poisson distribution to calculate expected frequencies Now we’re dealing with: Type of data Discrete 2 categories & Bernoulli process Use a Binomial model to calculate expected frequencies Continuous Other distributions > 2 categories Use a Poisson distribution to calculate expected frequencies Normal Distribution - bell curve Central Limit Theorem Any continuous variable influenced by numerous random factors will show a normal distribution. Normal curve is used for: 2) Continuous random data Weight, blood pressure weight, length, area, rates Data points that would be affected by a large number of random (=unpredictable) events physical activity genes age Blood pressure smoking diet stress Normal curves can come in different shapes So, for comparison between them, we need to standardize their presentation in some way Standarize by calculating a Z-Score Z = value of a random variable - mean standard deviation or Z=X-µ s Example of a z-score calculation The mean grade on the Biometrics midterm is 78.4 and the standard deviation is 6.8. You got a 59.7 on the exam. What is your z-score? Z=X-µ s Z = 59.7 - 78.4 = -2.75 6.8 If you look at the formula for z-scores: z = value of a random variable - mean standard deviation z is also the number of standard deviations a value is from the mean Each standard deviation away from the mean defines a certain area of the normal curve