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Statistics for Linguistics Students Michaelmas 2004 Week 3 Bettina Braun www.phon.ox.ac.uk/~bettina Overview • • • • • • Discussion of last assignment Z-scores Sampling distributions Confidence intervals Hypothesis testing Type I and Type II errors General comments • Please let every file you submit contain your initials and the week the assignment was given! • Please put your name somewhere on the page • Paste figures into the doc-file (or rtf-file) and only submit the .sav-files • Name the x- and y-axis of the figures and give them a title • Do not work with var0001 (name and label varia bles) • Scale figures so that numbers are readible Manipulating figures • If you want to copy SPSS-figures into your document, it is sensible to increase the font sizes (otherwise they’ll be too difficult to read). Also, you might want to change the title or legend, ... • Double click on any figure Measures of central tendency • Interval data, roughly normally distributed data (less appropriate for skewed distributions) mean (although mode and median should give same results!) • Interval data, strongly skewed mode, median • Categorical data (different versions, …) mode Sentence lengths Very likely that most of the sentences do not exceed 20 to 30 words but there will be few sentences that are very long… Sentence length N.B. It is likely that distribution of sentence lengths in Th. Mann are skewed to the left… Preference for 3 resynthesised versions • Suppose this were the outcome version # subjects a 20 b 37 c 18 Coding: a=1, b=2, c=3 mean = 1.97 Mode is more meaningful! If you are reporting a mean, one might think there is a normal distribution Merging datasets Year 1990 1990 1990 1990 … 2000 2000 2000 2000 2000 2000 results 21,00 64,00 48,00 64,00 58,00 33,00 8,00 55,00 47,00 61,00 Year 90 58 59 44 67 50 60 54 … year 00 30 54 45 67 54 60 45 … This is how to organise observations from the same person in different years Describe this distribution 25 Frequency 20 15 10 5 Mean = 773,3098 Std. Dev. = 268,96745 N = 163 0 200,00 400,00 600,00 800,00 ??? 1000,00 1200,00 Normal distribution (Gaussian distribution) • Example: IQ scores, mean=100, sd=16 Mean = Median = Mode z-scores • Z-score: deviation of given score from the mean in terms of standard deviations How likely is a given event? • Example: time to utter a particular sentence: x = 3.45s and sd = .84s • Questions: – What proportion of the population of utterance times will fall below 3s? – What proportion would lie between 3s and 4s? – What is the time value below which we will find 1% of the data? Sample mean and sd as parameter estimators • Mean and standard deviation of the population are unknown • But we can use the sample mean and sd as estimators for the parameters of the unknown population Sample mean and sd as estimators Population parameter Sample statistics mean Standard deviation Degrees of freedom: scores that contain new information; better estimator for parameter From sample statistics to population parameters • We only know the statistics of our sample • Sample statistics will differ from population parameters • Knowledge about sampling distribution of the statistic (i.e. how it behaved if large samples were taken) would tell us how well the statistic estimated the parameter (degree of confidence) Sampling distribution • Population (mean 4.9, sd 3.1) • 100 samples with n=50 3 examples: Taken from www.fw.umn.edu/FW5601/ ALAB/Lab5/LAB4_BA2.HTM Sampling distribution • Relative frequency of 100 means: sample mean: 4.9 sample sd: 0.46 Note: – Shape of sampling distribution roughly normal – Mean of sampling distribution is population mean – Sample sd smaller than population sd Central limit theorem Terminology: n=30 Standard deviation of the sampling distribution of the means is called standard error of the mean (SE) Experimental research • Often, we are interested if human behaviour is dependent on certain factors. E.g. – Is the speech rate dependent on the dialectal region? – Do foreigners and native speakers produce sentences with the same number of words? Dependent and independent variables • Independent variable: – Variable(s) manipulated by the experimenter – experimenter determines the values it will assume – Independent variables may have a number of different levels • Dependent variable: – Measure of behaviour (not manipulated or controlled by experimenter) Examples • What are the dependent and independent variables in the following questions? – Is the speech rate dependent on the dialectal region of the speakers? – Do foreigners and native speakers produce sentences with the same number of words? – Is the articulatory precision dependent on the part-of-speech? – Do different word orders influence the grammatiality judgements of subjects? Null-hypothesis H0 • Generally phrased to negate the possiblity of a relationship between the independent and dependent variables • If the null-hypothesis is true, there is no interaction between dependent and independent variables • Alternative hypothesis contradicts nullhypothesis Statistical tests of significance • Allows to evaluate the probability that the observed sample values would occur if the null hypothesis were true • If that probability is sufficiently low, the null hypothesis can be rejected • In other words: provide evidence for conlcuding (with a specified risk of error) that there are or are no real differences between conditions in the population p-value • Probability that values of the statistic like the one observed would occur if the null hypothesis were true • In other words: how unusual is the observed test statistic compared to what H0 predicts? • The smaller p, the more unusual the observed data if H0 were true (e.g. p=0.45 very usual, compared to p=0.001) Type I error • Type I error: – Rejection of a true null hypothesis – That is, in reality, there is no relationship between independent and dependent variable but you conclude there is – Probability of type I error is called α – α is usually determined before you run an experiment (often set at 5% or 1%) Type II error • Type II error: – Failure to reject a false null hypothesis – That is, in reality, there is a relation between the independent variable and the dependent one(s) but you conclude there is none – Probability of type II error is called β – In contrast to α, β cannot be precisely controlled Reducing the Type II error • β can be reduced by – Using an α-level of .05 (instead of a more stringent one) – Using as many subjects as can be reasonably obtained – Selecting the levels of the independent variable so as to maximise the size of the effect – Reducing variability (e.g. controlling more variables) Organise SPSS tables • Every independent variable and every dependent variable has its own column • Independent variables are often found before dependent variables • It is wise to compare the distributions of the conditions before statistical tests of significance (histograms, boxplots) – Either select the condition you are interested in – Or split the output according to the different levels – You can also compare boxplots for the different conditions Data exploration • Error bars show the 95% confidence interval for the mean (i.e. the mean and the area where 95% of the data fall in) Data exploration • Error bars show the 95% confidence interval for the mean (i.e. the mean and the area where 95% of the data fall in) • One independent variable – Error bar (simple, groups of variables) • Two independent variables – Error bar (clustered, groups of variables)