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Research Statistics (Psych 2040) Chapter Summaries

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University of Guelph
PSYC 2040
David Stanley

Research Statistics Chapter Summaries Chapter 1: Statistics, Computers and Statistical Packages Basic Definitions Samples and Populations  Samples refers to some subset of a population and a population is some large set of numbers o There are 3 types of populations  Finite population: has a definite number of individuals  Infinite population: has no limit on the number of individuals  Theoretical population: is simply an equation  The difference between a random sample and a non-random sample o Random sample is one in which every observation in the population has an equal opportunity of being selected  Inferential statistics are all based on the assumption of random sampling o A non-random sample is one in which this is not the case Statistics and Parameters  Statistic: is a number that describes a sample o Ex. the mean, standard deviation and the variance  Parameter: is a number that describes a population 2 o µ is the mean, σ is the standard deviation and σ is the variance  Difference between the parameter and the statistic o We are usually unable to assess the population o Instead we have to make do with a sample from the population and use the information we obtain on the sample to estimate the corresponding value of the population  We have a statistic and wish to estimate the parameter Unbiased and Biased Estimates  The difference between biased and unbiased estimates of parameters  A statistic is said to be unbiased if the mean of all possible values of that statistic is equal to the parameter o The mean is a statistic that is unbiased o This is because the mean of the means of all possible samples from the population equals the population mean  A statistic is said to be biased if the mean of all possible values is not equal to the population value o The variance is said to be biased because the mean of the variance of all possible samples from a population is less than the population variance Types of Statistics Statistics of Location  Statistics serve to locate the sample on the number line, which is a representation of all possible numerical values ranging from minus infinity to plus infinity  Any statistic that helps you locate where the sample is on this number line is a statistic of location  One class of such statistics are the 3 measures of central tendency o Median  The median is that value such that 50% of the values are greater than that value and 50% are less  It is the value that is at the centre of all the values o Mode  The mode is that value that occurs most frequently and the most popular one  If 2 values had the same highest frequency, the distribution of the scores are called bimodal and if more it is called multi-modal o Mean  The mean is that value such that the sum of the deviations of the scores from their mean and add to 0  It is identified as ̅  The formula is ̅ Statistics of Scale  Statistics of scale describe how much differentiation there is in a sample o Sometimes called statistics of variation or dispersion  If all the numbers are close together, the scale is small and if big, the scale is large  They give an indication of the amount of variability in a sample  There are a few statistics of scale o Range  The simplest measure of scale is range  It is defined as the difference between the highest and the lowest value  However that makes it a problem measure since it only takes 2 values into account  Measures that use all of the numbers in the sample would be expected to give much more stable answers o Semi-interquartile range  A type of range statistic that uses more information in the distribution is the semi-interquartile range, also known as the quartile deviation  Is it defined as the difference between the 75 and the 25 percentile divided by 2  This makes it more stable than the range o Absolute deviations  Mean absolute deviation  Mean absolute deviation is computed by summing the absolute deviations and dividing by the mean of deviations  This mean would give an indication of the relative size of the deviations of the values from the mean  Median absolute deviations  The median absolute deviation is simply the median of the absolute deviations  This tells us that roughly 50% of the values differ more than this much from the mean, while 50% of the values deviate less than this o Variance and standard deviation  Variance is the squared deviations of the values from the mean and then calculate the mean of these squared deviations 2  It is generally identified as S  The formula is biased formula is o If you wished to describe the variance or the standard deviation of the sample, you would use the biased estimate ( ̅  The formula for the unbiased estimate of the population variance is defined as o If you wished to used your statistic to estimate the population variance or the standard deviation of the sample, you would use the unbiased estimate ( ̅  The square root of the variance is referred to as the standard deviation and it is identified as S Statistic of Shape  A statistic of shape tells us how the values are distributed along the line, whether they are symmetrically distributed around the mean or skewed to one end of the other  A standard score is identified by the letter Z and it is defined as ̅ o S is the biased estimate of the variance and Z values are a transformation of the original X values, such that the mean of the Z is 0 and the variance is 1  The statistic of shape are: o Skewness  Skewness is a measure of asymmetry of the distribution of numbers  It is identified as1g and the formula is ( ̅ )  Cubing a large deviation yields a large number and retains the sign of the deviation, thus if 1 is  Positive it indicates that there is a long tail running out to the right (the larger values)  Negative, the tail runs out to the left (smaller values)  Is 0, it means the distribution is symmetrical o Kurtosis  Kurtosis is a measure of the presence of extreme values in the distribution  Is it defined by the notation 2 Statistics of Association  The statistics of association are association, correlation and regression Types of Parameters  For each type of statistic, there are the corresponding parameters  We expect that there will be a sampling distribution of sample means around the population mean Standard Error and Statistical Inference  The standard deviation of the sampling distribution is referred to as the standard error o It is defined as  ̅ √ o It is the standard deviation of all possible means for a population  Normal distribution o If a population is normally distributed, the sampling distribution of means will also be normally distributed, and will have a mean equal to the mean in the population (µ) and a standard deviation equal to √ . o 95% of the standard normal distribution falls between -1.96 and +1.96 where 5% falls out of that range  The central limit theorem o States that the distribution of means will tend to be normal regardless of the shape of the distribution in the population provided that the population variance is finite, and sample size is large o A sample size of 30 is often considered reasonable Overview of SPSS 14  The SPSS system is a general purpose program that permits you to permits you to perform many statistical tests and to conduct graphs of the results  This system is made up of three major components, a data editor, a syntax editor and a viewer o The data editor permits you to type in the data or to inspect it if it already exists o You can enter the data and variable names into the table directly or create a data file (text file) and open it in the data editor Types of SPSS files  There are 3 types of files in SPSS o .SAV file produced by the data editor o .SPS file that is produced by syntax editor and consists of the instructions given to the computer, and serves as a handy reminder of precisely what operations were executed Inputting Data  Data can be typed into the Data Editor directly o Each row represent a different individual participant o Each column represents a different variable  Data can be typed as an text file Running Jobs  The standard way of running jobs is to first of all make sure the data are in the Data Editor, then click Analyse on the Menu bar and choose the program you want Inputting ASCII (Text files) into SPSS14 Types of ASCII files  There are 2 types of ASCII data files that can be considered o A Delimited file is one that separates the data by some form of a delimiter, and in SPSS, you are provided with a number of options such as tab, space, etc…  Ex. 1 2 3 4  Follows the same rules as inputting data, rows = participant, column = variables o A fixed width file contains the numbers in given fields Steps in Using SPSS Frequencies  Many of the statistics can be computed using the SPSS frequencies program which can be run by: o Opening the data/inputting the data o Click on Analyze and pick Descriptive statistics o Click on frequencies o Click the statistics you want from the window, such as mean, median, mode, Skewness, kurtosis, standard deviation, variance and range Chapter 2: The T-test  We have the information about the means of two samples and want to know whether they are different from each other  These two samples can be the experimental condition or the control condition  We can use a t-test to do this and try to conclude that the null hypothesis is false if the t- statistic is less than .05 General Rationale Inferences Involving a Single Sample Mean  How likely is it that a sample with a given mean (statistic) could originate from a population with a given mean (parameter)? The T-test and Comparison of Independent Means  A t-test is performed to compare two independent means, and it is found that the two sample variance are themselves more variable than reasonably can be attributed to chance, it is necessary to compute the Welch estimate
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