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Date Created: 04/30/16
Intro to Statistics AKA PY211 Table of Contents The Absolute Basics Central Tendency Distributions Variability More on Distributions A Quick Word on Probability Z Scores Hypothesis and Hypothesis Testing P Values T Test v Z Test Effect Size and Statistical Power One Sample T Test Dependent Sample T Test Independent Sample T Test One Sample v Dependent v Independent ANOVA F Tests Correlation The Absolute Basics You learned this in kindergarten. Please bear with me. A Variable is defined as anything that can vary, like hair color, temperature, or age. A Value is defined as a possible number or category that a score can be, like 28 degrees or blonde. A Score is defined as a particular person’s value on a variable. For example, my score for hair color is brown. Categorical Scales are scales where the values are names or categories, not numbers. Quantitative Scales are scales where the values have to be numbers. There’s several kinds. o Interval Scales are scales where there is no absolute zero, and the numbers are equally spaced. o Ratio Scales are scales where there is an absolute zero, and the numbers are equally spaced. o Ordinal Scales are organized by rank, so the numbers aren’t necessarily spaced equally. Frequency Tables are mapped out tables showing where scores fall throughout the different values for the variable. Histograms are graphs of these tables. Distributions of scores these scores can be unimodal (having only one mode or high point), bimodal (having two high points), or multimodal (having more than two high points). Central Tendency Mean, median, and mode are measures of central tendency, also called averages. You usually hear mean and average used interchangeably, but all three of these measurements are averages. o The mean is heavily influenced by extreme scores, and in distributions with large skews the mean is an exceedingly poor measurement of the average. It’s only functional for ratio and interval scales. It can’t compare Coke and Pepsi. The mean is called the balance point of the distribution. The mean is the best measure of central tendency. Deviation Scores are found by subtracting the mean from the score. If you take all of these scores and add them, you get zero. That’s because the mean is the middle of the distribution. The trimmed mean throws out an equal number of top scores and bottom scores, which makes the mean more powerful because it isn’t drug around so much by outliers. o The median is where exactly half of the scores are higher than the mean and half of the scores are lower than the mean. The median is less influenced by extreme scores than the mean is, it’s harder for a positive or negative skew to drag it around. It can be used on ordinal data, but it isn’t useful for many statistical tests. o The mode is effectively useless in most statistical tests, but it’s defined as the most common score in the distributions. It’s the highest point on a histogram. Distributions In a symmetrical distribution, the mean, median, and mode are all the same line right through the high point of the distribution. In a negatively skewed distribution, the hump is on the right half of the graph and there’s a ‘tail’ dragging out towards the left half of the graph. In this kind of distribution, the mean sits furthest to the left, the median sits to its right, and the mode sits farthest to the right, at the high point. In a positively skewed distribution, the hump sits to the right half of the graph and the tail drags out towards the right half of the graph. This is exactly the opposite of a negatively skewed distribution, in this graph the mode comes first with the hump, then the median, then the mode, which is drug way out to the right by the positive skew. Variability Variability refers to the difference between scores. As long as all of the scores are not zero, variability cannot be zero. o Range is hyper sensitive to outliers, it’s the difference between the highest score and the lowest score. For example, of the scores 1, 5, 6, 9, 26, the range is 25. o Interquartile Range is the trimmed version of the range, it’s less sensitive to outliers. To find the interquartile range, throw out the bottom forth of the scores as well as the top fourth, and take the range of the middle half that’s left over. For example, of the scores 6, 12, 18, 29, the interquartile range is 15. o Sum of Squares is a step up from deviation scores. To find the sum of squares, take the deviation score of every score in the distribution and then square every single deviation score, and then add them all together. For example, of the scores 8,12,16,24, the mean is 15, the deviation scores are -7, -3, 1, and 9. Those squared are 49, 9, 1, and 81. The sum of squares then is 140. o Variance is a step up from sum of squares. The variance is the average of squared deviations, so to find the variance we just divide by the number of terms we had. We can only do that if we have every single data point in the data pool, so unless we took all 8 billion data points humanity has to offer, we can’t use the number of terms we had. To compensate for not having all of the data in the world, we divide by the number of terms we had minus one. Dividing by one less than the number of terms is a better estimate of the population than using just the number of terms we have. So using the data from the sum of squares example, the variance would be 140/(4-1), so 140/3, which is 46.67. o Standard Deviation is one more step from variance. Standard deviation is proportional, which means that if you were to take 5 away from every single data point in the distribution, the standard deviation wouldn’t change. Standard deviation is the average amount that each score is away from the mean. It’s a standard unit to the whole distribution, it gets us back into the units of the data we started with. All you have to do to find the standard deviation is take the square root of the variance. So with the data we’ve used for the last two problems, that’s the square root of 46.67, which is 6.83. To summarize those last three; x2 Sum of Squares (SS) = ∑ Variance(s) = SS/n-1 2 Standard Deviation ( o¿ = √s More on Distributions Skewness is the amount of lopsidedness of the data. Remember, the direction of the skew is the direction of the tail, ie the short part of the distribution, not the crest. Kurtosis is the type of peak the crest of the data has, a Mesokuric peak is a normal peak, a Platykuric peak is a flat peak, and a Leptokuric peak which is tall and pointy. A normal distribution has a mesokuric peak and no skew at all, so the mean equals the median equals the mode. The sampling distribution is also called the distribution of means. It’s made of tons and tons of same size samples randomly taken from the same population of individuals. The Central Limit Theorum says that given a mean and a variance, the sampling distribution of the mean approaches a normal distribution, given there are enough samples. Basically, in any set of data, regardless of how skewed it was to begin with, looks normally distributed with enough samples. It usually takes about 25-30 samples for the distribution to approach normal. A Quick Word on Probability Probability is defined as the number of possible outcomes we want divided by the total number of possible outcomes. This is usually expressed as a decimal, but occasionally as a percentage. For example, the probability of flipping a coin ‘heads’ is ½, or .5. There’s only one possibility we want, and only two possibilities total, hence .5. Occasionally it gets more complicated than that. For example, the probability of flipping a coin at least twice out of three flips. The possibilities are HHT, HTH, HTT, HHH, TTT, THT, THH, or TTH. Four of those options have two Hs in them. There’s eight options total. So the probability of flipping a coin at least twice is also .5. Z Scores Z scores are also called ‘standard scores’. It uses the mean and the standard deviation to describe how far above or below a score is from the mean. Because it has nothing to do with the units of the data it describes, it can be used to compare across different kinds of data. This is where growth curve (height/weight) charts come from. To move a raw score to a Z score, take the mean out of the raw score, and then divide that number by the standard deviation. To move from a Z score to a raw score, do the inverse. Multiply the Z score with the standard deviation, and then add the mean. To simplify, that’s (X−M)/o to get from raw scores to Z scores (Z)(o)+M And to get from a Z score to a raw score Because Z scores tell us how far above or below the mean a score is, it’s very important to pay attention to positives and negatives here. A raw score above the mean will always have a positive Z score. A raw score below the mean will always have a negative Z score. Z scores are normally distributed on a Bell Curve. On this curve, frequency is represented on the y axis and the x axis is marked by standard deviations away from the mean, represented in the center. 99.7% of scores fall within 3 standard deviations of the mean in the positive or negative direction. 95.4% of scores fall within 2 standard deviations of the mean in the positive or negative direction. 68.3% of scores fall within 1 standard deviation of the mean in the positive or negative direction. Because the distribution is symmetrical, you can divide these numbers by 2 to find out the percentage of scores exclusively 3 standard deviations above the mean (49.85%) or exclusively 1 standard deviation below the mean (34.15%). This information can also be accessed more readily from a Z chart, or a Z table, which has all possible Z values in one column matching up to the percent under the curve in another column. Hypothesis and Hypothesis Testing Hypothesis must consist of a dependent variable and an independent variable, and be specific enough to test. Hypothesis based study is a way to take a sample and see if it applies to a population, a way of seeing if what is true for a sample is true for a population. For example, if it’s below 32 degrees, fewer students will go to PY211. This hypothesis is based on informal observation. The dependent variable is the number of students in attendance, and the independent variable is the temperature. We would express this hypothesis symbolically as µ1 (our sample) < µ2 (the population) We never test the thing we’re interested in. We test the probability that we’re wrong about the thing we’re interested in. What we test instead of the thing we actually care about is called the null hypothesis. The specific hypothesis that refers to the thing we care about is called the alternate hypothesis. The null hypothesis says the exact opposite of what the alternate hypothesis says. If our alternate hypothesis says ‘if it’s below 32 degrees, fewer students will go to PY211’, our null hypothesis says ‘if it’s below 32 degrees, the normal amount of students or more will go to PY211’. We reject or fail to reject the null hypothesis based on the probability our data could have come from a random sample of the population. If there is no good proof the two groups are different, we fail to reject the null. Our results never prove or disprove our alternative hypothesis. If we fail to reject the null, our results are inconclusive. If we do reject the null, the results support our alternate hypothesis. There are 5 steps to hypothesis testing o 1) State the null hypothesis and the alternate hypothesis. (µ1=µ2, µ1<µ2, or µ1>µ2) o 2) Determine a comparison distribution (Distribution of means, t distribution, sampling distribution, distribution of differences, etc.) o 3) Determine a cut off score (for Z tests, your cut off scores will only ever be one of these four things; for a 1 tailed alpha of .05 -> a cut off of 1.64, a 2 tailed alpha of .05 -> 1.96, a 1 tailed alpha of .01 -> 2.33, and finally, a 2 tailed alpha of .01 -> 2.57) o 4) Find the sample score on the distribution. The formula for this is (M- µM)/ oµ . That means taking the sample mean out of the population mean, and then dividing that by standard error. Standard error is the standard deviation of standard deviation, so to find that, just take the square root of standard deviation over the number of terms. That’s the full number of terms, no n-1 for standard error. Formulaically that looks o like this √ n o 5) Make a decision about the hypothesis. If the sample score is more extreme than the cut off score, then reject the null. If the sample score is less extreme than the cut off score, then you have failed to reject the null. P Values The p value is the likeliness we got the result that we did assuming the null hypothesis to be true, meaning assuming the two groups are actually not different or separate at all. This is where we get alpha from. The level at which we set alpha is the level to which we’re okay with being wrong. Usually, that’s .05, which means we’re willing to be wrong about 1 out of 20 times. For clinical trials, alpha usually moves to .01, they’re only willing to be wrong 1 out of 100 times. There’s always a chance that even when we think we’re right we’re actually wrong, and then we’ve made either a type 1 or a type 2 error. o Type 1 error means that because we set alpha too low, we rejected the null when we shouldn’t have rejected the null. Called an alpha error. o Type 2 error means that because our sample was skewed weirdly, we failed to reject the null when we should have rejected the null. Called a beta error. Z Test v T Test In a Z test, we need to know the population variance. In reality, we rarely know the population variance. T tests don’t need the population variance The only statistical tests that allow us to actively compare means are o 1 sample t tests o Dependent samples t tests o Independent samples t tests o ANOVA tests (Analysis of Variance) o Multivariate Analysis of Variance T curves are shaped differently than Z curves. Because there’s less information available in t tests (no population variance) the tails of the distribution are fatter for t tests than they are for z tests. This means we have to use a t table for everything, and it tells us the cut off score for each distribution we need instead of making us actually think which is nice. There is an infinite number of t distributions, because there’s a slightly different distribution for each degree of freedom. Degrees of Freedom is the number of terms that are free to vary. This is found by using n-1. N-1 is an unbiased estimator, meaning it’s just as likely to be a high estimator as a low estimator. T distributions aren’t normal. Every Degree of Freedom has exactly one t distribution. The higher the degrees of freedom, the more the distribution approaches normality. Because the tails are so fat, the cut off score has to be higher to keep the same number of scores in the tails. You find a t score by taking the population mean out of the sample mean and then M−Mµ divide all of that by standard error; oµ . Effect Size and Statistical Power µ1−µ2 Effect Size is most often calculated by using Cohen’s D, o , which just says the difference between the means divided by the standard deviation. Effect Size is the standardized measure of population difference. A small effect size is anything .2 or smaller, a medium effect size is in the general vicinity of .5, and a large effect size is anything larger than .8. Statistical power is the likelihood the study will give a significant result. If there is a difference between the two groups, the statistical power of the experiment is the probability of finding it. Can be increased by increasing the effect size, the number of people in the study, the alpha, and by making it a one-tailed test. One Sample T Test Uses raw scores, and makes two very important assumptions; that the samples are truly random, and that the population follows a normal distribution. One Sample T Test can survive if this assumption isn’t met, meaning it’s ‘robust’ to the assumption. One sample t tests include a group being compared to a standard. The only major difference between this and a z test is that you have to use a t distribution chart. Dependent Sample T Test The biggest difference between a one sample test and a dependent test is that a dependent samples t test assumes a population mean of zero and uses difference scores. These are found by subtracting the difference from the two consecutive scores and using that number as the data. Once you’ve calculated the difference score, the raw data is effectively useless. Repeated Measures study design means you take data from a group of people, then induce a change, and then take their data again. This is also called a “paired samples” test. The upside is that you don’t have to worry about outside factors like intelligence or social fluency or blindness influencing your data, because all the people retain their characteristics from one trial to the next. It also requires fewer participants to be statistically powerful. The downside is that because there’s a lower degree of freedom, the cut off scores are high, as well as the fact that the people get better at the trials from one trial to the next because they’ve practiced, which messes with the data. There’s another downside of repeated measures; treatments overlap. When you have participants trying things one way and then trying them another way, the first way is still in their heads and it effects the way they engage in the second way. There’s always going to be a high degree of correlation between the two scores, because they’re from the same person. The power of a repeated measures design is high because it needs less people to be as statistically strong as other designs. Independent Samples T Test In an independent samples t test we compare two sample means. We can’t compare the means directly because we care about the population, not the sample. The sample mean is the best predictor of the population mean. In the Independent Samples test, we don’t have the population mean or the population variance. Because the scores are so different, we can’t use difference scores to interpret the data. It’d be like comparing peaches and fireflies. To build this distribution, create a distribution of means for each sample, then randomly pick a mean from each distribution, then graph it. Repeat this process to infinity. Independent Samples work on the assumptions that The population is normally distributed (t tests are robust to this one, especially with a high number of terms, because of central tendency) The scores are independent of each other The variances are the same across the board (t tests are robust to this also, especially when there’s the same number of terms in each group) When doing tests, we take the average of the variances (with weight towards the bigger grouping, if there is a bigger grouping) because neither is a better estimate than the other. There is a formula for finding variance when the groups are of different sizes. S = DF 1 S + DF 2 (S ) DFtot ( 1 DFtot 2 In English, that’s Pooled Estimate of Population Variance equals the degrees of freedom for group one divided by the total degrees of freedom times the variance of group one plus the degrees of freedom for group two divided by the total degrees of freedom times the variance of group two. The standard error is the square root of variance, so it’s the square root of all of that. The formula for that is; S2 S 2 S x1−x 2 + √ n1 n2 In English, that’s the pooled estimate of population variance divided by the number of terms in group one plus the pooled estimate of population variance divided by the number of terms in group two, the sum of which is then square rooted. One Sample Test v Dependent v Independent A One Sample T test is when you compare a sample mean to a population mean without a population variance, so you use the sample variance to predict the population variance. If we knew the population variance, we’d do a z test A Dependent Sample T test is when you don’t know the mean of the population or the variance of the population. An Independent Sample T test is when we compare two sample means. We can’t compare the means directly because they’re too different. The sample mean is the best predictor of the population mean. One Sample Dependent Independent Sample Sample Population No No No Variance Population Mean Yes No No Number of Scores 1 2 1 ComparisonDistrib Means Difference Scores Difference b/w ution Means Shape T distribution T distribution T distribution F Tests An ANOVA test is an Analysis of Variance, it compares the means of 2 or more samples. There’s a few new terms for F tests, Sum of Squares between, Sum of Squares within, Degrees of Freedom between, Degrees of Freedom within, Mean Square between, and finally Mean Square within. SSbw is the explainable variance due to the independent variable, and is calculated by taking the mean form the grand mean (mean of the means for each data set) and squaring the result, and because both of these numbers are constants for the set, the number is the same for every data point in the set. SSw is the variance due to personal characteristics because of things like motivation or intelligence or ability of the subject and is therefore unexplainable. This is calculated the same way regular sum of squares, by subtracting the mean from each data and squaring the result. DFbw is the degrees of freedom between the groups, and is calculated by taking the number of groups minus one. DFw is the degrees of freedom within the group, and is calculated the way we’re used to calculating DF; by taking the number of participants and subtracting one. MSbw is found by dividing SSbw by DFbw. MSw is found by dividing SSw by DFw Your F score is the Means Square between divided by Means Square within. Correlation Correlation is defined as the relationship between two variables. Correlation is not causation, but causation is correlated. There are 3 types of Correlation; X causes Y, Y causes X, or Z causes X and Y. Correlation is between -1 and 1. A correlation of 0 means the variables are not correlated. A negative correlation means a negatively sloped data set with high y values and low x values, whereas a positive correlation means a positively sloped data set with high y values and high x values. Z Scores let us compare across sets of unlike data. To do this, take the z score of each score in each category, and multiply them together. For example, if your x category had values 3,6, and 9 and your y category had scores 2,4, and 8, the z scores you would multiply together would be the Z for 3 together with the Z for 2, and the Z for 6 together with the Z for 4. To arrive at the correlation coefficient, r, add up all of the Z cross products together and then divide by the number of data points you have. r The t test for correlation is 2 it’s n-2 because there are 2 groups; x √(1−r )/(n−2) and y Good Luck!!!
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