PSYCH 240 1st Edition Lecture 4Sec. 2: Descriptives and Distributions cont.Variability-Now we want a single number to summarize how divergent the scores are from one another-If the scores all have very similar values, then variability is low-If some scores have values that are very different than other scores, then variability is high-Range: difference between the largest and the smallest number-Instead of just looking at the range, we more frequently use statistics that are based on all of the scores, not just the highest and lowest-One way to think about this is, "How much do the scores deviate from the mean on average?"-But, as we learned, the average deviation from the mean is always zero - the negative deviations perfectly cancel out the positive-We can get around this by squaring the deviations so they are all positive-Variance: (approximately) the average squares deviation from the mean oIf we were taking the true average we would divide by N-We do N-1 instead because this has some nice statistical propertiesoThe variance is in squared unite, so its not always easy to interpret. We usually wotk with the standard deviation, which is simply the square root of the varianceoThis lets us work with the original units, and you can interpret it as how much the scored deviate from the mean on averageoSo in the example S=2.37 Shape-Researchers don’t typically use summary statistics to describe the shape of a distribution. Instead, they mostly use graphs to visualize the distribution Histogram-Histogram: graph that shows the frequency (number) of scored in a data set that equal agiven value or fall in a given range of values-The x-axis will have each possible value of the variable, and the y-axis will have the frequency of scores at that value. -There will be a bar for each possible value that shows the frequency of scores that have that value-For continuous variable, each histogram bar represents not one specific value but a range of values. The bat height is the frequency of scored within the range Distribution Shape-Symmetrical Distribution: if you draw a line in the middle of the range of scores, one side of the distribution looks very similar to the other side-Positively Skewed Distribution: if you draw a line in the middle of the range of scores, scores will be concentrated on the left side of the line, with fewer scores (a "tail") on the right side-Negatively Skewed Distribution: if you draw a line in he middle of the range of scored, scores will be concentrated on the right side of the line, with fewer scores (a "tail") on the left side Floor and Ceiling Effects-Floor Effects: many of the scores are concentrated near the lowest possible value for thevariable. oOften leads to positively skewed distributions-Ceiling Effects: many of the scores are concentrated near the highest possible value for the variableoOften leads to negatively skewed distributions Mean vs. Median-The mean is a misleading measure of central tendency for distributions with extreme skew and/or outliersoThe median is preferred for skewed variable, since it is less sensitive to extreme scores-For a positively skewed distribution, the mean will almost always be higher that the median-For a negatively skewed distribution, the mean will almost always be lower than the median Unimodal vs. Bimodal-Mode: technically the one most common score, we also use it to mean a "high point" in the distribution more generally-Unimodal Distribution: only one peak in the distribution-Bimodal Distribution: two peaks in the distribution with a clear dip in betweenoOften arise when there is a mixture of different types of scores in the data set Uniform Distribution-Uniform/Regular Distribution: distribution with an approximately equal number of scores across the entire range of values Distributions-You will need to have information about a variable to help yourself make decisions. Every time this happens, ask yourself "What does the distribution look like?"-People tend to give you a single measure of central tendency, but you'll often need information about variability and shape to make good decisions Quantiles-Quantiles: set of cutoff scores that divides a distribution into N regions with an equal proportion of scores in each region (when it isn't convenient to show an entire graph)-Quartiles: quantiles that divide distribution into 4 regions, each with 25% of the scoresoGive you all of the major characteristics of a distribution in just 3 numbersoThe middle quartile is the median - a good measure of central tendencyoInter-Quartile Range: the distance between the 1st and 3rd quartiles (the range in which half of the scores fall around the center of distribution)-You compute it by subtracting the 1st quartile from the 3rd quartile-Higher values indicate more variable distributions-For symmetrical distributions, the median will fall right in the middle of the inter-quartile range-For positively skewed distributions, the median will fall closer to the left of the inter-quartile-For negatively skewed distributions, the median will fall closed to the right of the inter-quartile range-Deciles: quantiles that divide distribution into 10 regions, each with 10% of the scores-Percentiles: quantiles that divide distribution into 100 regions, each with 1% of the