Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38A bar chart of a quantitative variable with only a few categories (called a discrete variable) communicates the relative number of subjects with each of the possible responses. However, the bar chart does not graphically distinguish between quantitative and qualitative variables.Once we looked at the variable label and the values, we would realize that this is a quantitative variable, but it would take that extra work to understand it. 01/14/19 Slide 1If the quantitative variable has a large number of categories (called a continuous variable), the bar chart provides little information beyond the fact that there are a lot of different values, and some occur more frequently than others.01/14/19 Slide 2Histograms are used as the preferred graph for quantitative variables. While the bars resemble those of a bar chart, histograms are distinguished by the absence of gaps between consecutive bars.For continuous variables, values are grouped in equally spaced intervals to convey a sense of what the distribution looks like.01/14/19 Slide 3While we used counts and percents to describe the distribution of a qualitative variable, we use statistical measures to describe the center, spread, and shape of a quantitative variable.Measures of central tendency identify a value in the center of the distribution.Measures of central tendency identify a value in the center of the distribution.Measures of variability or dispersion summarize how the values for individual cases are spread out around the measure of central tendency.Measures of variability or dispersion summarize how the values for individual cases are spread out around the measure of central tendency.01/14/19 Slide 4There are two measures of the shape of the distribution: skewness and kurtosis.Many of the statistics we will use assume that the distribution of a variable is bell-shaped, i.e. the normal distribution.Skewness measures the symmetry of the distribution on both sides of the average score for the distribution. Having overlaid a blue normal curve on the distribution of this variable, we can see that the bars on either side of the red center line are similar as one moves away from the center.Kurtosis measure the degree to which the distribution is peaked or flat compared to the normal distribution. In this example, the bars at the center of the distribution are close to what would be expected for a normal distribution and the frequencies decrease as we move away from the center.01/14/19 Slide 5Both of these variables have a problem with skewness, caused by atypical scores at one end of the distribution. Skewness is characterized as negative or positive, depending on which side, or tail, of the distribution has the unusual scores.This is an example of negative skewness, where a few small scores have elongated the left tail of the distribution. The tail on the right is truncated.This is an example of positive skewness, where a few large scores have elongated the right tail of the distribution. The tail to the left is truncated.01/14/19 Slide 6Both of these variables have a problem with kurtosis, caused by either too few cases in the center of the distribution, or too many cases in the center of the distribution.This is an example of negative kurtosis, where the scores are uniformly distributed through the range of scores. The kurtosis statistic will have a negative value.This is an example of positive kurtosis, where the scores are heavily concentrated in the center of the distribution. The kurtosis statistic will have a positive value.01/14/19 Slide 7When the distribution has minimal skewness and is symmetric, both the red mean line and the green median line fall in the center of the distribution.There are two measures of central tendency for quantitative variables: the mean and the median. The mean is the average score.The median is the middle score, i.e. half of the scores are higher and half are lower.While both measures reflect the center of the distribution, the mean is the preferred measure because it uses information for all of the cases in the distribution.For each measure of centrality, there is a corresponding measure of spread. The standard deviation is used with the mean, and the interquartile range is used with median.01/14/19 Slide 8When skewing is present, the red mean line moves away from the center of the distribution as identified by the green median line in the direction of the skewness.At some level of skewness , the median becomes more effective at representing the center of the distribution.The issue is selecting a defensible rule for deciding the dividing line between acceptable skewness and problematic skewness.The rule of thumb that we will use is that skewness less than -1.0 or greater than +1.0 is problematic and indicates that the median is the preferred measure.01/14/19 Slide 9Kurtosis does not affect the location of the measure of central tendency. Kurtosis indicates that there are either more cases than expected in the middle of the distribution (positive kurtosis), or fewer cases than expected (negative kurtosis).01/14/19 Slide 10The bars extending about the normal curve overlay indicate that there is positive kurtosis. A distribution with positive kurtosis is characterized as a “peaked distribution.”When the bars fall below the center of the normal curve overlay, the distribution has negative kurtosis, and is referred to as a flat distribution.01/14/19 Slide 11•The homework problems on central tendency and variability focus on describing the distribution of quantitative variables.•The counts and percents that we used for qualitative variables are not effective for quantitative variables that can have many different scores in the distribution.•We describe the distribution of quantitative variables with summary statistics that try to communicate the value on which the distribution is centered, the spread of the values from the center of the distribution, the symmetry of the distribution around the center measure, and the degree to which the distribution is bell-shaped or flat.01/14/19 Slide 12•The center, or central tendency, of the distribution is usually represented by the mean (average score)