Distributions of variables have certain characteristics that describe the shape of the distribution. For example, a distribution may be symmetric which means that the right side of the distribution is a mirror image of the left side of the distibution.
Generally, the three important distribution characteristics are modality, symmetry and asymptoticness. Modality refers to how many HUMPS the distribution has. Symmetry describes how the right side of a distribution comapres to the left side. Asymptoticness refers to how far the distribution extends out to the left and to the right.
Modality refers to how many HUMPS (or modes) the distribution has. The distribution below is called UNIMODAL because it has one high point (or mode).
Unimodal distributions are typical when the distribution is describing a single population of individuals. For example, the distribution of heights of 25 year old males is unimodal because there one value (the average height) which is most common. However, the distribution that comes from two different populations would be BIMODAL because there would be a high point at the average value of the two populations. This might look something like the graph below which describes the weightlifting capacity of a large group of college students.
Notice that in this case, the two modes have a big valley between them because in general, there is not much overlap between the weightlifting capacities of the two genders. Most men can lift quite a bit more than the average women, even though there may be a few women who can lift more than the average men and a few men who can lift less than the average women. Of course, some bimodal distributions have humps that are less distinct. For example, the graph below shows the average heights of a large group of students.
In this case, the two modes have a small dip between them because there are quite a few women who are taller than the average man and quite a few men that are smaller than the average women. The size of the valley between the two humps is a function of the width (measured by standard deviation) of the two population distributions that are contributing to the bimodal distribution.
In the bimodal distributions shown so far, the humps have always been equally high, but this does not have to be the case. For example, the graph below shows the average heights of a large group of students where there is more women in the group.
Notice that the higher hump that comes from the heights of the men is smaller than the lower hump that comes from the heights of the women. At some point, of the size of the men is very small compared to the size of the women,then the combined distributions will not even be bimodal, but rather just skewed like this one..
Ultimately, it may be difficult to determine exactly what constitutes a bimodal distribution as opposed to a skewed distribution. Fortunately for students, most examples on exams are clearly bimodal or not, but the wise student rememebbrs that in the real world things can get a little more complicated.
Of course, distributions may have more than 2 modes if they are a combination of three or more populations. The badly drawn distribution below is called TRIMODAL because it has three high points (or modes).
In summary, modaility describes how many humps show in the distribution where each hump usually indicates a separate population.
Symmetry refers to how the left and right side of the distributions compare to each other. If the left and right side are equal, the distribution will is symmetric which looks like this.
NON-symmetric distributions come in two types, positively and negatively skewed. The skew refers to where the long tail is heading. So positively skewed distributions look like this..
Positively skewed distributions are associated with FLOOR EFFECTS because the values on the left side of the distributions are squished up against the floor. For example, the distribution of how many times students have taken the drug ecstasy is postively skewed because
Negatively skewed distributions have a tail that goes negative like this...
Negatively skewed distributions are associated with CEILING EFFECTS because the values on the right side of the distributions are squished up against the ceiling For example, the distribution of scores on a very easy exam is negatively skewed because
In summary, symmetry describes how the two sides of the distribution compare to each other
Asymptoticness refers to how far the distribution extends out from the mean. For a distribution to be perfectly asymptotic, the distribution has to extend out to infinite on both sides. Of course, the probability of values that are far away from the mean is very low, but these values are still possible.
Almost all distributions have a weak violation of asymptoticness. For example, the distributions of heights of males has a maximum and a minimum value. Thus, if the average height of a male is 68 inches and the standard deviation is 4 inches, then the lowest possible height of 0 inches is 17 standard deviations below the mean. This weak violation of asymptoticness is not important because values that are 17 standard deviations below the mean occur with such incredibly low probability that they have no real effect on the size of the area under the curve (which corresponds to probability). The chart below shows a weak violation of asymptoticness where the lowest and highest values are 4 standard deviations away from the mea. As the graph shows, it is nearly impossible to know this by looking at the graph because the probability of values 4 standard deviations away from the mean is so low
Strong violations of asymptoticness are when the maximum or minimum value is not too far away from the mean. For example, if the minimum value is only 1 standard deviation away from the mean, then this distribution is a said to be have a strong violation of asymptotic. The line between weak and strong does not really exist, but certainly 1 standard deviation are considered to be strong violations. The chart below shows a strong violation of asymptoticness
In summary, asymptotic describes how far the distribution extends out from the mean, although almost all distributions have a weak violation of asymptoticness, but only strong violations are considered to be important.