The Distribution of a Variable Psych 240 The distribution of a variable is a description of the values that the variable has and the frequency with which it has each of these values Frequency the number of times each value occurred is on the y axis In the example bellow the frequency of brown is about 17 Bar graphs are usually used to represent categorical variables However pie charts are not because the usually do not have numbers We can also use bar graphs to show the joint distribution of two categorical variables Example of joint distribution You can also make a table to show the joint distribution This is called a contingency table or 2 way table Quantitative Variable A histogram is the most common type of graph for showing the distribution of a quantitative variable Step 1 Determine the range of values of your variable Step 2 Divide the range into bins of equal size Steps 3 and 4 put the values in order count how many are in each bin Step 5 Make a graph Unlike with a bar graph there is no space between bars We want bins that enable us to see the shape of the distribution but without useless detail Not too big not too small In this example the 10 oz bin size looks most appropriate Shapes of Histogram one hump we call this unimodal is fairly symmetrical This means that most of the values are bunched up in one part of the range and that they are about equally spread out on one side as on the other Some unimodal distributions are not symmetrical but skewed If it has a long tail on the right we call it right skewed If it has a long tail to the left we call it left skewed If a histogram has two humps we call that a bimodal distribution And sometimes there s no hump at all We call that a uniform or rectangular distribution It is often possible to identify outliers by looking at a histogram We will learn to use a few different numbers a measure of the distribution s central tendency a measure of the distribution s spread The most basic measure of central tendency is the mean To find the mean you must add the values and divide by the number of observations Two very closely related measures of spread The variance s2 and the standard deviation s The variance is an average of the squares of the deviations of the observations from their mean The standard deviation id the square root of the variance Note that the variance will reflect how far the individual values are from the mean If they are clumped near the mean the variance will be small If they are very spread out from the mean the variance will be large Another measure of central tendency the median The less symmetrical a distribution is the more the mean and the median will differ The mean will be pulled in the direction of the long tail while the median will not The mean is also strongly pulled in the direction of any outliers while the median is not Q1 is the midpoint of the observations that fall below the overall median To find Q1 simply find the median of the values that are below the overall median Q3 is the midpoint of the observations that fall above the overall median To find Q3 simply find the median of the values that are above the overall median The range from Q1 to Q3 is sometimes called the inter quartile range or IQR boxplot or a box and whisker plot And here s a boxplot of the distribution of the ages of the fathers Notice how the right skew shows up in the boxplot