A zero measure of skewness will indicate a symmetrical distribution. A positive value of skewness signifies a distribution with an asymmetric tail extending out towards more positive X and a negative value signifies a distribution whose tail extends out towards more negative X. It is a pure number that characterizes only the shape of the distribution. While the mean and standard deviation are dimensional quantities (this is why we will take the square root of the variance ) that is, have the same units as the measured quantities X i X i, the skewness is conventionally defined in such a way as to make it nondimensional. The skewness characterizes the degree of asymmetry of a distribution around its mean. Similarly, skewed right means that the right tail is long relative to the left tail. By skewed left, we mean that the left tail is long relative to the right tail. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. The skewness for a normal distribution is zero, and any symmetric data should have skewness near zero. While a variance can never be a negative number, the measure of skewness can and this is how we determine if the data are skewed right of left. The variance measures the squared differences of the data from the mean and skewness measures the cubed differences of the data from the mean. The second moment we will see is the variance, and skewness is the third moment. Where s s is the sample standard deviation of the data, X i X i, and x ¯ x ¯ is the arithmetic mean and n n is the sample size.įormally the arithmetic mean is known as the first moment of the distribution. Again looking at the formula for skewness we see that this is a relationship between the mean of the data and the individual observations cubed.Ī 3 = ∑ ( x i − x ¯ ) 3 n s 3 a 3 = ∑ ( x i − x ¯ ) 3 n s 3 The most common exceptions occur in sets of discrete data.Īs with the mean, median and mode, and as we will see shortly, the variance, there are mathematical formulas that give us precise measures of these characteristics of the distribution of the data. It is not, however, true for every data set. This is an important connection between the shape of the distribution and the relationship of the mean and median. In symmetric distributions, we expect the mean and median to be approximately equal in value. When the distribution is skewed to the right, the mean is often greater than the median. Therefore, when the distribution of data is skewed to the left, the mean is often less than the median. The mean is affected by outliers that do not influence the mean. Again, the mean reflects the skewing the most. Of the three statistics, the mean is the largest, while the mode is the smallest. The mean is 7.7, the median is 7.5, and the mode is seven. A positive measure of skewness indicates right skewness such as Figure 2.13. If the skewness is negative then the distribution is skewed left as in Figure 2.12. The greater the deviation from zero indicates a greater degree of skewness. The mathematical formula for skewness is: a 3 = ∑ ( x i − x ¯ ) 3 n s 3 a 3 = ∑ ( x i − x ¯ ) 3 n s 3. We can formally measure the skewness of a distribution just as we can mathematically measure the center weight of the data or its general "speadness". A distribution of this type is called skewed to the left because it is pulled out to the left. The right-hand side seems "chopped off" compared to the left side. The histogram for the data: 4 5 6 6 6 7 7 7 7 8 shown in Figure 2.11 is not symmetrical. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median. This example has one mode (unimodal), and the mode is the same as the mean and median. In a perfectly symmetrical distribution, the mean and the median are the same. The mean, the median, and the mode are each seven for these data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The histogram displays a symmetrical distribution of data.
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