When a distribution is positively skewed the relationship of the mean, median and the mode from the left to right will be?

Consider the following data set.
4; 5; 6; 6; 6; 7; 7; 7; 7; 7; 7; 8; 8; 8; 9; 10

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Central Tendency Measures in Positively Skewed Distributions
Positively Skewed Distribution in Finance
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This data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval.

Figure 2.11

The histogram displays a symmetrical distribution of 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 mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.

The histogram for the data: 4; 5; 6; 6; 6; 7; 7; 7; 7; 8 shown in Figure 2.11 is not symmetrical. The right-hand side seems "chopped off" compared to the left side. A distribution of this type is called skewed to the left because it is pulled out to the left. 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". The mathematical formula for skewness is: a3=∑(xi−x¯)3ns3a3=∑(xi−x¯)3ns3. The greater the deviation from zero indicates a greater degree of skewness. If the skewness is negative then the distribution is skewed left as in Figure 2.12. A positive measure of skewness indicates right skewness such as Figure 2.13.

Figure 2.12

The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so.

The histogram for the data: 6; 7; 7; 7; 7; 8; 8; 8; 9; 10 shown in Figure 2.13, is also not symmetrical. It is skewed to the right.

Figure 2.13

The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most.

The mean is affected by outliers that do not influence the mean. Therefore, when the distribution of data is skewed to the left, the mean is often less than the median. When the distribution is skewed to the right, the mean is often greater than the median. In symmetric distributions, we expect the mean and median to be approximately equal in value. This is an important connection between the shape of the distribution and the relationship of the mean and median. It is not, however, true for every data set. The most common exceptions occur in sets of discrete data.

As 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. 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.

a3=∑(xi−x¯)3ns3a3=∑(xi−x¯)3ns3

where ss is the sample standard deviation of the data, XiXi , and x¯x¯ is the arithmetic mean and nn is the sample size.

Formally the arithmetic mean is known as the first moment of the distribution. The second moment we will see is the variance, and skewness is the third moment. The variance measures the squared differences of the data from the mean and skewness measures the cubed differences of the data from the mean. 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 skewness for a normal distribution is zero, and any symmetric data should have skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. The skewness characterizes the degree of asymmetry of a distribution around its mean. 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 XiXi, the skewness is conventionally defined in such a way as to make it nondimensional. It is a pure number that characterizes only the shape of the 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. A zero measure of skewness will indicate a symmetrical distribution.

Skewness and symmetry become important when we discuss probability distributions in later chapters.

A type of distribution in which most values are clustered around the left tail of the distribution

In statistics, a positively skewed (or right-skewed) distribution is a type of distribution in which most values are clustered around the left tail of the distribution while the right tail of the distribution is longer. The positively skewed distribution is the direct opposite of the negatively skewed distribution.

Central Tendency Measures in Positively Skewed Distributions

Unlike with normally distributed data where all measures of the central tendency (mean, median, and mode) equal each other, with positively skewed data, the measures are dispersed. The general relationship among the central tendency measures in a positively skewed distribution may be expressed using the following inequality:

Mean > Median > Mode

In contrast to a negatively skewed distribution, in which the mean is located on the left from the peak of distribution, in a positively skewed distribution, the mean can be found on the right from the distribution’s peak. However, not all negatively skewed distributions follow the rules. You may encounter many exceptions in real life that violate the rules.

Since a high level of skewness can generate misleading results from statistical tests, the extreme positive skewness is not desirable for a distribution. In order to overcome such a problem, data transformation tools may be employed to make the skewed data closer to a normal distribution.

For positively skewed distributions, the most popular transformation is the log transformation. The log transformation implies the calculations of the natural logarithm for each value in the dataset. The method reduces the skew of a distribution. Statistical tests are usually run only when the transformation of the data is complete.

Positively Skewed Distribution in Finance

In finance, the concept of skewness is utilized in the analysis of the distribution of the returns of investments. Although many finance theories and models assume that the returns from securities follow a normal distribution, in reality, the returns are usually skewed.

The positive skewness of a distribution indicates that an investor may expect frequent small losses and a few large gains from the investment. The positively skewed distributions of investment returns are generally more desired by investors since there is some probability of gaining huge profits that can cover all the frequent small losses.

Additional Resources

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