What are skewness and variance statistics?

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Skewness and variance statistics are concepts in probability theory. Probability theory is a branch of applied mathematics that analyzes random phenomena by collecting and interpreting quantitative data and estimating population parameters. Some of the important concepts of probability theory are random variables, stochastic processes and events. These concepts represent mathematical abstractions that include both non-deterministic events or quantities. They can be measured as well as either single occurrences or events that evolve over time and might appear at first to be merely random occurrences.

Statistical analysis shows that if seemingly random events are repeated and come to be measured over time, they will reveal a sequence. This sequence can be shown to have certain statistical patterns. These patterns have been formalized and they are known, for example, as the central limit theorem and the law of large numbers. Their patterns can be predicted and general statements can be made about them that can generally be shown to be true.

Probability theory is also the mathematical foundation for statistics. It provides a background for analyzing quantitatively various human activities that involve large data sets. They have led to the development of modern theories such as statistical mechanics and quantum mechanics that employ probability theory.

Among the important concepts of probability theory is the concept of skewness that measures asymmetry of the probable distribution of a random variable with a real value. Skewness values can be positive, negative or undefined. Values on one side of this distribution have the tendency to be further from the middle than are values on the other side of that distribution. The usual measures of location will provide different values for skewed data. For example, mode<median<mean shows right or positive skewness. Positive or right skewness is usually more common than negative or left skewness.

A positive skew shows that the tail on the right side is longer than the left side and the bulk of the values can be found to the left of the mean. A negative skew means that the tail on the left side of the probability density function is longer than the right side and most of the values are to the left of the mean. If statistical data shows evidence of skewness, then transformations can be applied. An example of this is taking logarithms of right (or positive) skewed data. A zero value, on the other hand, means that the values are more or less evenly distributed on both sides of the mean. Often, but not always, this implies that the distribution is symmetric.

Skewness gives a measure of the direction and the degree of asymmetry. A normal symmetric distribution has a skewness value of zero and a distribution that is skewed to the left. This might be the case, for example, when the mean is less than the median and has a negative skewness.

Variance, on the other hand, measures variability. It is a non-negative number that indicates how widely spread the values of the random variable will most likely be. Thus the larger the variance, the more widely scattered will be the observations. The variance shows how closely concentrated the distribution will be around the expected value. It measures the spread of a distribution around its average value.

Variance represents the sum of the squared distances of data value from the mean which is then divided by the variance divisor. Variance is not used as an index of spread because it is in squared units. Rather than using variance, standard deviation is used. Standard deviation is the square root of the variance and it measures the spread of a set of observations. The observations are more spread out for larger standard deviations. Both the variance and the standard deviation of a random variable will always be non-negative.

Resource by A.Moore

I just moved to Michigan from South Carolina. This is my first winter here and so far it's freezing. I enjoy traveling, drawing, camping, writing, and doing community service in my spare time. I work for a education company and help them with their outreach to teachers and students. Hope everyone enjoys my articles.

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