Hi, and welcome to this video about skew. In this video, we will discuss the differences in data distributions with respect to symmetry.

The distributions of quantitative data using a histogram tell us about the frequency of numeric outcomes in a sample of data. The graph lists the possible outcomes in increasing order along the x-axis, and the counts of each occurrence on the y-axis. The pattern that is revealed can take on many shapes that are all helpful in statistical analysis. When a sample size is large enough, we can use theoretical models to make generalizations about the data collected. Analysis of data distributions focuses on the concepts of center, shape, and spread.

The statistics having to do with the center of the data set are mean, median, and mode. The median is the value that is in the exact center of the data-set. Fifty-percent of the values are higher than the median and fifty-percent are lower. The mean is the average value of the data, calculated by adding all values together and dividing by the number of values in the sample. The mode is the value that occurs most often. The variance and standard deviation are statistics that describe the spread of the data. These are quantifiable values to represent how far individual data points are from the mean.

The shape of the data can be bell-shaped, as seen in a “normally” distributed model. The symmetry of the model indicates that the median and the mean values are comparable. The data tapers off in a consistent pattern away from the center to “tails” on either side.

When distributions are not symmetrical, they may be described as skewed. A distribution that is “skewed right” would have a longer tail on the right side of the center. This occurs when lower values are more frequent in the data set, but the few larger values are “pulling” the tail to the right. As a result, calculation for the mean value will be affected. The mean value will be higher than the median in a right-skewed distribution. Similarly, if a data set had a longer tail on the left side, that would indicate that there would be a greater frequency of high valued data points than low valued data points. The mean would then be lower than the median in a left-skewed distribution.

Here is the formula used to quantify skew:

In this formula, “n” represents the size of the sample, xi represents each individual data value, and \(\overline{x}\) represents the sample mean. Note that this formula reflects the spread of the data, by comparing the magnitude of each data value to the mean. Cubing the differences of data points from the mean in the numerator maintains the sign. For example, a difference that is negative when cubed will remain negative, which is essential for capturing the “direction” of the skew.

The size of the sample should be kept in mind when interpreting a skew value. As with all statistics, a larger sample size generates more reliable results. In general, skew values close to zero result from distributions that are nearly normal. The larger the skew value is in the negative direction, the more extreme the left skew is in the data. Larger positive skew value indicate a stronger right skew.

Thank you for watching this review on skew. See you next time!



by Mometrix Test Preparation | Last Updated: July 29, 2020