12 Skewness and also the Mean, Median, and Mode

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

This data set can be stood for by complying with histogram. Each interval has width one, and also each worth is located in the middle of an interval.

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. The better the deviation from zero suggests a greater degree of skewness. If the skewness is an adverse then the circulation is skewed left together in (Figure). A confident measure the skewness suggests right skewness such as (Figure).


The typical is 7.7, the mean is 7.5, and also the mode is seven. The the 3 statistics, the average is the largest, when the mode is the smallest. Again, the average reflects the skewing the most.

To summarize, normally if the distribution of data is skewed to the left, the average is much less than the median, which is often less 보다 the mode. If the circulation of data is skewed to the right, the setting is frequently less 보다 the median, which is less than the mean.

As through the mean, median and also mode, and also as we will see shortly, the variance, there room mathematical recipe that provide us specific measures the these features of the circulation of the data. Again looking in ~ the formula for skewness we watch that this is a relationship in between the mean of the data and the individual observations cubed.


is the sample standard deviation that the data, , and also
is the arithmetic mean and also
is the sample size.

Formally the arithmetic typical is recognized as the first moment that the distribution. The second moment we will check out is the variance, and skewness is the third moment. The variance measures the squared differences of the data native the mean and also skewness steps the cubed distinctions of the data from the mean. If a variance deserve to never be a an unfavorable number, the measure up of skewness can and this is how we determine if the data room skewed ideal of left. The skewness for a normal distribution is zero, and also any symmetric data should have actually skewness near zero. An unfavorable values because that the skewness suggest data that room skewed left and positive worths for the skewness suggest data that room skewed right. By it was crooked left, we mean that the left tail is lengthy relative come the ideal tail. Similarly, it was crooked right means that the right tail is long relative come the left tail. The skewness characterizes the level of asymmetry that a distribution approximately its mean. While the mean and standard deviation are dimensional quantities (this is why we will certainly take the square source of the variance ) that is, have the same units together the measured amounts , the skewness is conventionally defined in together a means as to do it nondimensional. That is a pure number that characterizes only the shape of the distribution. A confident value that skewness signifies a circulation with an asymmetric tail expanding out towards much more positive X and a negative value signifies a distribution whose tail extends out towards an ext negative X. A zero measure of skewness will suggest a symmetry distribution.

Skewness and also symmetry become important when we talk about probability distribution in later chapters.

Chapter Review

Looking in ~ the circulation of data have the right to reveal a lot around the relationship between the mean, the median, and also the mode. There room three species of distributions. A right (or positive) skewed circulation has a shape prefer (Figure). A left (or negative) skewed distribution has a shape choose (Figure). A symmetrical distrubtion looks like (Figure).

Formula Review

Formula because that skewness: Formula because that Coefficient that Variation:


Use the following information to answer the next three exercises: State even if it is the data are symmetrical, skewed to the left, or it was crooked to the right.

The data space symmetrical. The mean is 3 and also the mean is 2.85. They space close, and the setting lies close come the center of the data, for this reason the data room symmetrical.

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The data room skewed right. The typical is 87.5 and also the mean is 88.2. Even though they room close, the setting lies to the left that the center of the data, and also there space many an ext instances of 87 than any type of other number, so the data are skewed right.