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 adhering to histogram. Each interval has width one, and each value is situated in the center of an interval.You are watching: i m sorry of the following is most likely to have a median that is smaller sized than the median

. The better the deviation native zero shows a greater degree of skewness. If the skewness is an unfavorable then the circulation is skewed left as in (Figure). A hopeful measure of skewness suggests right skewness such together (Figure). The mean is 7.7, the median is 7.5, and also the mode is seven. Of the 3 statistics, the typical is the largest, when the mode is the smallest. Again, the mean reflects the skewing the most.

You are watching: Which of the following is likely to have a mean that is smaller than the median

To summarize, typically if the distribution of data is it was crooked to the left, the mean is much less than the median, i beg your pardon is regularly less than the mode. If the distribution of data is it was crooked to the right, the mode is often less than the median, i beg your pardon is less than the mean.

As v the mean, median and also mode, and also as us will check out shortly, the variance, there room mathematical formulas that give us an exact measures the these qualities of the distribution of the data. Again looking in ~ the formula because that skewness we watch that this is a relationship between the mean of the data and the individual observations cubed.

where   is the sample size.

Formally the arithmetic median is well-known as the an initial moment that the distribution. The 2nd moment us will view is the variance, and also skewness is the 3rd moment. The variance steps the squared distinctions of the data from the mean and skewness measures the cubed distinctions of the data native the mean. While a variance can never be a an unfavorable number, the measure of skewness can and also this is how we recognize if the data are skewed best of left. The skewness because that a normal circulation is zero, and also any symmetric data should have skewness close to zero. An adverse values for the skewness indicate data that are skewed left and also positive values for the skewness show data that room skewed right. By it was crooked left, we typical that the left tail is lengthy relative to the ideal tail. Similarly, it was crooked right method that the ideal tail is lengthy relative to the left tail. The skewness characterizes the level of asymmetry that a distribution about its mean. If the mean and also standard deviation room dimensional quantities (this is why we will certainly take the square source of the variance ) that is, have actually the same units together the measured quantities , the skewness is conventionally identified in such a method as to make it nondimensional. It is a pure number the characterizes only the shape of the distribution. A hopeful value the skewness signifies a circulation with an asymmetric tail extending out towards an ext positive X and a an adverse value signifies a circulation whose tail extends out towards much more negative X. A zero measure of skewness will suggest a symmetrical distribution.

Skewness and also symmetry end up being important once we comment on probability distributions in later on chapters.

### Chapter Review

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

### Formula Review

Formula for skewness: Formula because that Coefficient the Variation: Use the adhering to information to answer the following three exercises: State even if it is the data space symmetrical, skewed to the left, or it was crooked to the right.

The data room symmetrical. The mean is 3 and the median is 2.85. They room close, and the mode lies close to the middle of the data, so the data room symmetrical.See more: In The Spaces Below, perform Two ways In Which man D. Rockefeller and Andrew Carnegie were Similar.

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