5.3: The Mean, Median, and Mode in Normal and Skewed Distributions
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In a normal distribution, the mean, median, and mode are at the same point in the distribution; stated differently, the mean, median, and mode have the same value.
Normal distributions are rare because most distributions have imperfections in their distribution, and they become skewed. The skewed distribution is the reason we have three measures of central tendency. There are two skewed distributions, positive and negative.
A positively skewed distribution will ALWAYS have the following characteristics:
- The tail is on the high end and to the right of the distribution
- The order of the central tendency values (from left to right): Mode, Median, Mean.
- Mean is pulled to the high end, where there are high extreme scores
- The mean is higher than the median
- The median is the midpoint. Think of the median as ‘in the middle of the road.” The median is always in the middle between the mean and the mode.
- The mode is the highest point in the distribution, or the “hump” in the distribution
In a negatively skewed distribution:
- The tail is on the low end and to the left of the distribution
- The order of the central tendency values (from left to right): Mean, Median, Mode.
- Mean is pulled to the low end, where there are low extreme scores
- The mean is lower than the median
- The median is the midpoint. For example, the median is in the middle of the road. The median is always in the middle between the mean and the mode.
- Mode, which is the highest point in the distribution, or the “hump” in the distribution.
Figure One depicts a skewed distribution with the mean, median, and mode. The positively skewed distribution is on the top, and the negatively skewed distribution is on the bottom. Identifying skewed distributions was addressed in Chapter III. For now, you only need to know the characteristics of the positive and negative skewed distributions. The only reason you need to know this information is for your knowledge base for a licensing exam that addresses this topic. Otherwise, there is little need for you to apply this knowledge when conducting statistics.
Conclusion
The central tendency is an important statistic because it establishes the point of comparison. Statistics is all about comparison, and we need a central tendency to establish how to compare something to something. Establishing that central tendency is not easy, and the presence of outliers, leading to skewed distributions, means that we need three versions of central tendency. We have the mean, median, and mode, and there are pros and cons to using each mean or the median, except the mode, which is completely useless.