3.4: Shapes of Distributions
- Page ID
- 58896
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When we visualize data — especially using histograms, dot plots, or boxplots — the shape of the distribution gives us important information about how the values are spread and where they tend to cluster. Recognizing the shape helps us choose the right summary statistics, spot unusual values, and prepare for more advanced ideas like probability and inference.
In this section, we’ll explore the most common distribution shapes: symmetric, right-skewed, left-skewed, uniform, and bimodal.
Side Note: Skewness and Kurtosis
In more technical contexts, statisticians use formal formulas to describe the shape of a distribution using skewness (how lopsided the data is) and kurtosis (how "peaked" or "flat" it is). We won’t calculate those here, but you can learn more on Wikipedia: Skewness(opens in new window) and Wikipedia: Kurtosis(opens in new window).
Symmetric Distribution
A distribution is symmetric if the left and right sides are roughly mirror images of each other. The most famous example is the bell-shaped normal distribution.
- The mean and median are approximately equal.
- The data is balanced; there's no lean to either side.
- This shape sets the foundation for many statistical models.
Figure: A symmetric (normal-like) distribution. Values are evenly spread around the center.
Right-Skewed (Positively Skewed) Distribution
A distribution is right-skewed when there’s a long tail on the right. This often happens with data like income, where many people earn moderate amounts, but a few earn much higher than average.
- The mean is pulled right (higher) compared to the median.
- Most values cluster on the left side, with a few high values stretching the distribution.
Figure: A right-skewed distribution. A few high values pull the mean upward.
Left-Skewed (Negatively Skewed) Distribution
A distribution is left-skewed when the tail stretches out to the left. This might occur with variables like exam scores when almost everyone does well, but a few students score much lower.
- The mean is pulled left (lower) compared to the median.
- Most values cluster on the higher end of the scale.
Figure: A left-skewed distribution. A few low values pull the mean down.
Uniform Distribution
A uniform distribution occurs when all values are roughly equally common. There are no obvious peaks, and each interval in the range has about the same frequency.
- Can indicate a random, equal probability across intervals.
- Looks “flat” — no shape or center stands out.
Figure: A uniform distribution. All bins are roughly equal in height.
Bimodal or Multimodal Distribution
A bimodal distribution has two clear peaks; a multimodal distribution has more than two. This may suggest that the data come from more than one group or process.
- Can result from combining groups (e.g., heights of children and adults)
- May signal the need to break the data into subgroups
Figure: A bimodal distribution. Two distinct groups are present in the data.
Summary
Understanding the shape of a distribution helps us choose the best descriptive statistics and prepares us for more advanced statistical tools.
- Symmetric → Consider using the mean and standard deviation.
- Skewed → The median and IQR might give a clearer picture.
- Visuals like histograms and boxplots are a key first step in shape recognition.
Related Videos
In the next section, we’ll explore how data can change over time and how visual tools like time plots help us track shifts, trends, and patterns.