Exploratory Data Analysis
Graphical Techniques: Alphabetic
Summarize a Univariate Data Set
The purpose of a histogram
is to graphically summarize the distribution of a univariate
The histogram graphically shows the following:
- center (i.e., the location) of the data;
- spread (i.e., the scale) of the data;
- skewness of the data;
- presence of outliers; and
- presence of multiple modes in the data.
These features provide strong indications of the
proper distributional model for the data. The
probability plot or a
goodness-of-fit test can be
used to verify the distributional model.
section shows the appearance of a number of common features
revealed by histograms.
The above plot is a histogram of
the Michelson speed of light
The most common form of the histogram is obtained by splitting
the range of the data into equal-sized bins (called classes).
Then for each bin, the number of points from the data set that
fall into each bin are counted. That is
The classes can either be defined arbitrarily by the user or
via some systematic rule. A number of theoretically
derived rules have been proposed by Scott
- Vertical axis: Frequency (i.e., counts for each bin)
- Horizontal axis: Response variable
The cumulative histogram is a variation of the histogram
in which the vertical axis gives not just the counts for a
single bin, but rather gives the counts for that bin plus
all bins for smaller values of the response variable.
Both the histogram and cumulative histogram have
an additional variant whereby the counts are
replaced by the normalized counts. The names for these variants
are the relative histogram and the relative cumulative
There are two common ways to normalize the counts.
- The normalized count is the count in a class divided by
the total number of observations. In this case
the relative counts are normalized to sum to one
(or 100 if a percentage scale is used).
This is the intuitive case where the height of
the histogram bar represents the proportion of the
data in each class.
- The normalized count is the count in the class
divided by the number of observations times the
class width. For this normalization, the area
(or integral) under the histogram is equal to one.
From a probabilistic point of view, this normalization
results in a relative histogram that is most akin to
the probability density function and a relative
cumulative histogram that is most akin to the
cumulative distribution function. If you want to
overlay a probability density or cumulative
distribution function on top of the histogram, use
this normalization. Although this normalization is
less intuitive (relative frequencies greater than 1
are quite permissible), it is the appropriate
normalization if you are using the histogram to model
a probability density function.
The histogram can be used to answer the following questions:
- What kind of population distribution do the data come from?
- Where are the data located?
- How spread out are the data?
- Are the data symmetric or skewed?
- Are there outliers in the data?
- Symmetric, Non-Normal,
- Symmetric, Non-Normal,
- Symmetric and Bimodal
- Bimodal Mixture of 2 Normals
- Skewed (Non-Symmetric) Right
- Skewed (Non-Symmetric) Left
- Symmetric with Outlier
The techniques below are not discussed in the Handbook.
However, they are similar in purpose to the histogram.
Additional information on them is contained in the
Stem and Leaf Plot
The histogram is demonstrated in the
heat flow meter
data case study.
Histograms are available in most general purpose statistical
software programs. They are also supported in most general
purpose charting, spreadsheet, and business graphics programs.