# Histograms

## Visual representation of data on a histogram and the many graphs associated with this display

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Histograms

Histogram is a way to represent data, in the form of a bar chart with zero space between each bar, a bar area that is proportional to the frequency it represents, and a bar width equal to the data interval.

To create a histogram, first separate the data from your independent variable into intervals.  Then create a data table, with the independent variable containing the interval you chose, and the dependent variable containing the frequency in that interval.

Here is an example of a data table, containing data comparing the year, and the number of 4.0 GPA's:

[Figure1]

After you have created a data table, create a simple bar chart, with your independent variable intervals on the x-axis, and your dependent variable on the y-axis.  Each bar should go up to the frequency of the dependent variable in the specified interval, with no space between intervals.  Here is a histogram created from the data above:

[Figure2]

Vocabulary

An Interval is a range of data. Grouping data into intervals can be beneficial in a number of ways, including simplifying the appearance and minimizing the effect of individual measurement errors.

Range is a value representing the difference between the least and greatest value in a data set. The range can be found by subtracting the smallest value from the largest.

Binning is the process of grouping data ranges into appropriate intervals. There is no all-around best number of bins, and different numbers of bins can reveal different things about a data set.

Class limits are, collectively, the upper and lower limit of an interval. A class mark is the middle value, or average of the class limits.

How To Interpret a Histogram

• Range: a description of the difference between the greatest and least values in a given data set. On a histogram, this is important in two particular ways:
• How widely dispersed are the frequencies of each bin? Extremely large frequency ranges (particularly as a percentage) may indicate data that is fundamentally unreliable.
• How wide are the bins themselves? Specifically, how broad are the intervals or how descriptive are the classes? Unusually large or small intervals, or unusually broad or narrow categories may indicate important observations about the data as a whole.
• Frequency Density: On a histogram, the frequency is measured by the areaof the bar. What that means it that you can use a histogram with different interval or class widths to represent data with varying densities.
• Shape: The shape of a histogram can lead to valuable conclusions about the trend(s) of the data. In fact, the shape of a histogram is something you should always note when evaluating the data the histogram represents. Some common shapes and their indications are:

a. Bell-Shaped:

• unimodal-1 mode
• symmetrical
• mean=median=mode

b. Uniform:

• multimodal-multiple modes
• frequency of each class is very similar to that of the others

c. Right-Skewed:

• peak that is left of center and a gradual tapering to the right side of the graph
• unimodal
• mode<median<mean

d. Left-Skewed:

• peak that is right of center and a gradual tapering to the left side of the graph
• unimodal
• mean<median<mode

e. Undefined Bimodal:

• shape not defined
• bimodal-2 modes

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