What is a Histogram? A histogram is "a representation of a frequency

distribution by means of rectangles whose widths

represent class intervals and whose areas are

proportional to the corresponding frequencies."

Online Webster's Dictionary

Sounds complicated . . . but the concept really is

pretty simple. We graph groups of numbers

according to how often they appear. Thus if we have

the set {1,2,2,3,3,3,3,4,4,5,6}, we can graph them

like this:

This graph is pretty easy to make and gives us some

useful data about the set. For example, the graph

peaks at 3, which is also the median and the mode of

the set. The mean of the set is 3.27—also not far

from the peak. The shape of the graph gives us an

idea of how the numbers in the set are distributed

about the mean: the distribution of this graph is wide

compared to size of the peak, indicating that values

in the set are only loosely bunched round the mean.

How is a Real Histogram Made? The example above is a little too simple. In most real data sets almost all numbers will be unique.

Consider the set {3, 11, 12, 19, 22, 23, 24, 25, 27, 29, 35, 36, 37, 45, 49}. A graph which shows

how many ones, how many twos, how many threes, etc. would be meaningless. Instead we bin the

data into convenient ranges. In this case, with a bin width of 10, we can easily group the data as

below.

Note: Changing the size of the bin changes the apprearance of the graph and the conclusions you

may draw from it. The Shodor histogram activity allows you to change the bin size for a data set

and the impact on the curve.

Data

Range Frequency

0-10 1

10-20 3

20-30 6

30-40 4

40-50 2

Note that the median is 25 and that there is no mode;

the mean is 26.5.

How Shall We Look at Histograms? Of course, part of the power of histograms is that they allow us to analyze extremely large

datasets by reducing them to a single graph that can show primary, secondary and tertiary peaks

in data as well as give a visual representation of the statistical significance of those peaks. To get

an idea, look at these three histograms:

This plot represents data with a well-defined

peak that is close in value to the median and

the mean. While there are "outlyers," they are

of relatively low frequency. Thus it can be said

that deviations in this data group from the

mean are of low frequency. If this were a mass

plot in particle physics, we'd say the mass is

understood with good precision.

In this plot the peak is still fairly close to the

median and the mean but it is much less

defined. It is harder to tell from the plot what

the exact location of the peak is. There are

almost as many values close to the peak as at

the peak itself and outlyers are frequent. As a

particle physics mass plot, this gives an

imprecise and undertain mass of a particle.

Where are the median and the mean? It is hard

to tell; it also may not be relevant. There are

two peaks in this plot: a taller primary peak as

well as a shorter secondary peak. This could

indicate either very poor definition of one

signal in the data or, more likely, two signals.

In particle physics, this could show two

separate particles or, as is often the case, a

large signal with "background" particles and a

smaller signal (sometimes very small), called a

"bump," which shows the actual particle under

study.

Resources

Sample Histogram - This is another example of how a histogram is made, with a focus on

the effect of bin size.

Shodor Histogram Page - This is a nice interactive histogram page in which you can

choose different sample histograms and vary the bin size.

Excel Help - To work with large datasets, it helps to use a spreadsheet. This tutorial

walks you through the process of making a histogram in MS Excel.

Histogram Problems - These are practice problems (with solutions) so that you can

construct and analyze histograms on your own.