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Histograms: Construction, Analysis andUnderstanding

Conservation Laws - Data Analysis Using Graphs - Histograms - Units or Vectors in Particle Physics

What is a Histogram? A histogram is "a representation of a frequencydistribution by means of rectangles whose widthsrepresent 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 numbersaccording to how often they appear. Thus if we havethe set {1,2,2,3,3,3,3,4,4,5,6}, we can graph themlike this:

This graph is pretty easy to make and gives us someuseful data about the set. For example, the graph

peaks at 3, which is also the median and the mode ofthe set. The mean of the set is 3.27 also not far

from the peak. The shape of the graph gives us anidea of how the numbers in the set are distributedabout the mean: the distribution of this graph iswide compared to size of the peak, indicating thatvalues in the set are only loosely bunched round themean.

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 thedata 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 youmay draw from it. The Shodor histogram activity allows you to change the bin size for a data setand the impact on the curve.
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DataRange Frequency

0-10 1

10-20 320-30 6

30-40 4

40-50 2

Note that the median is 25 and that there is nomode; 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 extremelylarge datasets by reducing them to a single graph that can show primary, secondaryand tertiary peaks in data as well as give a visual representation of the statisticalsignificance 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 andthe mean. While there are "outlyers," they areof relatively low frequency. Thus it can be saidthat deviations in this data group from the

In this plot the peak is still fairly close to themedian and the mean but it is much lessdefined. It is harder to tell from the plot whatthe exact location of the peak is. There arealmost as many values close to the peak as at


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mean are of low frequency. If this were a mass plot in particle physics, we'd say the mass isunderstood with good precision.

the peak itself and outlyers are frequent. As a particle physics mass plot, this gives animprecise and undertain mass of a particle.

Where are the median and the mean? It is hardto tell; it also may not be relevant. There aretwo peaks in this plot: a taller primary peak aswell as a shorter secondary peak. This couldindicate either very poor definition of onesignal in the data or, more likely, two signals.In particle physics, this could show twoseparate particles or, as is often the case, alarge signal with "background" particles and asmaller signal (sometimes very small), called a"bump," which shows the actual particle understudy.

Resources

Sample Histogram - This is another example of how a histogram is made, witha 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. Thistutorial walks you through the process of making a histogram on MS Excel.

Histogram Problems - These are practice problems (with solutions) so that youcan construct and analyze histograms on your own.

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