Data Presentation as a Histogram or a Frequency Polygon

Data Presentation as a Histogram or a Frequency Polygon

Histogram

A histogram shows the distribution of numerical data in the form of a graph. However, it is very similar to a bar chart, a histogram groups data into intervals. To construct a histogram, you need to establish all the intervals of data, commonly known as bins. The intervals should capture all the data points and also be non-overlapping.

The intervals appear on the horizontal axis, while the absolute frequencies appear on the vertical axis. For a histogram with equal intervals in size, a rectangle should be erected over the interval, with its height being proportional to the absolute frequency. If intervals are unequal in size, the erected rectangle has an area proportional to the absolute frequency of that particular interval. We would have the vertical axis labeled as ‘density’ instead of frequency in such a case. There should be no space between bars to indicate that the intervals are continuous.

Example 1: Histogram

Consider the previous example of the returns offered by a stock. To bring you up to speed, these were the intervals and the corresponding frequencies:

$$ \begin{array}{c|c} \textbf{Interval} & \textbf{Tally} & \textbf{Frequency} \\ \hline -30\% \leq R_t \leq -20\% & \text{II} & \text{2} \\ -20\% \leq R_t \leq -10\% & \text{I} & \text{1} \\ -10\% \leq R_t \leq 0\% & \text{III} & \text{3} \\ 0\% \leq R_t \leq 10\% & \text{IIIIII} & \text{6} \\ 10\% \leq R_t \leq 20\% & \text{IIIIIII} & \text{7} \\ 20\% \leq R_t \leq 30\% & \text{IIIII} & \text{5} \\ 30\% \leq R_t \leq 40\% & \text{I} & \text{1} \\ \textbf{Total} & \text{} & \textbf{25} \\ \end{array} $$

Frequency Polygon

It is also used to represent the distribution of data graphically. However, it has a major difference when compared to the histogram. Instead of having the class intervals on the horizontal axis clearly showing their upper and lower limits, a frequency polygon uses the midpoints of the class intervals.

$$ \text{Midpoint of a class interval} =\text {Lower limit} + \cfrac { (\text{Upper limit} – \text{Lower limit}) }{ 2 } $$

The vertical axis features the absolute frequencies, which are then joined using straight lines and markers.

Example 2: Frequency Polygon

Going back to the stock return data, we could come up with a frequency polygon.

To come up with the midpoints, we use the formula above. As an example, the midpoint of the interval -30% ≤ Rt ≤ -20% is:

$$ \text{Midpoint} = -30 + \cfrac {(-20 – – 30)}{2} = -25 $$

We can calculate the midpoints for the other intervals in a similar manner. The final frequency polygon should look like this:

The frequency polygon is important because it shows the shape of a distribution of data. It can also be very useful when comparing two sets of data side-by-side.

Note: The endpoints touch the X-axis. The vertical scale can also be positioned at the left margin.


 

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep


    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.