Histogram
Histograms or Frequency Distribution Diagrams are bar charts showing the distribution pattern of observations grouped in convenient class intervals and arranged in order of magnitude. Histograms are useful in studying patterns of distribution and in drawing conclusions about the process based on the pattern.
The Procedure to prepare a Histogram consists of the following steps :
- Collect data (preferably 50 or more observations of an item).
- Arrange all values in an ascending order.
- Divide the entire range of values into a convenient number of groups each representing an equal class interval. It is customary to have number of groups equal to or less than the square root of the number of observations. However one should not be too rigid about this. The reason for this cautionary note will be obvious when we see some examples.
- Note the number of observations or frequency in each group.
- Draw X-axis and Y-axis and decide appropriate scales for the groups on X-axis and the number of observations or the frequency on Y-axis.
- Draw bars representing the frequency for each of the groups.
- Provide a suitable title to the Histogram.
- Study the pattern of distribution and draw conclusion.
The Histogram is normal if the highest frequency is in the central group and there is symmetrical tapering on either side of the central group as in diagram 1. The natural or normal distribution would indicate that the process being studied is under control. Let us see a few instances of Histograms that are not normal and what conclusions one can drawn from them.
Diagram 2 shows two peaks with a little valley between them. Such a distribution is known as Bimodal Distribution. It indicates that the lot being examined is mixed. It may be due to pooling of production from two machines or two shifts, each having a different central value. When one encounters such a distribution, one should study the two lots separately if the identity of the two lots can be ascertained. For instance if one is examining the dimensions of containers and encounters bimodal distribution, the containers from the two moulds can be separated by the mould mark on them. Segregating the data from different lots is known as Stratification of data.
We will see more about this soon. Each lot may then exhibit a normal distribution. In that case one only needs to find why the two lots are different and eliminate the cause. Diagram 3 shows a High Plateau that reminds one of Empire State building. One often encounters this type of distribution in incoming materials if the supplier has sorted out and removed items showing wide variation. It can also occur in finished products if there is inspection and sorting at the end of the production line. Such a distribution indicates that the actual variation in the process is more than what is seen in the Histogram. The items with wider variation have been removed before sampling. Such a process generates waste, is uneconomical and needs to be improved to reduce the variation.
Cliff pattern seen in Diagram 5 may arise due to inspection and sorting out the items only at one end - those below a specific value, but not at the other end. It can also occur when the lower end is zero, and the variation on the higher end goes well beyond twice the value at the peak. Alternate peaks and vales as shown in Diagram 4 is an unnatural pattern that may arise even if the process is under control if the figures have been rounded off incorrectly or class intervals have been selected wrongly. Examine the process of rounding of figures in the data and regroup the data in correct classes and you may get a normal pattern.
A Histogram with an unnatural pattern may indicate that there is possibly something unusual with the process, but is not an evidence of a process being out of control. For instance a Histogram depicting the distribution of age of all citizens will not peak at the centre. It will start with a cliff tapering gradually till around the life expectancy then dropping a little faster and once again tapering into along tail. The distribution of ages of students in a school shows a high plateau as there is a specific age at which children are admitted into a school and they would pass out of the school by a specific age. Still one needs to consider the reasons for every unnatural pattern and draw conclusions based on the pattern, one's knowledge about the process being studied and judgment based on common sense.
