A histogram is a graphic summary (a bar chart) of variation in a specific set of data. The idea of the histogram is to present the data pictorially rather than as columns of numbers so that the readers can see ‘the obvious conclusions’ which are not always easy to see when looking more or less blindly at columns of numbers. This attribute (simplicity) is an important asset in QC circle activities.
The construction of the histogram may be done directly after the collection of data, i.e. in combination with the construction and use of a check sheet or the construction may be done independently of the use of check sheets, i.e. when analyzing data which have been collected by other ways.
The data which are presented in histograms are variables data, i.e. time, length, height, weight. An example will show how to construct a histogram.
A fictive company with 200 employees (the data has been constructed from several companies of that average size) has had some success with the involvement of the
employees in continuous quality improvements. The employees have been educated in using the seven QC tools and a suggestion system has been set up to handle the suggestions which are expected to come up. During the first year the total number of suggestions was 235.
From the beginning it had been decided that the suggestion committee should have meetings once a week (every Monday morning) in order to make sure that suggestions were evaluated almost continuously so a quick feedback could be given to the individual or to the group which had written the suggestion. The members of the suggestion committee realized that the response time was an important ‘checkpoint’ for the number of suggestions and they realized that the higher the response time the fewer suggestions would be the result of the suggestion system.
A standard for the response time was discussed and a so-called ‘loose standard’ of 13 working days (= two weeks and three working days) was decided. The standard of the 13 working days was decided because it was expected that the complex and difficult questions would need detailed analysis and discussions at perhaps one to two meetings of the suggestion committee.
It was also decided that the response time for each suggestion should be measured and after a year the collected data from the first year should be analysed in order to better understand the system for suggestions and to decide on a fixed standard for the following year. Table 7.5 shows the collected data arranged in groups of five and in the order of suggestions.
From Table 7.5 it can be seen that the response time varies from seven days to 20 days. It is also apparent that ‘the loose standard’ of 13 working days has been difficult to meet.
To achieve a deeper understanding of the variation in the data it was decided to construct a histogram.
The following four steps are recommended when constructing a histogram.
Step 1: Plan and collect the data. The data has been collected and shown in Table 7.5. Step 2: Calculate the range of the data. The range is equal to the difference between the highest and the smallest number in the data set. In the case example the range is equal to (20–7) days=13 days.
Step 3: Determine intervals and boundaries. The purpose of this step is to divide the range into a number of equal broad intervals in order to be able to calculate the frequencies in each interval. The number of intervals depends on the number of data but both too few intervals and too many intervals should be avoided. A number of intervals between eight and 12 is normally a good rule of thumb.
When you have determined the desired number of intervals then the width of each interval can be calculated by dividing the range by the desired number of intervals.
In the example it would be natural to construct a histogram with 13 intervals, so the width of each interval would be equal to one day. If the variation in the data had been higher the width of each interval would have been higher. For example if the range had been 24 days than each interval would have been equal to two days. The intervals are usually calculated by a computer program.
Table 7.5 Response time (days) of suggestions
Group Response time
1 14, 14, 11, 13, 10 2 19, 10, 11, 11, 14 3 11, 11, 17, 10, 11 4 13, 10, 13, 10, 13 5 11, 13, 10, 13, 10 6 11, 16, 12, 10, 13 7 11, 16, 10, 10, 9 8 9, 14, 12, 10, 13 9 13, 14, 10, 10, 11 10 10, 13, 11, 9, 11 11 13, 9, 11, 10, 10 12 10, 9, 11, 11, 10 13 14, 11, 11, 9, 10 14 10, 11, 9, 14, 11 15 14, 11, 17, 10, 11 16 11, 11, 9, 16, 10 17 10, 11, 10, 10, 14 18 14, 13, 9, 11, 14 19 10, 10, 10, 14, 11 20 11, 14, 11, 10, 11 21 8, 11, 11, 11, 11 22 9, 11, 11, 10, 10 23 10, 11, 9, 10, 13 24 11, 11, 10, 20, 14 25 10, 10, 11, 10, 11 26 11, 9, 11, 14, 11 27 11, 14, 17, 14, 9 28 9, 12, 11, 11, 14 29 16, 16, 13, 11, 15 30 16, 14, 13, 9, 16 31 18, 16, 14, 9, 16 32 15, 13, 13, 10, 10 33 13, 13, 11, 18, 9 34 11, 10, 14, 7, 14 35 10, 14, 9, 9, 13 36 11, 10, 11, 10, 9 37 9, 9, 10, 14, 10 38 13, 14, 16, 17, 14 39 10, 16, 19, 11, 11 40 9, 12, 13, 14, 11 41 11, 10, 14, 11, 11 42 11, 10, 13, 16, 10
43 11, 11, 11, 11, 11
44 9, 14, 14, 13, 13
45 10, 13, 16, 11, 14
46 13, 9, 11, 14, 14
47 11, 13, 14, 14, 11
Fig. 7.9 Response time for suggestions
(in days).
Step 4: Determine the frequencies and prepare the histogram. Now the data in each interval shall be tallied, i.e. the frequencies have to be calculated so that the histogram can be constructed. Today when data is usually stored in a computer this step is unnecessary. For most software packages steps 2 to 4 are done interactively with a computer program.
Figure 7.9 shows the histogram of the subgroup averages constructed on a computer package.
The following are easily concluded from the histogram:
1. The standard of 13 days was met only in approximately two out of three cases. 2. There seem to be two different distributions mixed in the same histogram. Perhaps the
left distribution is the result of simple suggestions and the right distribution is the result of more complex suggestions.
One weakness of the histogram is that you do not see a picture of the variation in time. For example the variation shown in a histogram may be the result of a combination of two or more different distributions. In Figure 7.9 the response time data may have come from a distribution with a higher mean in the first half-year than in the second half-year, or the data may have come from distributions where the means have changed following a
decreasing trend. To analyse if that is the case you have to construct a control chart. This will be examined in section 7.7.2.