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Histograms for Process Improvement

Creating histograms and utilizing to understand and improve products and processes.

When making a histogram you first need to identify the data to be utilized. The data can be discrete or continuous, however, in the vast majority of situations continuous data will be used.

Histograms require a minimum of 50 to 100 data points, the greater the number of data points collected, then the more accurate the final graph, however, with an initial 50 to 100 readings from a process, excellent results will be obtained.

There are now four basic steps in making a histogram.

Step 1: Determine the range of the data (Highest Data point – Lowest Data Point)

Step 2: Determine the number of divisions (“cells”) to use.

Step 3: Count the number of data points per cell.

Step 4: Plot and interpret the data

Histogram “Range” and “Cells”.

With the initial set of data recorded, determine the “range” of the data, i.e. identify the highest and lowest data points, subtract the lowest from the highest. This provides the total range for the histogram.

With the range known and the number of data points also known, then the optimum number of “Cells” (Cells are also frequently referred to as Divisions, Classes or Bars) within the Histogram can be determined.

The number of Cells along the X axis needs to relate to the sample size. A general rule for determining the optimum number of cells is as follows:

2^k ~ n, where

n is the number of data points,

k is the number of cells.

Histograms for Process Improvement – Example:

If the number of data points =30, then for n = 30,

k ~ 5, i.e. you have approximately 5 cells within the histogram.

for n = 60 ~ 6 cells

n = 150 ~ 7 cells

n = 250 ~ 8 cells

With the number of cells determined and the data range also known, then the value ranges for each call can be identified.

Example, take a distribution of data which details distances in metres. If the range of data is from 0 to 100 metres, and there are 5 cells, then:

cell #1 will span from 0 to 20 metres,

cell #2 will span from 20 to 40m,

cell #3 will span from 40 to 60m,

cell #4 will span from 60 to 80m,

cell #5 will span from 80 to 100m.

In the above, there is overlap where the cells meet, therefore, the cell limits need to be modified to remove the overlap. Normally, the cell limits will be adjusted up or down, to an order of magnitude greater than the measurement accuracy. Taking the above example, if we determine that measurements will be to one decimal place, i.e. measurements may be as follows, 10.2m, 15.6m, 87.0m, etc. Therefore, we will set the cell limits to two decimal places as follows:

cell #1 will span from 0 to 19.99 metres,

cell #2 will span from 20 to 39.99m,

cell #3 will span from 40 to 59.99m,

cell #4 will span from 60 to 79.99m,

cell #5 will span from 80 to 100m.

Count the number of data points per cell.
We can now start to allocate the individual data points to specific cells. For large numbers of data, data spreadsheets (such as excel) can be applied, for smaller sample sizes, it may be simpler to manually count. The end result will be in the form of a distribution table as follows.

Cell #1 – “x” data points

Cell #2 – “y” data points

Cell #3 – “z” data points

Cell #4 – “a” data points

Cell #5 – “b” data points

Plot and interpret the data.
With the above complete, we can now make the histogram. On the X-axis the range and cell boundaries will be detailed. On the Y-axis the number (or frequency) of data points within each cell will be detailed.

The cells can now be entered onto the graph, where the height of each cell equates to the number of data points within each cell.

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“Ideal” Histograms for Process Improvement.

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Statistical Process Control Full Details

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SPC & Statistical Methods for Process Improvement.

Process Capability. Variability Reduction. Statistical Process Control.
Pre-Control. R&R Studies.
Process capability indices Cp, Cpk, Cpm, Capability ratio.
Performance indices Pp and Ppk.
Variable Control Charts.
Attribute Charts.
Pareto Charts.
Individual – X Charts.
Histograms / Process Capability Analysis.
Scatter Diagrams.
Etc. … Etc. …
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