## Information & Training. | SPC and Statistical Methods for Process Improvement.

# Creating a Histogram

## Example of using Histogram Analysis to deliver Product and Process Improvement.

**Creating a Histogram – Example:**The following data, presented in tablet format, relates to the length of metal beams (in meters) produced from an automated process.

The specification for the process allows the beams to range from 6.7 to 7.4 meters.

Create a histogram to determine if the process is correctly centered and operating within specification.

## Measurements recorded:

6.9 | 7.1 | 6.7 | 7.0 | 7.0 | 6.8 | 6.9 | 7.1 | 7.0 | 7.0 | 6.9 | 6.9 | 7.0 | 6.9 |

6.8 | 6.9 | 6.8 | 6.8 | 7.0 | 6.8 | 7.0 | 7.0 | 6.8 | 6.9 | 7.1 | 7.1 | 7.1 | 7.1 |

7.1 | 6.8 | 7.0 | 7.1 | 6.9 | 7.2 | 6.9 | 6.9 | 7.2 | 6.9 | 6.9 | 6.9 | 6.9 | 7.0 |

7.0 | 6.9 | 6.9 | 7.0 | 7.0 | 7.0 | 6.8 | 7.1 | 7.1 | 7.1 | 7.0 | 6.9 | 7.1 | 6.9 |

## The first step is to determine the Range and optimum number of Cells. The Range is obtained from identifying the Highest and Lowest Values.

In this example the Range = 7.2 – 6.7 = 0.5 meters.

Using the formula:

2^{k} ~ n, where

n is the number of data points,

k is the number of cells.

n = 56, therefore k ~ 6, therefore 6 looks to be a good number of cells.Note: Using the formula: 2

^{k}~ n, the number of cells based on the number of data points could be estimated as follows.

# Data Points # Cells

Under 100 5-7

101 – 200 8

201 – 500 9

501 – 1000 10

Over 1000 11-20

Continuing with example #2, we now need to allocate limits to each of the cells:

Cell #1: 6.7 – 6.79

Cell # 2: 6.8 – 6.89

Cell # 3: 6.9 – 6.99

Cell # 4: 7.0 – 7.09

Cell # 5: 7.1 – 7.19

Cell # 6: 7.2 – 7.29

Then count the number of occurrence of each data point within each cell:

Cell #1: 6.7 – 6.79 – 1

Cell # 2: 6.8 – 6.89 – 8

Cell # 3: 6.9 – 6.99 – 18

Cell # 4: 7.0 – 7.09 – 15

Cell # 5: 7.1 – 7.19 – 12

Cell # 6: 7.2 – 7.29 – 2 (Frequently this may be labelled as 7.2+ )

Now proceed to creating a Histogram. First detail the cell limits onto the X-axis, then detail the frequency of occurrences onto the Y-axis, label each axis appropriately. Then draw in the cells, with the height of the cells relating to the frequency of occurrences.

With the histogram created, it is possible to answer questions regarding the process performance. The process specifications have been given as ranging from 6.7 to 7.4 meters. These specification limits can now be added onto the histogram graph.

Per the specifications, the process should be centred midway between the specifications, i.e. halfway between 6.7m and 7.4m. This centre point is 7.05m

(7.4-6.7 = 0.7m. Therefore add half of 0.7 = 0.35 onto 6.7m to get the process nominal centre of 7.05m).

Add the process specifications and nominal specification mid-point onto the histogram.

Reviewing the histogram, the process looks to be reasonably well centred, however, the process looks to be skewed to the left, meaning there is an increased likelihood of metal bars being outputted from the process that will be below specification. Further analysis could be performed on the data, via calculating standard deviations which can be applied to determine the likely percentage of product that will be outputted below (and above) specifications. Looking at the histogram, it could be expected that due to the relatively broad span of data, versus the specification limits, that the process will output defects per both specification limits. This observation could be verified via standard deviation analysis.

## Information & Training.

## 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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