# Histograms for Process Improvement

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

## Test your understanding.

**Question.**

The following measurements were recorded off a plastic cutting process, where lengths of plastic were automatically cut to specification. Specifications are 7.0mm minimum to 8.5mm maximum. Twenty five samples were recorded as follows:

7.1; 7.3; 7.4; 7.6; 7.5

7.8; 8.1; 8.3; 8.5; 8.3

8.1; 8.0; 7.6; 7.9; 7.8

8.0; 8.0; 8.0; 7.9; 7.8

7.9; 8.2; 8.0; 8.1; 8.3

**Calculate:**

a) Mean,

b) Median,

c) Mode,

d) Range,

e) Plot the histogram.

f) Interpret the results.

*Note: Normally 50 to 100 data points would be necessary, however, for simplicity 25 data points are being used.*

**a) Mean**

= “Sum of all the measurements” divided by the “Number of measurements recorded”

= 197.5 / 25 = 7.9

**b) Mode**

= The mode is the most frequently occurring value in a group of data = 8

**c)Median**

= The median represents the “middle” value when the data is arranged in ascending or descending order.

If the data is arranged in ascending order as follows:

7.1; 7.3; 7.4; 7.5; 7.6; 7.6; 7.8; 7.8; 7.8; 7.9; 7.9; 7.9; 8.0; 8.0; 8.0; 8.0; 8.0; 8.1; 8.1; 8.1; 8.2; 8.3; 8.3; 8.3; 8.5

Then the “middle” value or median = 8

**d) Range**

= Difference between the LARGEST and the SMALLEST values.

= 8.5mm – 7.1mm = 1.4mm

**e) Plotting the histogram.**

The range is known, the next step is to determine the number of cells (or divisions or bars). Using the formula:

2^{k} ~ n, where

n is the number of data points,

k is the number of cells.

4 to 5 cells can be used. We will use 5 in this example. The 5 cells will go from 7.1mm to 8.5mm a distance of 1.4mm. Therefore the cell will have a width of 2.8mm, which we will round up to 0.3mm.Cell #1: 7.0 – 7.29 mm

Cell #2: 7.3 – 7.59 mm

Cell #3: 7.6 – 7.89 mm

Cell #4: 7.9 – 8.19 mm

Cell #5: 8.2 + mm

Frequency of occurrence is as follows:

Cell #1: 7.0 – 7.29 mm occurs 1 time

Cell #2: 7.3 – 7.59 mm occurs 3 times

Cell #3: 7.6 – 7.89 mm occurs 5 times

Cell #4: 7.9 – 8.19 mm occurs 11 times

Cell #5: 8.2 + mm occurs 5 times

Histograms for Process Improvement – The histogram can now be plotted.

**f) What inference can be taken from a review of the histogram.**

The data looks to be skewed to the right, meaning that the process outputs are not correctly centred. The data is widely distributed within the specifications. Ideally the histogram should be a tight bell shaped curve with very few or no data points near the specification limits. While no data points recorded during the study lie outside the specification limits, the histogram tells us that the process will output “out of specification” product if a large enough number of data points were to be recorded.

Those responsible for the process need to make process improvements, namely:

i) Center the process outputs so that the mean, mode and median all lie centred between the specification limits.

ii) Reduce process variability so that the range is significantly decreased.

## 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. …
**Information & Training presentation >>>**