Calculation formulas

This document contains information about the statistical formulas used in KensoBI panels and data sources.

Table of Contents

1. Sample Mean Formula
2. Average Range Formula
3. Standard Deviation Formulas
4. Mean of Means Formula
5. Capability Indices Formulas
6. Control Limits Calculation
7. Shewhart individuals control chart

Sample Mean Formula

The sample mean ($\bar{X}$) is a measure of central tendency, representing the average value of a set of sample data.

Formula

The formula for the sample mean is given by:

$\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}$

Where:

• $\bar{X}$ is the sample mean,
• $n$ is the number of data points in the sample,
• $X_i$ represents each individual data point.

In this formula, $\sum$ represents the sum, and $\bar{X}$ denotes the mean.

Average Range Formula

The average range ($\bar{R}$) in statistical process control is calculated as the average of the individual ranges. The range is the difference between the maximum and minimum values in a sample.

Formula

The formula for average range is given by:

$\bar{R} = \frac{R_1 + R_2 + \ldots + R_k}{k}$

Where:

• $\bar{R}$ is the average range,
• $R_1, R_2, \ldots, R_k$ are the individual ranges in each of the $k$ samples, and
• $k$ is the number of samples.

This formula represents the central tendency of the range values and is commonly used in control chart calculations, such as in the context of an R-chart in statistical process control.

Standard Deviation Formulas

The standard deviation ($\sigma$) is a measure of the amount of variation or dispersion in a set of values.

Population Standard Deviation Formula

The formula for the population standard deviation is given by:

$\sigma = \sqrt{\frac{\sum_{i=1}^{N}(X_i - \mu)^2}{N}}$

Where:

• $\sigma$ is the population standard deviation,
• $N$ is the number of data points in the population,
• $X_i$ represents each individual data point,
• $\mu$ is the mean of the population.

Sample Standard Deviation Formula

The formula for the sample standard deviation is given by:

$s = \sqrt{\frac{\sum_{i=1}^{n}(X_i - \bar{X})^2}{n-1}}$

Where:

• $s$ is the sample standard deviation,
• $n$ is the number of data points in the sample,
• $X_i$ represents each individual data point,
• $\bar{X}$ is the mean of the sample.

In both formulas, $\sqrt{}$ denotes the square root, $\sum$ represents the sum, and $\bar{X}$ denotes the mean.

Sample Range Formula

The sample range ($R$) is a measure of the spread or dispersion of a set of sample data.

Formula

The formula for the sample range is given by:

$R = X_{\text{max}} - X_{\text{min}}$

Where:

• $R$ is the sample range,
• $X_{\text{max}}$ is the maximum value in the sample,
• $X_{\text{min}}$ is the minimum value in the sample.

In this formula, $X_{\text{max}}$ represents the maximum value, and $X_{\text{min}}$ represents the minimum value in the sample.

Mean of Means Formula

The mean of means ($\bar{\bar{X}}$) is a measure that represents the average of sample means across multiple groups or samples.

Formula

The formula for the mean of means is given by:

$\bar{\bar{X}} = \frac{\sum_{i=1}^{k} \bar{X}_i}{k}$

Where:

• $\bar{\bar{X}}$ is the mean of means,
• $k$ is the number of groups or samples,
• $\bar{X}_i$ represents the mean of the $i^{th}$ group or sample.

In this formula, $\sum$ represents the sum, and $\bar{\bar{X}}$ denotes the mean of means.

Capability Indices Formulas

Capability indices are statistical measures used to assess the ability of a process to meet specifications.

Cp Formula

The Cp (Process Capability) index is calculated as:

$Cp = \frac{{\text{{USL}} - \text{{LSL}}}}{{6 \times \sigma}}$

Pp Formula

The Pp (Potential Process Capability) index is calculated as:

$Pp = \frac{{\text{(nominal + USL)} - \text{(nominal + LSL)}}}{{6 \times \sigma_x}}$

Cpk Formula

The Cpk (Process Capability Index) index is calculated as:

$Cpk = \min\left(\frac{\text{USL} - \bar{\bar{x}}}{{3 \times \sigma}}, \frac{\bar{\bar{x}} - \text{LSL}}{{3 \times \sigma}}\right)$

Ppk Formula

The Ppk (Potential Process Capability Index) index is calculated as:

$Ppk = \min\left(\frac{\text{(nominal + USL)} - \bar{\bar{x}}}{{3 \times \sigma_x}}, \frac{\bar{\bar{x}} - \text{(nominal + LSL)}}{{3 \times \sigma_x}}\right)$

Control Limits Calculation

It's important to note that X-bar charts can have up to four control limits, depending on the type of chart and the desired analysis.

X-bar charts

The control limits for X-bar charts are calculated as follows:

$UCL_{Rbar} = \bar{x} + A_2\bar{R}$

$UCL_{Sbar} = \bar{x} + A_3\bar{S}$

$LCL_{Rbar} = \bar{x} - A_2\bar{R}$

$LCL_{Sbar} = \bar{x} - A_3\bar{S}$

R-bar charts

For R-bar charts, the control limits are calculated as follows:

$UCL_R = D_4\bar{R}$

$LCL_R = D_3\bar{R}$

S-bar charts

For S-bar charts, the control limits are calculated as follows:

$UCL_S = B_4\bar{S}$

$LCL_S = B_3\bar{S}$

Shewhart individuals control chart

In statistical quality control, the individual/moving-range chart is a type of control chart used to monitor variables data from a business or industrial process for which it is impractical to use rational subgroups.

Process variation chart

Chart type: Moving-range chart

Calculation of moving range

The difference between data point, $x_i$ and its predecessor, $x_{x-1}$, is calculated as

$MR_i = \vert{x_i - x_{i-1}}\vert$.

For $m$ individual values, there are $m-1$ ranges. Next, the arithmetic mean (Center line) of these values is calculated as

$\overline{MR}=\frac{\sum_{i=2}^{m} MR_i}{m-1}$.

Calculation of moving range control limit

The upper control limit for the range (or upper range limit) is calculated by multiplying the average of the moving range by $D_4$ for n = 2;

$UCL_r=D_4*\overline {MR}$

The lover control limit for the range is calculated by multiplying the average of the moving range by $D_3$ for $n = 2$;

$LCL_r=D_3*\overline {MR}$

Process mean chart

Chart type: Mean chart

Calculation of individuals control limits

First, the average (Center line) of the individual values is calculated:

$\overline {x}=\frac {\sum _{i=1}^{m}{x_i}}{m}$

Next, the upper control limit ($UCL$) and lower control limit ($LCL$) for the individual values (or upper and lower natural process limits) are calculated:

$UCL={\overline {x}}+3 \frac{{\overline {MR}}}{d_2}$.

$LCL={\overline {x}}-3 \frac{{\overline {MR}}}{d_2}$.

where $d_2$ is constant for subgroup size $n=2$

Constants

The values $A2, A3, D3, d2, D4, B3, B4$ are constants configured for calculations in the SPC panel. The values are also available as an SQL script.