# Xbar-R Chart

## What is Xbar-R Chart?

The Xbar-R chart is a statistical control chart used to monitor and control the variability of a process over time. It is used to detect any variability in the process mean and range of data collected over time. The chart consists of two components: the Xbar-chart and R-chart.

The Xbar-chart shows the average value of a sample of data taken from the process for each point in time. It is used to monitor the process mean and detect any shifts in the process mean. The R-chart shows the variation within the sample for each point in time. It is used to monitor the process variation and detect any changes in the process variability.

## When to use Xbar-R Chart?

The Xbar-R chart is used when the subgroup size is greater than one and the sample size is small or medium (less than 10). It is used to monitor the central tendency (mean) and variability (range) of a process over time when normality can be assumed. The Xbar-R chart is helpful in detecting changes in the process mean and variability, which can indicate the need for process improvement. It is commonly used in manufacturing and other industries where quality control is important.

## Guidelines for correct usage of Xbar-R Chart

• Use continuous data for variable control charts, and attribute data for P Chart or U Chart.
• Enter data in time order, with oldest data at the top.
• Collect data at equally spaced time intervals.
• Use rational subgroups, which are small samples of similar items produced under the same conditions.
• Subgroup size should be 8 or fewer for Xbar-S Chart, and I-MR Chart for no subgroups.
• Collect an appropriate amount of data based on subgroup size.
• Control charts can still work with non-normal data if collected in subgroups.
• Avoid correlated data points within subgroups to prevent control limits being too narrow.

## Alternatives: When not to use Xbar-R chart

• For subgroups with 9 or more observations, opt for the Xbar-S Chart, unless there is a consistent source of variation within the subgroups, in which case, use the I-MR-R/S Chart.
• If there are no subgroups, use the I-MR Chart.
• If your data pertains to counts of defectives or defects, employ an attribute control chart like the P Chart or U Chart.

## Example of Xbar-R Chart?

Let us understand how to make Xbar chart and R chart with the help of an example. Suppose an engineer wants to monitor a manufacturing process that produces camshaft for their automotive OEM customer. 5 samples were collected in each subgroup. A subgroup consists of  samples collected within a short period, typically consecutive parts produced from a single production machine. The subgroup data were collected approximately once every 30 minutes.

One of the dimensions of the samples, lets say a diameter is as follows:

1. If the engineer were to calculate the Xbar-R chart manually or using excel, after gathering the data, she needs to find the mean and range in each row or subgroup. Mean can also be written as Xbar and range can also be written as R.
2. The centreline of Xbar chart is the mean Xbar values of all the subgroups above, and we call that X_double_bar.
3. The centreline of R chart is the mean of R values of all subgroups, and we call that Rbar.
4. The upper and lower control limits of Xbar Chart & R chart are calculated using the estimated "within standard deviation" of the process, and an approximation using constants, assuming the data is normally distributed. The formula for calculating the estimated std.dev is a bit involved and will be covered separately.

Alternatively, a software like Zometric Q-Tools can be used to easily create the control charts. The engineer analyses the data with the help of https://qtools.zometric.com/

After using the above mentioned tool, she fetches the useful graphs as follows:

## How to generate Xbar-R Chart?

The guide is as follows:

1. Login in to your Zometric Q-Tools account at: https://qtools.zometric.com/
2. On the home page, one will see Xbar-R Chart under control charts.
3. Click on Xbar-R Chart and will reach the dashboard.
4. Next, update the data manually or can completely copy (Ctrl+C) the data from excel sheet and paste (Ctrl+V) it here.
5. Next, you need to choose Sd estimation method along with the desired Check Rules.
6. Finally, click on calculate at the bottom of the page and you will get desired results.

On the dashboard of Xbar-R chart, the window is separated into two parts.

On the left part, Data Pane is present. In the Data Pane, each row makes one subgroup. Data can be fed manually or the one can completely copy (Ctrl+C) the data from excel sheet and paste (Ctrl+V) it here.

On the right part, there are many options present as follows:

• Process mean: If process mean is provided, this value is considered to be the centerline. If not, Zometric Q-Tools calculates the centerline from the data provided.
• Process sd: If process mean is provided, this value is considered to be the centerline. If not, Zometric Q-Tools calculates the centerline from the data provided.
• Sd estimation method: This leaves the user with two choices for the calculation. Choosing Rbar as the estimation method or pooled standard deviation method changes the result.
• Rbar Sd estimation method: Here the Rbar is calculated by taking an average for all the R of each subgroup. All R’s are of equal weightage.
• Pooled Sd estimation method: Here the Rbar is calculated by taking a weighted average for all the R of each subgroup. All R’s are of different weightage.
• Use unbiasing constants for pooled sd: This option is applicable only when Stdev is estimated using Pooled Stdev method.
• Check Rule 1: 1 point > K Stdev from center line:  If a data point is K standard deviations from the center line, it means that it is K times the standard deviation away from the mean. This is important in statistical process control because it indicates whether a data point is within acceptable limits or whether there may be a problem with the process that needs to be addressed. Typically, data points that are more than three standard deviations from the center line are considered outliers and may require further investigation.
• Check Rule 2: K points in a row on same side of center line: If there are K points in a row on the same side of the center line in a dataset, it suggests that there may be a bias or trend in the data that is causing the values to cluster together. This could be due to a variety of factors, such as measurement error, sampling bias, or a true underlying pattern in the data.
• Check Rule 3: K points in a row, all increasing or all decreasing: If there are K points in a row, it is certain that at least one of two things must be true:
• The points are all increasing (i.e. each point has a greater y-coordinate than the one before it)
• The points are all decreasing (i.e. each point has a smaller y-coordinate than the one before it)
• Check Rule 4: K points in a row, alternating up and down: If the trend is upwards, it indicates that the process is becoming less consistent and more variable over time. This can be caused by factors such as equipment deterioration, operator error or changes in raw material quality. If the trend is downwards, it indicates that the process is becoming more consistent and less variable over time. This could be due to process improvements or tighter control measures being implemented.
• Check Rule 5: K out of K + 1 points > 2 standard deviation from center line (same side):  According to the statement, if K out of K+1 data points fall on the same side of the center line and are more than two standard deviations away from it, it suggests that the process might be out of control, and special causes should be investigated to identify and fix the problem.
• Check Rule 6: K out of K + 1 points > 1 standard deviation from center line (same side): In this statement, K represents the number of consecutive observations that are above the center line (on the same side) and are greater than one standard deviation away from it. This indicates a potential shift in the mean of the process. The K+1 point serves as a reference point to compare the K consecutive observations against.
• Check Rule 7: K points in a row within 1 standard deviation of center line (either side): If K points in a row are within 1 standard deviation of the center line, it suggests that the data points are clustered around the expected value, and there is no significant trend or deviation from the expected pattern.
• Check Rule 8: K points in a row > 1 standard deviation from center line (either side): If K points in a row are more than 1 standard deviation away from the center line, it suggests that there may be a trend or pattern in the data that is moving away from the expected value.

## Xbar-R Chart Formula

In this section we describe how the control limits of Xbar chart and R charts are calculated. To begin with, note the description of the terms used in the calculations that will follow.

Term Description
x_{ij} jth observation in the ith subgroup
n_i number of observations in subgroup i
\sum x sum of all individual observations
\sum n total number of observations
μ process mean
k parameter for Test 1 (The default is 3)
σ process standard deviation
d_2 (.) value of unbiasing constant d2 that corresponds to the value specified in parentheses
d_3 (.) value of unbiasing constant d3 that corresponds to the value specified in parentheses
r_i range for subgroup i
m number of subgroups
\overline x_i mean of subgroup i
µ_ν mean of the subgroup variances
c_4 (.) value of the unbiasing constant c4 that corresponds to the value that is specified in parentheses
c_5 (.) value of the unbiasing constant c5 that corresponds to the value that is specified in parentheses
Γ() gamma function

• ### Formula for Xbar Chart

• Plotted Points: Each plotted point, \overline x_i , represents the mean of the observations for subgroup, i .

\overline x_i = \frac{\sum\limits_{j=1}^{n_i} x_{ij}}{n_i}

• Center Line: The center line represents the process mean (µ).

\overline{\overline{X}} = \frac{∑x}{∑n}

• Control Limits:
• Lower Control Limit: The value of the lower control limit for each subgroup, i , is calculated as follows:

LCL_i= µ- \frac{kσ}{√(n_i )}

• Upper Control Limit: The value of the upper control limit for each subgroup, i , is calculated as follows:

UCL_i= µ+ \frac{kσ}{√(n_i )}

• ### Formula for R Chart

• Plotted Points: Each plotted point,  r_i , represents the range for subgroup  i .
• Center Line: The value of the center line for each subgroup, \overline R_i , is calculated as follows:

\overline R_i= d_2(n_i) × σ

• Control Limit:
• Lower Control Limit: The value of the lower control limit for each subgroup i is equal to the greater of the following:

LCL_i= [d_2(n_i) × σ] - [kσ×d_3(n_i)]

or

LCL_i= 0

• Upper Control Limit: The value of the upper control limit for each subgroup  i  is calculated as follows:

UCL_i= [d_2(n_i) × σ] + [kσ×d_3(n_i)]

• ### Formula for estimation of sigma (within standard deviation)

• Many times people mistakenly assume the σ used in the above calculations is the usual sample or population standard deviation that can be obtained using standard spreadsheet calculations. However, note that its an estimated standard deviation based on the principle that we need to use the standard deviation that is only inherent to the process, and we need to exclude the between subgroup variation. There are two popular methods of estimating the within subgroup variation.
• Using Rbar Method: We use the range of each subgroup, r_i , to calculate  S_r , which is an unbiased estimator of σ:

S_r = \frac{\sum\limits_{i}^{} (\frac{f_i r_i}{d_2 (n_i)})}{\sum\limits_{i}^{} {f_i}}

where

f_i = \frac{[d_2 (n_i)]^2}{[d_3 (n_i)]^2}

When the subgroup size is constant, the formula simplifies to the following:

S_r = \frac{\overline R}{d_2 (n_i)}

where

\overline R = \frac{\sum r_i}{m}

• Using pooled standard deviation method: The pooled standard deviation (Sp) is given by the following formula.

S_p = \sqrt\frac{\sum\limits_{i}^{} \sum\limits_{j}^{} (x_{ij}-\overline x_i)^2 }{\sum\limits_{i}^{} (n_{i}-1)}

When the subgroup size is constant, Sp can also be calculated as follows:

S_p = \sqrt µ_ν

• ### Formulae for unbiasing constants

• The uniting constants used in the above formulas are usually obtained from standard statistical tables. However, the following formula can also be used to calculate the values of the unbiasing constants.
• d_2 () For values of N from 51 to 100, use the following approximation for d2(N):

d_2 (N) = 3.4873 + 0.0250141 × N - 0.00009823 × N^2

• d_3 () and d_4 () For values of N from 26 to 100, use the following approximations for d3(N) and d4(N):

d_3 (N) = 0.80818 - 0.0051871 × N + 0.00005098×N^2 - 0.00000019 × N^3

d_4 (N) = 2.88606 + 0.051313 × N - 0.00049243×N^2 + 0.00000188 × N^3

• c_4 () and c_5 () :

c_4 (N) = {\sqrt \frac {2}{N-1}}\frac{Γ\frac{N}{2}}{Γ\frac{N-1}{2}}

c_5 (N) = \sqrt {1 - c_4(N)^2}