**Subject :**Math -3

**Unit :**Statistical Techniques - II

## Control Charts for Variables

**Control Charts for Variables:**

Control charts for variables monitor characteristics that can be measured and have a continuous scale, such as height, weight, volume, or width. When an item is inspected, the variable being monitored is measured and recorded. For example, if we were producing candles, height might be an important variable.We could take samples of candles and measure their heights. Two of the most commonly used control charts for variables monitor both the central tendency of the data (the mean) and the variability of the data (either the standard deviation or the range). Note that each chart monitors a different type of information. When observed values go outside the control limits, the process is assumed not to be in control. Production is stopped, and employees attempt to identify the cause of the problem and correct it.

**Mean (X-Bar) Charts:**

A mean control chart is often referred to as an x-bar chart. It is used to monitor changes in the mean of a process. To construct a mean chart we first need to construct the center line of the chart. To do this we take multiple samples and compute their means. Usually these samples are small, with about four or five observations. Each sample has its own mean, x¯_{i }. The center line of the chart is then computed as the mean of all N sample means, where N is the number of samples:

To construct the upper and lower control limits of the chart, we use the following formulas:

Upper control limit (UCL)

Lower control limit (LCL)

σ ¯ x = standard deviation of the distribution of sample means, computed as

σp = population (process) standard deviation

n = sample size (number of observations per sample)

Another way to construct the control limits is to use the sample range as an estimate of the variability of the process. Remember that the range is simply the difference between the largest and smallest values in the sample. The spread of the range can tell us about the variability of the data. In this case control limits would be constructed as follows:

Upper control limit (UCL)

Lower control limit (LCL)

**Range (R) Charts:
**

Range (R) charts are another type of control chart for variables. Whereas x-bar charts measure shift in the central tendency of the process, range charts monitor the dispersion or variability of the process. The method for developing and using R-charts is the same as that for x-bar charts. The center line of the control chart is the average range, and the upper and lower control limits are computed as follows:

CL = ¯R

UCL = D_{4} ¯R

LCL = D_{3} ¯R