# CUSUM Chart

### Cumulative Sum Control Chart

CUSUM Charts is a type of a Moving Average chart that is typically used when plotting variables data to detect small changes over a small period of time. The underlying distribution of the data is assumed normal.

They detect shifts of less than or equal to 2 sigma faster than Xbar and I-MR charts but are slower in detecting larger abrupt shifts in the process mean than Xbar and I-MR charts.

The moving average smooths the variation of time therefore should not be used when looking for a point that is outside of the process control limits.

There are several variations of moving average, also called a "rolling average" or "running average". There are other charting types listed below in addition to EMWA that use a central tendency to highlight loner term fluctuations and trends, rather than short term. In other words, they give less and less weight to earlier data as more data is gathered.

• Simple moving average
• Weighted moving average
• Moving median

These calculations analyze data points by creating a series of averages of different subsets of the full data set. A moving average chart is commonly used with time series data to smooth out short-term fluctuations and help identify longer-term trends or cycles.

The data must be obtained and plotted in sequential order. Since the data is smoothed it is used to predict performance in the next period of change or instability.

Most statistical software programs the capability and may offer the option to enter various "memory" and weight values. The most recent data point is given the most weight and as time progresses the weight of the older points decreases.

CUSUM charts use equals weights for previous data points. As opposed to EMWA charts which the weights of the older points decrease exponentially with time.

These charts are applicable when a I-MR or X-bar & R control chart appears out of control due to wear as would be the case on perishable tooling or dies.

### Assumptions

This chart plots variable data and assumes a normal distribution.

Why not just use an I-MR or X-bar & R chart?

The I-MR and X-bar & R chart are used under the assumption that the mean is constant and observations are independent.

Tools and dies are going to wear and shifts in the performance are expected and this may be common cause variation in reality but showing up as special cause.

When the tool or die is replaced or adjusted, the control chart would exhibit the same pattern as before. This is case where there is correlation between consecutive points and the assumption of independent measurements is most likely violated.

In this case, the CUSUM (or EWMA) chart could show the pattern and make replacement schedules more predictable before there is a failure.

The centerline of the CUSUM chart is the targeted value for the particular characteristic being measured over time. The x-axis are sample measurements measured sequentially so the chart represents a history of readings (similar to EWMA chart).

### Control Limits

The control limits can be determined for CUSUM charts the process in not discussed here.

The CUSUM chart relies on the specification of a target value (which can be directly stated or calculated) and a known estimate of standard deviation. CUSUM charts are best applied only after the process has been proven to be in control.

The CUSUM chart typically signals an out-of-control process by an upward or downward drift of the cumulative sum until it crosses the control limit boundary. An assignable cause is suspected whenever the CUSUM chart indicates an out-of-control process.

### Applications

This chart (or other moving averages) is frequently used in stock modeling software packages for analysts trying to predict the next day performance based on the last couple weeks or months of performance.

Stock price values are most often not totally independent observations as each day is based on a previous performance and outside factors and are often correlated between successive dates.

## EWMA and CUSUM

Both accumulate information from successive readings and signal a change when a shift occurs, even if the change is relatively small so that a Shewhart Xbar or I-MR chart fails to detect it or fails to detect change as fast as a EMWA or CUSUM would detect the change.

Both types of charts are generally equally accepted options; pick one and move forward.

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