# Statistical Quality Control Questions and Answers – Variable Charts – Shewhart Control Chart for Individual Measurements

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This set of Statistical Quality Control Multiple Choice Questions & Answers (MCQs) focuses on “Variable Charts – Shewhart Control Chart for Individual Measurements”.

1. The Shewhart control charts for sample size n=1 are called _________
a) Single Sample control charts
b) Stationary control charts
c) Control charts for zero variance
d) Control charts for individual measurements

Explanation: The Shewhart control charts for sample size n=1 are called, Control charts for individual measurements. It has no basis of rational sub grouping in this case.

2. In the case of automatic manufacturing, the sample size is ___________
a) 1
b) 2
c) 4
d) Greater than 5

Explanation: In the case of automated inspection and measurement technology, every unit manufactured is analyzed and so there is no basis of rational sub grouping. So the sample size here, is 1.

3. In the case of individuals control charts, which of these is used?
a) Relative range
b) Process standard deviation
c) Mean of the highest observations
d) Moving range

Explanation: In the case of individual measurements, we use the moving range of two successive observations as the basis of estimating the process variability.

4. MR (Moving Range) is defined as
a) The difference of highest observation and the lowest observation
b) The difference between any two successive observations
c) The difference between the highest observation and mean of observations
d) The difference between the lowest observation and mean of the observations

Explanation: The difference between any two observations in the data is called the Moving range or MR of the two observations. This helps in estimation of the process variability in the case of the individuals control chart.

5. What is the correct expression for the UCL for an Individuals Control Chart?
a) UCL=$$\bar{x}+\frac{3\overline{MR}}{d_2}$$
b) UCL=$$\bar{x}+\frac{2\overline{MR}}{d_2}$$
c) UCL=$$\bar{x}+\frac{3\overline{MR}}{d_3}$$
d) UCL=$$\bar{x}+\frac{\overline{MR}}{d_2}$$

Explanation: The UCL for any individuals control chart is given by the following expression,
UCL=$$\bar{x}+\frac{3\overline{MR}}{d_2}$$

6. The center line for any individuals control chart represents the value equal to __________
a) The process mean
b) The moving range
c) The mean of moving ranges
d) The process standard deviation

Explanation: The center line of any individuals control chart represents the value equal to x or the process mean for that particular process.

7. The value of ARL for any x bar control chart having 3-σ limits is __________
a) 482
b) 310
c) 370
d) 270

Explanation: For any shewhart x bar control chart having 3 – sigma limits, we have p=0.0027. The average run length is calculated by formula,
$$ARL=\frac{1}{p}=\frac{1}{0.0027}=370$$

8. The average number of points that must be plotted before an out-of-control point is plotted on an Individuals control chart?
a) ARL
b) ATS
c) MTBF
d) Hazard rate

Explanation: The average number of points that must be plotted before a point indicating an out-of-control situation is plotted, is called the Average run length of any control chart.

9. The ARL for any individuals control chart with the 1 sigma limits, and the β=0.972, is ____
a) 2
b) 6.3
c) 43.96
d) 13

Explanation: We know that for any individuals control chart,
ARL=$$\frac{1}{1-\beta}$$
Putting the values of β=0.9772, we get ARL=43.96.

10. Which of this stands correct for the relation between the ARL for a shewhart control chart and the individual control chart ARL?
a) ARLindividual control chart=1/ ARLshewhart control chart
b) ARLindividual control chart= ARLshewhart control chart
c) ARLindividual control chart≪ARLshewhart control chart
d) ARLindividual control chart≫ ARLshewhart control chart

Explanation: The Average run length for any individuals control chart with 3 sigma limits is considerably lower than the value of Average run length for any Shewhart control chart.

11. What is the correct expression for average moving range for any individuals control chart?
a) $$\overline{MR} = \sum_{i=2}^m \frac{MR_i}{m-1}$$
b) $$\overline{MR} = \sum_{i=1}^m \frac{MR_i}{m}$$
c) $$\overline{MR} = \sum_{i=2}^{m-1} \frac{MR_i}{m-1}$$
d) $$\overline{MR} = \sum_{i=1}^m \frac{MR_i}{m-1}$$

Explanation: The average moving range for any individuals control chart is given by,
$$\overline{MR} = \sum_{i=2}^m \frac{MR_i}{m-1}$$

12. The estimator of the process standard deviation for any individuals control chart is written as?
a) $$\frac{\overline{MR}}{2}$$
b) $$\overline{MR}/0.8865$$
c) $$\overline{MR}/1.128$$
d) $$1.128\overline{MR}$$

Explanation: The estimation of the process standard deviation for any individuals control chart is written as,
$$\widehat{σ_1}=\overline{MR}/d_2$$
For individuals control chart, n=2. For n=2, we have d2=1.128. So we get,
$$\widehat{σ_1}=\overline{MR}/1.128$$.

13. The estimation of the process standard deviation for any individuals control chart is expressed as ____ in the terms of sample standard deviation.
a) $$\widehat{σ_2}=\frac{s}{c_4}$$
b) $$\widehat{σ_2}=\frac{s}{c_1}$$
c) $$\widehat{σ_2}=\frac{s}{c_2}$$
d) $$\widehat{σ_2}=\frac{s}{c_3}$$

Explanation: The estimation of the process standard deviation for any individuals control chart is expressed as $$\widehat{σ_2}=\frac{s}{c_4}$$ in the terms of sample standard deviation, s.

14. Moving range can only be calculated by the two successive observations.
a) True
b) False

Explanation: The moving range is mostly called as MR of span two because it uses 2 successive observations difference for MR calculation. The span can be increased, MR can be calculated by more than 2 observations.

15. “$$\widehat{σ_3} = 1.047\overline{MR}$$; where $$\overline{MR}$$ is the median of the span-two moving ranges.”
a) True
b) False

$$\widehat{σ_3} = 1.047\overline{MR}$$