# Statistical Quality Control Questions and Answers – Variable Charts – Control Charts for x̅ and S – 1

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This set of Statistical Quality Control Multiple Choice Questions & Answers (MCQs) focuses on “Variable Charts – Control Charts for x̅ and S – 1”.

1. What is the estimator of standard deviation in the x bar and R charts?
a) Mean of one sample
b) Mean of whole process
c) Range
d) Process capability ratio

Explanation: In x bar and R charts, process standard deviation is estimated indirectly through the use of the range R. x bar is used as an estimator of mean.

2. What does “s” denote in x bar and s charts?
a) Sample
b) Sample standard deviation
c) Process standard deviation
d) Statistics

Explanation: Process standard deviation in the x bar and s charts, is estimated directly instead of indirectly through the use of Range as in x bar and R charts. Here “s” denotes the sample standard deviation.

3. What is an unbiased estimator of unknown variance of a probability distribution?
a) Sample mean
b) Sample standard deviation
c) Sample variance
d) Sample range

Explanation: If σ2 is the unknown variance of a probability distribution, then an unbiased estimator must be used to estimate σ2. In this case, sample variance is used as the required estimator.

4. What is the standard formula of sample variance?
a) $$\frac{\sum_{i=1}^n (x_i-\bar{x})^{1/2}}{n-1}$$
b) $$\frac{\sum_{i=1}^n (x_i-\bar{x})^{2}}{n-1}$$
c) $$[\frac{\sum_{i=1}^n (x_i-\bar{x})^{2}}{n-1}]^{1/2}$$
d) $$\frac{\sum_{i=1}^n (x_i-\bar{x})^{2}}{n}$$

Explanation: The sample variance is given by the following formula.
s2 = $$\frac{\sum_{i=1}^n (x_i-\bar{x})^{1/2}}{n-1}$$.

5. Which of these formulas gives the exact equation for the UCL of s chart with a std. value for σ given?
a) B6 σ
b) B5 σ
c) c4 σ
d) c3 σ

Explanation: The UCL parameter of the s chart with a std. value for σ given, is expressed by
UCL=B6 σ.

6. The center line of the s chart with a standard value for σ given, denotes the value of _____
a) B6 σ
b) c4 σ
c) B5 σ
d) c5 σ

Explanation: The center line of the s chart with a standard value for σ given, denotes the value equal to,
CL=B6 σ

7. If the sample standard deviations for a process are 1.567, 1.429, 1.323, 1.525, 1.989, 1.457, what will be the mean standard deviation?
a) 1.548
b) 1.858
c) 1.327
d) 1.967

Explanation: The mean standard deviation of the sample standard deviations is given by,
$$\bar{s} = \frac{1}{m} \sum_{i=1}^m s_i$$
Where si denotes the standard deviation of ith sample. Calculating the mean using the above formaula gives, s=1.548.

8. What is the value of B5 in the terms of c4?
a) $$c_4-3\sqrt{(1-c_4^2)}$$
b) $$c_4+3\sqrt{(1+c_4^2)}$$
c) $$c_4+3\sqrt{(1-c_4^2)}$$
d) $$c_4-3\sqrt{(1+c_4^2)}$$

Explanation: The value of B5 in the terms of c4 is given by,
B5 = $$c_4-3\sqrt{(1-c_4^2)}$$

9. The center line of the s chart denotes ____
a) Standard deviation of the process
b) Mean of m number of standard deviations, where m is the number of samples
c) c4 s
d) B5 s

Explanation: The center line of the s chart denotes the mean of m number of standard deviations, where m is the number of samples. This is the desired value of the sample standard deviation for the process to be in control.

10. What is the value of LCL for the s chart when the standard value for σ is not given?
a) B5 s
b) B4 s
c) B6 s
d) B3 s

Explanation: The LCL of the s chart gives the value equal to, B3 s when the standard value for σ is not given. This is the lowest the value of s can be, for the process to be in-control.

11. What is the value of B3 in the terms of c4?
a) $$c_4-3\sqrt{(1-c_4^2)}$$
b) $$c_4+3\sqrt{(1+c_4^2)}$$
c) $$1-\frac{3}{c_4} \sqrt{(1-c_4^2)}$$
d) $$1-\frac{c_4}{3\sqrt{(1-c_4^2)}}$$

Explanation: The value of B3 in the terms of c4 is given by,
B3 = $$1-\frac{3}{c_4} \sqrt{(1-c_4^2)}$$

12. What is the formula for UCL for x bar chart when s is known?
a) $$UCL = \bar{\bar{x}} + A_3 \bar{s}$$
b) $$UCL = \bar{\bar{x}} – A_2 \bar{s}$$
c) $$UCL = \bar{\bar{x}} – A_3 \bar{s}$$
d) $$UCL = \bar{\bar{x}} + A_2 \bar{s}$$

Explanation: The formula of UCL for x bar and s chart construction when s is known, is given by
$$UCL = \bar{\bar{x}} + A_3 \bar{s}$$

13. For mean of all sample standard deviations=0.0094 and the sample size= 5, what will be the estimate of process standard deviation?
a) 100
b) 0.01
c) 0.0094
d) 94

Explanation: We know that estimate of the process standard deviation,
$$\hat{\sigma} = \frac{\bar{s}}{c_4}$$
Here for sample size=5, c4=0.94, and s = 0.0094, we get σ = 0.01.

14. Process standard deviation is the mean of all sample standard deviations.
a) True
b) False

Explanation: It is not necessary that process standard deviation is the mean of all sample deviations. This is because there is some inherent and natural variability in the process. This may or may not appear in every sample.

15. X bar and R charts are highly favorable when the sample size is high.
a) True
b) False

Explanation: X bar and R charts are not used for high sample sizes- say, n>10 or 12. This is because the range method for estimating σ loses its efficiency for moderate to large samples.

16. Which of this is a situation when x bar and s charts should be utilized instead of x bar and R charts?
a) When sample size is constant
b) When sample standard deviation is less than 1
c) When sample range is more than 1
d) When sample size is variable 