# Statistical Quality Control Questions and Answers – Variable Charts – Control Charts for x̅ and S – 2

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This set of Statistical Quality Control Question Bank focuses on “Variable Charts – Control Charts for x̅ and S – 2”.

1. When the sample size is variable, which one of these can be used to evaluate the value of x double bar?
a) $$\frac{\sum_{i=1}^m n_i \bar{x_i}}{\sum_{i=1}^n n_i}$$
b) $$\frac{\sum_{i=1}^n n_i \bar{x_i}}{\sum_{i=1}^m n_i}$$
c) $$\frac{\sum_{i=1}^n n_i \bar{x_i}}{\sum_{i=1}^n n_i}$$
d) $$\frac{\sum_{i=1}^m n_i \bar{x_i}}{\sum_{i=1}^m n_i}$$

Explanation: When sample size is variable, the formula to evaluate the process mean is given by,
$$\bar{\bar{x}} = \frac{\sum_{i=1}^m n_i \bar{x_i}}{\sum_{i=1}^m n_i}$$

2. Which one of these is correct to evaluate the mean standard deviation of the process samples?
a) $$\bar{s} = \frac{\sum_{i=1}^m (n_i-1) s_i^2}{\sum_{i=1}^m n_i}$$
b) $$\bar{s} = \left[\frac{\sum_{i=1}^m (n_i-1) s_i^2}{\sum_{i=1}^m n_i-m}\right]^2$$
c) $$\bar{s} = \left[\frac{\sum_{i=1}^m (n_i-1) s_i^2}{\sum_{i=1}^m n_i-m}\right]^{1/2}$$
d) $$\bar{s} = \frac{\sum_{i=1}^m (n_i-1) s_i^2}{\sum_{i=1}^m n_i-m}$$

Explanation: The mean of sample standard deviations in the case of variable sample size, is evaluated by following formula,
$$\bar{s} = \left[\frac{\sum_{i=1}^m (n_i-1) s_i^2}{\sum_{i=1}^m n_i-m}\right]^{1/2}$$

3. Which of these is taken as the sample size while estimating process standard deviation, where sample size is variable?
a) ni; Which is the highest among all sample sizes
b) ni; Which is the lowest among all sample sizes
c) ni=5
d) ni; Which is the most frequently occurring among all the sample sizes

Explanation: When there are samples of variable sample sizes and we have to estimate process standard deviation, we use the sample size which is most frequenty occurring among all sample sizes.

4. The control charts based directly on the sample variance are called ____________
a) s Control charts
b) σ2 Control charts
c) s2 Control charts
d) x Charts

Explanation: The variable control charts are based on the range method or standard deviation method, but some practitioners use charts based directly on the sample variance, which are called the s2 control charts.

5. What is the value of the center line of the sample variance control chart?
a) $$\bar{s}^2$$
b) s2
c) $$\bar{s}$$
d) $$\overline{s^2}$$

Explanation: The center line in the case of sample variance control chart or s2 control chart denotes the value of $$\bar{s}^2$$.

6. Which of these is a name of s control chart?
a) s2 Chart
b) Process standard deviation chart
c) σ Chart
d) σ2 Chart

Explanation: The s chart or sample standard deviation chart is also called σ chart by some practitioners. This is based on the control limits set by calculations using the sample standard deviations.

7. X bar chart should be interpreted before s chart if both are indicating out of control situations.
a) True
b) False

Explanation: The base rule of x bar and s chart is that if both charts indicate out-of-control situation then the s chart should be interpreted first as deleting assignable causes in the s chart will automatically delete assignable causes in x bar chart.

8. X bar and S chart are more accurate in predicting out-of-control situations than the x bar and R charts, in the case of high sample size.
a) True
b) False

Explanation: X bar and R charts have a high level of β- risk when the sample size is high. This means they lose their efficiency at high or moderate sample sizes. So x bar and s charts are more accurate than the former ones.

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