Variable control chart questions and solution

1. VARIABLE CONTROL
CHARTS
Prepared by Mahmoud Eltaweel

2. Control
Charts
Variable
Xbar S
N≥ 9
Xbar-R
2≤ 𝑁 ≤ 9
Moving
Range
N=1
Attributes
N= number of
observation in
one sample

3. Xbar- S charts
Data Given Standard Given
S chart
UCL
(upper control limit)
𝑈𝐶𝐿! = 𝐵"
̅𝑆 𝑈𝐶𝐿! = 𝐵# 𝜎$
CL
(center line)
CL= ̅𝑆 =
∑&
'
CL= ̅𝑆
LCL
(lower control limit)
𝐿𝐶𝐿! = 𝐵(
̅𝑆 𝐿𝐶𝐿! = 𝐵) 𝜎$
Xbar Chart
UCL 𝑈𝐶𝐿*+ = )𝑋 + 𝐴(
̅𝑆 𝑈𝐶𝐿*+ = 𝑋$ + 𝐴𝜎$
CL CL= )𝑋 =
∑ *+
'
CL= )𝑋
LCL 𝐿𝐶𝐿*+ = )𝑋 − 𝐴(
̅𝑆 𝐿𝐶𝐿*+ = 𝑋$ − 𝐴𝜎$

4. ■ M= number of samples
■ N= number of observations in one sample
■ 𝜎!=
̅#
$!
■ S stands for standard deviation which can be calcuated by
S=
∑ &"' (& #
)'*
Steps to calcaute S using calcuator
1. Press on mode
2. Choose STAT then press AC
3. Press SHIFT then 1
4. Choose Data
5. Enter the sample data then press AC
6. Press SHIFT then 1 then Var
7. You can find the avg and S
8. Repeart steps from 3 to 7 for each
sample

5. Question 1
■ The following table
presents the inside
diameter measurements
of forged automobile
engine piston rings. Each
sample or subgroup
consists of five piston
rings. Use these data to
construct !𝑿 and s charts.

6. Solution

7. From Table (factors for constracting
variables for control charts) @ n=5
B3= 0
B4= 2.089
A3= 1.427
Solution

9. S chartXbar chart
Solution

10. Question 2
■ Samples of size n = 5 are collected from a process every half
hour. After 50 samples have been collected, we calculate %𝑋
= 20.0 and ̅𝑠 = 1.5. Assume that both charts exhibit control
and that the quality characteristic is normally distributed.
a) Estimate the process standard deviation.
b) Find the control limits on the )𝑋 and 𝑠 charts.
c) If the process mean shifts to 22, what is the probability of
concluding that the process is still in control?

11. Solution
C4 from the table

12. 𝑍 =
𝑈𝐶𝐿 − 𝜇,-.
𝜎
𝑛
𝑍 =
𝐿𝐶𝐿 − 𝜇,-.
𝜎
𝑛
LCL UCL
20 22