application methods of statistics in business

1. Presentation on topic :-Control charts and
explain control charts for variables
Presented by :-
Sangeeta Saini
M.Com
(final)

2. Contents
Standard quality control.
Control charts.
Types of control charts.
Control chart for variables.
Mean chart( -chart)
Range chart(R-chart)
Standard deviation chart(σ-
chart)
Conclusion .
X

3. Statistical quality control
Statistical quality
control refers to
the use of
statistical
techniques in
controlling the
quality of

4. Control charts
Control charts are
graphic devices
developed by Walter
A. Shewhart for
detecting unnatural
pattern of variation
in a production

5. Control chart consists of three horizontal lines
1. Central line :- it is the middle line of chart . It indicates
the grand average of the measurements of the samples.
It is generally drawn as bold line.
2. Upper control limit (UCL):- It is usually obtained by
adding 3 sigma (3σ) from the process average. It is
denoted by mean + 3σ. It is generally drawn as dotted
line.
3. lower control limit (UCL):- It is usually obtained by
subtracting 3 sigma (3σ) from the process average. It is
denoted by mean -3σ. It is generally drawn as dotted line.

6. X

7. Control charts for variables
X - charts
These charts are used when the quality or
characteristics of a product is capable of being
measured quantitatively such as diameter of a
screw, tensile strength of steel pipe etc. Such
charts are of three types:-
This chart is
constructed for
controlling the
variations in the
average quality
standards of the
products in a production
process.

8. Procedure for construction of - chart X
Step 1. Compute the mean of each sample i.e.,
Step 2. Compute the mean of the sample means by dividing the sum
of sample means by number of samples i.e.,
Step 3. determine the control limits by using the formula .
(a)On the basis of standard deviation of population (σ)
UCL = + 3σ÷√n LCL = - 3σ÷√n
(b) On the basis of quality control factors A2 and
XX
R

9. Step 4. construct the mean chart by
plotting the sample number on x – axis
and sample mean , UCL,LCL and central
line on y - axis
Step 5. Interpret the mean chart . If
all sample means falls within control
limits , the production process is in a
state control otherwise it is beyond
control.

10. R- Chart
The R – Chart is constructed for controlling the
variation in the variability of the quality standard of
the product in the production process.
Procedure for construction of R – chart
Step 1. : compute the range (R) of each sample using the
formula :-
R= L-S
L= largest value
S= smallest value
Step 2. : compute the mean of ranges by dividing the sum
of samples range by the number of samples i.e.,
=R1+R2+R3+……..+RkR
K

11. Step 3. : find the UCL and LCL by using following formula .
(a)On the basis of quality control factors D3 and D4.
(b) On the basis of quality control factors D1 and D2 and
σ.
UCLr = D2 σ LCLr =D1σ
Step 4. construct the R- chart by plotting
the sample number on x – axis and sample
ranges, UCL,LCL and central line on y - axis
Step 5. Interpret the R- chart . If all
sample ranges falls within control limits ,
the production process is in a state control
otherwise it is beyond control.

12. Mean chart and range chart
Example :- construct mean chart and range chart for the following data of 5 samples with each
set of 5 items
Sample
no.
1 20 15 10 11 14
2 12 18 10 8 22
3 21 19 17 10 13
4 15 12 19 14 20
5 20 19 26 12 23
weights
Conversion factors for n= 5, A2= 0.577, D3=0 , D4= 2.115

13. Solution :
Construction of mean chart
Sample
no.
Total
weights
( )
=
/5
Range
R=L-S
1 20 15 10 11 14 70 14 10
2 12 18 10 8 22 70 14 14
3 21 19 17 10 13 80 16 11
4 15 12 19 14 20 80 16 8
5 20 19 26 12 23 100 20 14
K=5 =80 =57
 X
X X
X R
Weights of items in each sample
= /K = 80 /5 =16X X R = / K = 57/5 = 11.4R

14. Mean chart
X = 16 (central line )
Control limits
UCL = 16+ 0.577× 11.4 = 22.577
LCL = 16 – 0.577 × 11.4 = 9.423
0
5
10
15
20
25
1 2 3 4 5 6 7 8
Samplemean
Sample number
upper control
limit
central line
lower control
limit
As all the mean
values fall within
the control
limits, so mean-
chart shows
that the given
process is in
statistical

15. Range chart
0
5
10
15
20
25
30
1 2 3 4 5 6 7 8
Samplerange
Sample number
upper control
limit
central line
lower control
limit
R = 11.4 (central line) Control limits
UCL = 2.115 × 11.4
=24.09
LCL= 0 × 11.4
=0
As all the range
points fall within
the control
limits, so R-
chart shows that
the given

16. Standard deviation chart (σ- chart)
This chart is constructed to get a better
picture of the variations in the quality standard
in the process.
Procedure of construction of σ - chart
Step 1. : find the standard deviation (S.D.) of each
sample, if not given.
Step 2. : compute the mean of S.D. by using the
formula :-
= S1+S2+S3+………..Sk/K
S

17. Step 3. : find the UCL and LCL by using following formula:-
(a)On the basis of quality control factors B1 and B2 and
population standard deviation.
UCL = B2.σ LCL = B1.σ
(b) On the basis of quality control factors B3 and B4 and
estimated population standard deviation.
UCL= B4.S LCL=B3.S
Step 4. construct the σ- chart by plotting the sample
number on x – axis and sample S.D., UCL,LCL and
central line on y - axis
Step 5. Interpret the σ- chart . If all sample S.D.
falls within control limits , the production process is in
a state control otherwise it is beyond control.

18. σ – chart
Example : Quality control is maintained in a factory
with the help of standard deviation chart. 10 items are
chosen in every sample. 18 samples in all were chosen
whose ∑S was 8.28. Determine the three sigma limits
of σ- chart. You may use the following factors :-
n=18 ,B3= 0.28 ,B4=1.72
Solution :
Given :- n=18, ∑S= 8.28
S = ∑S / n
= 8.28/18 = 0.46
σ- chart.
control limits
S= 0.46
UCL= B4S = 1.72 × 0.46 = 0.79
LCL=B3S = 0.28 × 0.46 = 0.13

19. 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 2 3 4 5 6 7 8
SampleS.D.
Sample number
UCL
CL
LCL
σ – chart
UCL= 0.79
CL =0.46
LCL = 0.13