Question

The no. of defects in 20 pieces of cloth each of 100 meter length is given below: 2, 1, 3, 3, 1, 6, 4, 3, 7, 10, 2, 2, 6, 4, 3, 2, 1, 5, 6, 4. Draw the appropriate control chart and interpret the result.

Answer 1

Application of statistics for management.    see diagram.

A Control chart is made for decision making by management reviewing the value of statistic if it is outside the threshold limits.

statistic here is number of defects: x.    we have N = 20

Average of x = μ = sum/N = 75/20 = 3.75
Standard deviation of x = σ = 2.657
Standard error = σ_ε = σ/√20 = 0.5916

Warning levels for x :
        x = μ + 2 * σ_ε   = 4.9332       and  μ - 2 * σ_ε  = 2.5668

A piece is accepted with a warning if
      μ - 3 * σ_ε < x < μ - 2 * σ_ε    or    μ + 2 σ_ε  < x < μ + 3 σ_ε

Accepted range :
          μ - 3 * σ_ε  <  x  <  μ + 3 * σ_ε        
           1.9752 < x < 5.5248
                  
Rejection threshold levels:  control limits.
      rejected if    :    x > μ + 3 σ_ε          or  x < μ - 3 σ_ε

So the pieces  with number of defects :  1,  6, 7, 10, will be sent for quality assurance and process verification, the cause analysis. 

Normally, for a good process, 99.7% of the values should be between the control limits.  Here we have 8 values of 20 being outside the control range.

The manufacturing process needs to be reviewed.