X bar R Chart explained with solved example.

Hi Readers! Today we will discuss about the Mean and Range chart.  It is also called X̅ – R chart. This type of chart is used for variable data or continuous data. Sample size should lie between 2<n<10.

Why is X bar R chart used for?

• X bar chart monitors the mean between sample values.
• R chart monitors the variation within samples and R chart is analysed before X-bar chart to determine out of control situations, as R chart reflects process variability, which should be brought into control.
• If R chart shows out of control range, then the X-bar chart is meaningless.

Steps to draw X̅ – R chart in Excel sheet.

Step 1. Calculate the sample mean and range for each sample

 Sub group Mean :

         X̅=  (X₁ + X₂ + X₃ + …….+X_n ) / n  

  n = Number of samples in a subgroup.      

Step 2.

Sub group Range :

  Ri = Xmax – Xmin      ( With in each group)   & Average range ( R bar)

 k =Number of subgroup used to determine the Average Range  

Step 3. Grand Average

Let us understand with an example.  Suppose we have taken sample size of 5 nos of pencils and measured the length of the sample.  USL & LSL of sample 6   +/- 2.  We did it for 25 Nos subgroup.  Then we have taken the sum of individual subgroup and average of subgroup. Additional infomation if Sample size n= 5 then A₂ = 0.577 , D₄ = 2.114 , D₃ = 0

Step – 1.

Sub group	X1	X2	X3	X4	X5	sum of individual	Average (total/5)
1	5.4	6.2	7.5	7.1	4.3	30.5	6.1
2	4.8	7.3	4.7	5.5	4.7	27	5.4
3	7.5	7.5	4.5	4.8	5.3	29.6	5.92
4	4.7	4.8	3.9	7.5	6.6	27.5	5.5
5	4.5	7.5	6.2	6.5	3.9	28.6	5.72
6	3.9	6.5	7.1	7.1	6.5	31.1	6.22
7	6.5	5.8	5.5	7.5	7.1	32.4	6.48
8	5.8	6.2	4.8	4.7	5.5	27	5.4
9	6.2	7.1	4.3	4.5	4.8	26.9	5.38
10	7.1	5.5	4.7	3.9	7.5	28.7	5.74
11	5.5	4.8	5.3	3.9	6.5	26	5.2
12	6	7.5	6.6	6.5	5.8	32.4	6.48
13	6.2	6.5	3.9	5.8	6.2	28.6	5.72
14	4.3	7.1	6.5	6.2	7.1	31.2	6.24
15	4.7	5.5	5.8	6	7.5	29.5	5.9
16	5.3	6	6.2	6.2	6.5	30.2	6.04
17	6.6	6.6	6	3.9	7.5	30.6	6.12
18	3.9	3.9	6.2	3.9	6.5	24.4	4.88
19	6.5	6.5	4.3	6.5	5.8	29.6	5.92
20	5.8	4.8	4.7	7.5	6.6	29.4	5.88
21	6.2	7.5	3.9	6.5	3.9	28	5.6
22	5.8	6.5	6.5	7.1	6.5	32.4	6.48
23	6.2	6.2	5.8	5.5	5.8	29.5	5.9
24	7.3	4.3	6.5	6	6.2	30.3	6.06
25	7.5	4.7	5.8	6.5	6.5	31	6.2

Step – 2 Sub group Range :

  Ri = Xmax – Xmin      ( With in each group)    &  Range Average

For Sub group 1 ,   Xmax = 7.5  ,   Xmin = 4.3

Range = 7.5 – 4.3 = 3.2

 Ranges of all sub groups are calculated in the same way. 

3.2, 2.6, 3, 3.6 ,3.6 ,3.2, 2, 1.5, 2.8, 3.6, 2.6, 1.7, 2.6, 2.8, 2.8, 1.2, 3.6, 2.6, 2.2 ,2.8, 3.6, 1.3, 0.7, 3, 2.8 (R1, R2, ..R25)

Step – 3 (Grand Average)

(6.10+5.40+5.92+5.50+5.50+5.72+6.22+6.48+5.40+5.38+5.74+5.20+6.48+5.72+6.24+5.90+6.04+6.12+4.88+5.92+5.88+5.60+6.48+5.90+6.06+6.20)/ 25 = 5.86

Interpretation of X bar R chart

From Above chart it is clearly seen that average of subgroup data is within the upper & lower control limit. No data point is outside of control limit. If any data falls out side of control limit then in that case revised control limit need to be defined.

Range of subgroup is also within upper & lower control limit.