Vijay Kumar Tomar

The Levene’s test is generally used to test for equality of variance in a dataset. It is used to determine if two or more samples have equal variances. If the results of the test indicate that the samples do not have equal variances, then it means that one sample has different variance than other samples. An advantage of Levene’s test is, it is highly stable for the data set which is not normally distributed.
Null Hypothesis: - Data Groups have equal variances.
Alternate Hypothesis: - Data Groups have different variances.
If the p-value for the Levene’s test is greater than .05, then the variances are not significantly different from each other and assumption of equal variance is met however If the p-value for the Levene's test is less than .05, then variances for one or more sample data set is not equal.
Difference Between the Levene’s test and Bartlett's Test: -
Both tests are used to test the assumptions of variance equality. However, the main difference is Bartlett test requires data of each group to be normally distributed and Levene’s test to be used when data is not normally distributed. For Normality check Anderson Darling can be performed.
Example: -
Data of cost of tickets sold in thousands in as how for a month are tabulated for five different competent Circus groups.
The P value for Levene’s test and bartlett test are highly different as Data is not normally distributed and Levene’s test is more stable for non-normal distribution.
Gem
Joyride
Starlite
Fantasy
Fun
39.3
23.3
7.3
10
36
42
60
11.3
180
40.7
40.7
150
18.7
36
40
43.3
36.7
30.7
120
46.7
44
70
38
48
56
47.3
110
44.7
52.7
60.7
48
53.3
49.3
54
64.7
49.3
52
48
54
64
48
20
40.7
50
58.7
46.7
40.7
33.3
43.3
51.3
42.7
5
21.3
36
42.7
40.7
80
12.7
150
38.7
 
 Levene’s Test Steps and Result in Minitab: -





 
Levene's Test.docx

An I-MR Chart  is a control chart which is used when data is in continuous category and is collected once at a time. It consists of two charts placed one over above, I Chart which is individual chart and MR chart is plotted for moving range which is absolute value of the difference between two consecutive points.
Data following normal distribution is an assumption while drawing I-MR chart however in practical or real-world problem data doesn’t follow normal distribution all the time hence Process stability follows major role. I-MR charts are very sensitive to Normality of the data. Non-Normal data if considered as normal data can cause unexpected behaviors including false alarm rates and difficulty in identifying the special cause variation.
If data is not normal, it is always advisable to do transformation using Box-Cox or Johnsons transformation to avoid the false alarm and get the right behavior of data for stability and control.
A normal distribution may have the value from minus to plus infinity.  In the real-world example this doesn’t occur physically very often.  For example, Cycle time cannot be in negative numbers.
 
Following is the Example for drawn I-MR charts when data is considered as normal however data is not normal and respective I-MR charts using data transformation: -
Cycle time in Minutes: -
Sl.No
Cycle Time (In Minutes)
S.No
Cycle Time (In Minutes)
S.No
Cycle Time (In Minutes)
1
3196
11
267
21
322
2
241
12
302
22
147
3
372
13
518
23
774
4
42
14
554
24
185
5
481
15
566
25
556
6
6081
16
900
26
555
7
131
17
158
27
361
8
26
18
109
28
556
9
1445
19
167
29
898
10
363
20
51
30
170
Table 1
Probability Plot of dats Using Mini-tab Normality test for data in Table 1: -
Normality test is done to illustrate whether data is normal or non-normal.
 
I-MR Charts drawn in Minitab for Table 1 mentioned assuming data following Normal distribution: -
The chart clearly illustrates that process is out of control, however out of control points are trigged due to false alarms

I-MR Chart drawn in Minitab for Table 1 mentioned after transforming days using Box-Cox transformation: -
The Chart clearly illustrates that process is in control, our of control data points mentioned earlier were due to false alarm in wrong assumption of data being normal.

 
 
 
 

In Statistic control Process, to analyze Variable Data, we need to use mean, range and standard deviation. Control charts are used to Monitor these Parameters to study whether the process is out of control or not. Any of the Parameter outside the Control limit means special cause variation is present in the process.
When we deal with any Variable, it is necessary to control the mean and dispersion for process stability. For Average we have X bar chart however to check the variability either Range (MR, R Chart) or Standard deviation (S Chart) to be calculated based on rational Subgrouping of Sample Data.
Hence two charts required to check the process means and variation over a time.
I-MR is chart is used when there no subgrouping in the Data. Following is chart depicts Sample of 500 ML size Bottle Filled over a period.

From the above I-MR chart analysis, MR chart is stable, however Individual chart is out of control. MR chart must be analyzed before analyzing I-Chart as I-chart control limits are calculated based on MR charts variation and average. MR chart depicts the variation of range calculated from consecutive data points. If MR chart is out of control, then it is irrelevant to Analyze the I-Chart as control limits for I chart will be incorrect.
X bar R (X bar R chart is required to calculate average and Range when Rational Subgroup Size is 2 to 8.)
The following charts (Average Call Handling time for 5 Calls for 30 Days) for X bar and R show the process is in control. R-chart should be in control before analyzing the X bar chart because X bar control limits are derived from R control chart. If R chart is out of control, then X bar charts limits are inaccurate, and it is irrelevant to analyze the X bar chart. For process to be in control, Both X bar and R charts must be in control.

 
X bar S (X bar S chart is required when subgroup size is greater than 8. When number of subgroups are large, S chart (Standard deviation) is more efficient than R (range) chart as X bar S charts can provide more accurate depictions of small variation a process.
The following charts (Average Call Handling time for 12 Calls for 30 Days) for X bar and R show the process is in control. S-chart should be in control before analyzing the X bar chart because X bar control limits are derived from S control chart. If S-charts value are out of control, then X bar charts limits are inaccurate. For process to be in control, Both X bar and S-charts must be in control.