Variables Charts
Overview
Variable data is interval or ratio data; variables charts are found more commonly in manufacturing. These charts are used in non-manufacturing when money or time is involved.
When unusual occurrences take place, the Process Measurement Chart will show some signals.
Types of Measurement
There are four basic types of measurement: nominal, ordinal, interval, and ratio. Attributes charts are for the first two types of measurement, while variables charts for the latter two.
Nominal measurement is when numbers are arbitrarily assigned to something, such as assigning numbers to colors. Typically, such classification is reduced to a binary system, such as yes/no or true/false.
Ordinal measurement assigns a ranking or scale to different items, thereby classifying which items has preference over another. Ordinal measurement does not, however, state specifically the degree of preference. An example is military ranking.
An interval scale defines any two points on a scale with a known and fixed relationship, such as distances on a ruler.
A ratio scale has all the properties of an interval scale, but has a true zero point as its origin.
Chart for Variables
In all cases, X-bar represents the mean and R represents the range in samples. X-bar is the average of the samples in each set. The range is the difference between the samples with the highest and lowest in each set.
X-double-bar is the overall average, which is found by finding the sum of all the set means, or X-bar, and dividing by the number of sets. R-bar is the average range, or the average of all the sets' ranges.
Computing for Charts
The center line of the chart is X-double-bar. The upper and lower control limits are X-double-bar +/- A(2) * R-bar. A(2) is found from a chart of averages factors. Go to the A(2) column and find the corresponding value with the size of your sets. For example, if you collected a bunch of sets, and each set has n=4 samples, you would find the value that matches n=2 with A(2).
n |
A2 |
D3 |
D4 |
d2 |
2 |
1.8800 |
- |
3.2665 |
1.1284 |
3 |
1.0233 |
- |
2.5746 |
1.6926 |
4 |
0.7285 |
- |
2.2820 |
2.0588 |
5 |
0.5768 |
- |
2.1145 |
2.3259 |
6 |
0.4832 |
- |
2.0038 |
2.5344 |
7 |
0.4193 |
0.0757 |
1.9243 |
2.7044 |
8 |
0.3725 |
0.1362 |
1.8638 |
2.8472 |
9 |
0.3367 |
0.1840 |
1.8160 |
2.9700 |
10 |
0.3083 |
0.2230 |
1.7770 |
3.0775 |
11 |
0.2851 |
0.2556 |
1.7444 |
3.1729 |
12 |
0.2658 |
0.2833 |
1.7167 |
3.2585 |
13 |
0.2494 |
0.3072 |
1.6928 |
3.3360 |
14 |
0.2354 |
0.3281 |
1.6719 |
3.4068 |
15 |
0.2231 |
0.3466 |
1.6534 |
3.4718 |
16 |
0.2123 |
0.3630 |
1.6370 |
3.5320 |
17 |
0.2028 |
0.3778 |
1.6222 |
3.5879 |
18 |
0.1943 |
0.3913 |
1.6087 |
3.6401 |
19 |
0.1866 |
0.4035 |
1.5965 |
3.6890 |
20 |
0.1796 |
0.4147 |
1.5853 |
3.7350 |
21 |
0.1733 |
0.4226 |
1.5774 |
3.7783 |
22 |
0.1675 |
0.4345 |
1.5655 |
3.8194 |
23 |
0.1621 |
0.4434 |
1.5566 |
3.8583 |
24 |
0.1572 |
0.4516 |
1.5484 |
3.8954 |
25 |
0.1526 |
0.4593 |
1.5407 |
3.9306 |
Negative control limits can exist for variables charts.
To make a range chart, R-bar is the center line. When n is less than or equal to 6, the lower control limit is set to zero. Otherwise, LCL is equal to D3 * R-bar, and UCL = D4 * R-bar.
If the size of subgroups are different, then use the corresponding values of n with each A2, D3, and D4.
The standard deviation, or sigma-hat is equal to R-bar divided by d2. To find sigma-hat when the size of subgroups vary, take the sum of the ranges for each n-size and divide that sum by its corresponding d2. Then take the sum of these calculations and divide by the number of sets. In ranges chart, if the size of subgroups vary, the center line varies as well. The center line is equal to R-bar(n) = d2(n) * sigma hat.
Again, if subgroup sizes are varying, the center line is a weighted average: take the sum of all samples and divide by the number of data points. The upper and lower control limits for averages charts with varying subgroup size is X-double-bar +/- A2(n) * R-bar-n. For range charts with varying subgroup sizes, UCL = D4 * R-bar-n and LCL = D3 * R-bar-n.
Points Falling on or Outside the Control Limits
Any point falling outside control limits needs investigation. The test to see whether if these points are a special cause of variation is non-parametric. In general, points outside the control limits occur with a very low probability because it is three sigma's away from the average. Dr. Deming warns that one should decide in advance which test to apply beforehand, because the more one looks for special causes, the more likely that one will find a result indicating a special cause when no such cause actually exists.
The first test is to see if there is a shift or run, which is when there is a string of points that are above or below the center line. For example, the probability of two points in a row occurring below the center line is 1 in 4, but the probability of 7 points in a row occurring below the center line is 1 in 128, which is very unlikely and thus, indicates a fundamental change has taken place.
A trend test is seeing if seven or more points in a row are rising or falling.
Periodicity occurs in charts where the value for subgroups depends on when the samples were taken. For example, when counting the amount of mail delivered, points near the first of the month will always be higher, indicating a special cause. In such circumstances, one should make separate charts for certain times of the month.
Another case is if all the points are near the center line but the upper and lower control limits are very far apart - "hugging the center line."
The zone test is done by dividing the chart into six sigma's, with zone A in the highest and lowest sigma's, zone B in the second highest and second lowest sigma's, and zone C being the the two sigma's above and below the center line. A special cause occurs when there are two out of three points in a row in Zone A, four out of five points in a row in Zone B or beyond, or eight points in a row on both sides of the center line with none in Zone C.