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a I Statistical Design Locating Important Sources of Variation A multistage process can be sampled to determine which stages contribute most variability in the product by W. S. Connor, The Research Triangle Institute
Om
OF THE most useful functions of statistics is to identify important sources of variation in a production process. A process may be divided into a number of subprocesses and these, in turn, may be divided. I t is then possible to distinguish several stages, or hierarchies, in the process. For example, a material may be obtained from a number of suppliers, in a sequence of shipments, from each of which several batches of product are manufactured. The product may have numerous variable characteristics, for each of which it is desired to determine the important sources of variation. Variation in product made from material from different suppliers is the sum of four components of variance:
. . . among suppliers . . . among shipments from the same
.. . ...
supplier among batches from the same shipment among samples from the same batch
By appropriately sampling the batches, it is possible to estimate these four components of variance. If the product is discrete items, like tires or bolts, a sample consists of a number of these items; but if the product is a bulk material, like cement or oil, a sample consists of several portions of the material. In either case, a batch can be thought of as composed of disjointed parts, some of which are taken into the sample. An example will be discussed to illustrate some of the concepts and techniques. I t is concerned with the wool content of part wool blankets. Sources of variability are colors, batches, looms, and pieces, which may be identified with suppliers, shipments, batches, and samples, respectively.
Variability of Wool Content
many batches of each color, and each batch was supplied to many looms. A sample was taken which consisted of 16 pieces, eight for each color, four for each batch, and two for each loom. The batches, looms, and pieces were selected randomly. Observations, in per cent wool,
A study was undertaken to determine the important sources of variation in the wool content of nominally 10% woo1 blankets. There were only two colors being manufactured-green and yellowTable I. Per cents Batch Loom Piece Per cent Sums Loom Balch Total
Observations in Per Cent Wool and Sums Green Material 1
2
1
2
1 10.1
2 9.7
3
3 14.2
19.8
4 15.1
4
5 14.7
29.3
6 14.9
8 14.4
7 10.1
29.6
24.5
49.1
54.1 103.2
Yellow Material Per cents Batch Loom Piece Per cent
Sums Loom Batch Total
3
4
5
7
6
9 10.6
11 12.5
10 10.3 20.9
12 12.4
8
13 10.3
24.9
14 12.8 23.1 48.3
94.1
Calculations for Per Cents Wool Green
Diff. Sq. 5 2
Looms
Diff. sq. S2
Batches
0.4 0.16 0.08
0.9 0.81 0.40
Diff.
Yellow
0.2 0.04 0.02
9.5 90.25 22.56
0.1 0.01 0
0.3 0.09 0.04
5.1 26.01 6.50
Diff.
2.5 6.25 3.12
4.0 16.00 4.00
0.4 0.16 0.08 2.1 4.41 1.10
2.5 6.25 0.78
I
5 2
9.1 82.81 5.18
sq.
82
Table Ill. Source of Variation
4.3 18.49 9.24
5.0 25 00 3.12
sq.
Total
16 12.4 25.2
45.8
Table II. Pieces
15 12.8
DF
Colors Batches
1 2
Looms Pieces
8 -
4
15
Analysis of Variance for Per Cents Wool
. Quantity Estimated
52
MS
5.18 3.90
5.18 1.95
u2 u:
34.16
8.54
a;
12.98 -
1.62
a$
+ 2af + 40; + 8 ( G - Y)*/2 + 2af + 4uf + 201
56.22
VOL. 53, NO. 12
DECEMBER 1961
73 A
S T A T I S T I C A L DESIGN are shown in Table I. Also shown are sums of successive pairs. T h e statistical analysis is carried out by repeated application of the formula for the sum of squares, 5‘2
=
z(x - 2
ance within the batch. Such variances, defined for all 2B batches, added, and divided by 2B, define uf. T h e variance U; is defined similarly.
)2/m
Estimating Variances
PEN NSA LT ALKYL ALKANOL AMINES Semi-Commercial (PiI ot PIa n t Q u a n t iti e s)
N-ISOPROPYLETHANOLAMINE 2523 (CH3)2CHNHC2H40H.Mol. Wt. 103.2 Sp. Gr. @ 20°C.. . . . . . . . , 0 3 9 - 0 . 9 0 IBP 168°C.. . . . . .
. . . . . . . FBP 178OC.
N-ISOPROPYLDIETHANOLAMlNE 1338 ( C H ~ ) ~ C H N ( C ~ H ~ O H ) Wt. ~ . M 147.2 OI. Sp. Gr. @ 20°C.. . . . . . . . .0.98-0.99 IBP 255°C.. . . . . . . . . . . . . FBP 270°C.
N-BUTYLETHANOLAMINE 1097 C ~ H ~ N H C Z H ~.O. H . .Mol. . Wt. 117.2 Sp. Gr. @ 20°C.. . . . . , . . , 0 3 8 - 0 . 9 0 IBP 192°C.. . . . . . . . . . . . . FBP 2 10°C.
N-BUTYLDIETHANOLAMINE 1029 C ~ H ~ N ( C Z H ~ O. H . .Mol. ) ~ . Wt. 161.2 Sp. Gr. @ 20°C.. . . . . . . . .0.96-0.97 IBP 265OC.. . .
. . . . . . . . . . FBP 287OC.
DIETHYLAMINOETHOXYETHANOL 1480 ( C Z H ~ I ~ N C ~ H ~ O C Mol. Z H ~Wt.161.2 OH Sp. Gr. @ 20°C.. . . . . . . . .0.93-0.95 Distillation: 95% between 21 5-228OC. Technical literature and samples available on request. Market Development D e w *Industrial Chemicals Division
PENNSALT CHEMICALS CORPORATION Three Penn Center
Circle No. 32
74 A
Philadelphia 2, Pa.
where m is the number of observations added together to make x, and X is the average x. For the present case, which contains only two values in each set, the formula simplifies to s2
where X I and x2 are the two values (Table 11). Analysis of variance is shown in Table 111. The numbers of degrees of freedom (DF) indicate the amounts of information about each source of variation. S2’s are calculated by summing the individual values from Table 11, and the mean squares (MS) are obtained by dividing each S2 by its associated number of degrees of freedom. T h e mean square for pieces estimates o:, the variance among per cents for pieces from the same loom; the mean square for looms estimates u; 2u7, where a? is the variance among average per cents for different looms running the same batch; the mean square for batches estimates U; 2012 4& where U; is the variance among average per cents for different batches of the same color; and the mean square for colors estimates r,“ 2uf 40: 8C2,where C2 = (G - Y ) 2 / 2 ,and G and Y a r e the average per cents for all pieces of green and yellow material, respectively. The coefficients 2,4, and 8 are the numbers of pieces for which the per cents are averaged and would be different for a different sampling scheme. If there are B batches of each color, L looms per batch, and P pieces per loom, then up”, a?, and U : are defined by applying the first formula for S2, with m = 1. For n,; x is the per cent wool in a piece, and X is the average over the P pieces from a loom. T h e quantity S2, divided by P - 1, is the variance within the loom. Such variances, defined for each loom, added over-all 2BL looms, and divided by ZBL, define up”. For a : , x is the average over-all P pieces from a loom and X is the grand average over the L looms for a batch. T h e quantity Pdivided by L - 1 is the vari-
on Readers’ Service Card
INDUSTRIAL
= (XI - x2)2/2m
A N D ENGINEERING CHEMISTRY
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The unknown variances are estimated by equating the mean squares to the quantities which they estimate and solving the equations. Thus, 1.62 is the estimate of ;:rc (8.54 - 1.62)/2 = 3.46 is the estimate of a?; (1.95 - 8.54)/4 = - 1.65 is the estimate of u i ; and (5.18 - 1.95)/8 = 0.40 is the estimate of (G - Y)”2. Because the variances cannot be negative, a negative estimate, such as - 1.65, is replaced by 0. An exception occurs when many such estimates are to be averaged, in which case positive estimates may offset negative ones and yield a positive average. T h e variance of a per cent, for a piece of green (or yellow) material, selected at random from any batch and loom, is the sum of the variances, estimated to be 1.62 3.46 0 = 5.08%. Estimated standard deviation is 2.3%. Expressed as percentages, the variation in green (or yellow) material is distributed as follows: between batches, 0% ; between looms within a batch, about 707,; and between pieces from the same loom, about 30%. T h e average difference between green and yellow material is estimated to be [2(0.40)]1/2= 0.9; or, alternatively, by direct comparison of averages, to be 103.2/8 - 94.1/8 = 1.17@. T h e estimates found here are imprecise, due to the small number of pieces in the sample. To obtain estimates which could be relied on would require many more batches, looms, and pieces.
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