Product Quality Improvement Case study 1 and Case study 2

Data Origin

The Wheelie Company has presented a table with data of how 100 bicycles got to the nearest 100 km before having one of the tires punctured or faulty.
Taking a glance to this table a great range of the product can be detected, from 20 to 77 km. The distribution is quite simetric (see Fig 1) with a peak between 40 and 50 km involving 42% of the samples.

From the table, we can deduce that tires have been faulty in a range of 57km. The first one at 20km, and the last one at 77km.

The Wheelie Company can say that their average tyre makes 44.45 km. For doing the data more comprehensive, it has been grouped in a frequency distriubution. This provides a better way of appreciating the data (Figure 1)

Figure 1: Distribution of
errors/data by groups and frequency and it’s bar chart.

Mean (X)=44.45 Km Standard Deviation=11.42
Range 77-20=52Km

Position	Num Of samples
Inside the limits (1σ)	69 (69%)
Out of 1σ	31 (31%)
Inside the limits (2σ)	97 (97%)
Out of 2σ	3 (3%)
Inside the limits (3σ)	100 (100%)

+ 1σ = 44.45 + 11.42 = 55.87

- 1σ = 44.45– 11.42 = 33.03

+ 2σ = 44.45 + 2*11.42 = 67.3

- 2σ = 44.45– 2*11.42 = 21.6

+ 3σ = 44.45 + 3*11.42 = 78.5

- 3σ = 44.45– 3*11.42 = 10.2

Once calculated the standard deviation (1σ
and 2σ) the measurements and its distribution can be displayed in the
charts below as a Gaussian or normal distribution. We can see that the
bell curve is not very tall, and it is quite wide. This means that the data is much
spread. Logothetis N. (1992)

Based on the previous calculations, we see that
The Wheelie Company would be between a 2σ and 3σ, where 99.73% of the
values would lie within the limits. In other words, a wheel from TheWhelie has 69% of chances of breaking between 33 and 56
km, and a 97 % of chances of breaking between 21 and 67 km

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Figure 2: In this histogram can be identified the products inside the limits for 1σ and 2σ. The
black line shows the trend given by the data, the characteristic bell curve
with the mean in the middle and the extremes approaching 0.

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Figure 3. We can see that the variability for 2σ is as big as 50 km, and the
variability for 3σ is more than 70 km. We can see also how powerful is the
mean (at 45km), with more than 8 samples.

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Figure 4: This is another way of showing the data and the different limits for 1 and 2
sigma. Rounded in circles are the errors escaping 2σ. For 3σ all the data would be within the area

This control chart diferenciates
between the special and common causes. In this case, for 2σ (for 3σ
all the samples would be normal) three of the samples have scaped
the control limits. 2 of the wheels have had a long endurance, and another one
failed before it was expected. It is important to analyze why this wheels
didn’t performed as expected, an “out of control situation of the average
performance”. It is important to find out why these errors have happened.

Logothetis mentions the idea that based in the principles
of normal distribution (like this case) actions have to be taken over the
samples concentrated around the limits, rather than in the ones near the mean.
By reducing the range of these points (here near the limits of 2 sigma) the
quality would be improved. “The guideline is that action should be taken when
at least 2 consecutive points fall out the warning limits” Logothetis,
N (1992)

Process control is a function in a manufacturing
process, which looks for deviations from the mean output and gathers changes
before the product quality is compromised. The use of SPC will involve the use
of control charts where the output of a certain process is gathered and
charted. When the process produces results outside the limits, it is said to be
out of control. Actions must be taken in this point to bring the process under
control again. Raisinghani, M.S. (2005