STEP #3 - ANALYZE CON’T
ORGANIZE & ANALYZE DATA

# Step #3 - ANALYZE - PART 2

SCATTER PLOTS
A scatter plot is a graph that helps you visualize
the relationship between two variables. It can be
used to check whether one variable is related to
another and is also a good way to illustrate the
relationship that may be found. I refer you to our
section on scatter plots for your further
information and development on this tool.
Why use a scatter plot?
·
Studying and identifying possible
relationships between the changes observed in
two different sets of variables.
·
Understanding relationships between
variables.
When to use a scatter plot?
·
To discover whether two variables are
related.
·
To find out if changes in one variable are
associated with changes in the other.
·
To test for a cause-and-effect relationship
(note: finding a relationship does not
always indicate causation).
Important notes regarding scatter plots:
1.
Each data point represents a pair of
measurements (for example, one dot
correlates between one item on the axis
versus another item on the axis.
2.
Two variables are represented - generally
the effect is on the vertical axis and the
potential cause is on the horizontal axis.
3.
Both axis are roughly equal in length, thus
the plot is square.
4.
The pattern formed by the scatter is an
important clue to how the two variables are
related, or possibly not related.
5.
Stratification using different symbols allows
you to look at multiple patterns at the
same time.
How to create a scatter plot:
1.
Collect paired data along with other
information. Other information could include
potential stratification factors.
2.
Determine which variable will be on the
horizontal axis (x) and which will be on the
vertical axis (y). By convention, place the
potential cause on the horizontal axis and
the effect on the vertical axis.
3.
Find the minimum and maximum of x and y.
4.
Set up the plot axis - each axis should be
about the same length.
5.
Plot all the x, y pairs on the graph.
6.
Label the graph.
To Interpret the Scatter Plot, you need to look for
outliers, but the emphasis is on the main pattern
formed by the scatter of the data points. The
tighter together that the points are clustered, the
stronger the correlation.
A pattern that slopes from the lower left corner to
the upper right corner means that as the variable
on the X-axis increases, so does the variable on
the Y-axis. This is a positive correlation. For
example, if you were studying that paint tends to
take longer to dry because of an increase in
humidity, your scatter plot may show a positive
correlation because as the humidity goes up, the
time it takes for paint to dry also lengthens.
A pattern than slopes from the upper left corner
down to the lower right corner means that as the
variable on the X-axis increases, the variable on
the Y-axis decreases. This would be an inverse or
negative correlation. To go back to our example
earlier, a scatter plot of the order lead time vs. the
number of operators available on the shift may
show a negative correlation because as the
number of operators goes up, the lead time goes
down.
Why use a scatter plot?
·
Studying and identifying possible
relationships between the changes observed in
two different sets of variables.
·
Understanding relationships between
variables.
When to use a scatter plot?
·
To discover whether two variables are
related.
·
To find out if changes in one variable are
associated with changes in the other.
·
To test for a cause-and-effect relationship
(note: finding a relationship does not
always indicate causation).
Correlation and Causation
Even strong correlations do not imply causation. If
there is a pattern on your scatter plot, it doesn't
necessarily mean that the two variables are
related. It is likely that there is a positive
correlation, but possibly not causation between
the occurrences of one item to another.
Conversely, no correlation does not mean also
that there is no causation; there may be
relationships over a wider range of data, or a
different portion of the range of data.
You should verify your cause review; do this by:
·
Selecting the most likely causes to verify.
·
Use existing data or collect new data to see
if these causes contribute to the
problem.
·
Use scatter plots, stratified frequency plots,
tables or experimentation to
understand the relationship between the
causes and the effects.
·
You may even try doing a Hypothesis test.
Hypothesis Tests
A hypothesis test is a procedure that summarizes
data so you can detect differences among groups.
It is used to make comparisons between two or
more groups.
How hypothesis tests work - since there is
variation, no two things will be exactly alike. The
question is whether the differences you see
between samples, groups, processes, etc., are due
to random, common-cause variation, or if there is
a real difference that exists. To help you make this
decision, various hypothesis tests provide ways of
estimating common cause variation for different
situations. These test whether a difference is
significantly bigger than the common cause
variation you would expect for the situation that
exists. If the answer is no, there is no statistical
evidence of a difference. If the answer is yes,
conclude that the groups are significantly
different.
Hypothesis tests take advantage of larger samples
because the variation among averages decreases
as the sample size increases.
We will now begin to discover the power of
statistical testing methods. Before we begin, there
are some definitions you need to understand.
H0 = no difference between groups or data sets
Ha = groups are different.
P value = the probability of obtaining the
observed difference given that the "true"
difference is zero.
A hypothesis test:
·
Tests the "null" hypothesis (no difference
between groups).
·
Against the alternative hypothesis (groups
are different).
·
Obtain a P-value for the null hypothesis.
·
Use the data and the appropriate
hypothesis; test statistic to obtain a P-value.
·
If P is less than .05, reject the H0 and
conclude the Ha.
·
If P is greater than or equal to .05, cannot
reject the H0.
Why use a hypothesis test?
·
To detect differences that may be pertinent
to your business.
·
You are unsure if minor difference in
averages is due to random variation or if it
reflects a true difference.
When to use a hypothesis test?
·
When you need to compare two or more
groups: on average, in variability, or in
proportion.
·
You are not sure if a true difference exists.
Assumptions for hypothesis testing - if data are
continuous, we assume the underlying
distribution is normal. You may need to transform
non-normal data (such as cycle times).
When comparing groups from different
populations, you can assume:
·
Independent samples
·
Achieved through random sampling.
·
The samples are representative (unbiased)
of the population.
When comparing groups from different processes
we assume:
·
Each process is stable.
·
There are no special causes or shifts over
time (that is, no time-related differences).
·
The samples are representative of the
process (unbiased).
P-value definitions:
·
Hypothesis tests compare observed
differences between groups.
·
The P-value equals the probability of
obtaining the observed difference given that
the "true" difference is zero (= the null
hypothesis).
·
P-values range from 0.0 to 1.0 (0% chance to
100% chance).
·
By convention, usually treat P as less than
.05 as indicative that the difference is
significant.
·
If P is less than .05, conclude there is little
chance that the true difference is 0.
Referring back to the types of data, again there
are two types - Discrete and Continuous:
Discrete - proportions.
Continuous - averages, variation, and shapes or
distributions.
How to use a Hypothesis Test:
1.
Determine the type of test suited to your
data and question.
2.
Select the appropriate test.
3.
Obtain p value; declare statistically
significant difference if p <.05.
Two types of errors in hypothesis testing
There are four possible outcomes to any decision
we make based on a hypothesis test - We can
decide the groups are the same or different, and
we can be right or wrong.
Type I error - Deciding the groups are different
when they aren't (the difference is due to random
variation).
Type II error - Not detecting a difference when
there really is one.
P-value - the probability of making a Type I error.
You choose what level of Type I error you're
willing to live with, by convention it is usually set
at .05 or 5%; thus it is said there is 95% confidence
level. The probability of making a type II error can
be calculated given an assumed true difference.
Practical Implications of type I and type II errors:
·
Both errors are important.
·
Guarding too heavily against one error
increase the risk of the other error.
·
Increasing the sample size reduces the risk
of type II errors.
·
Allows you to detect small differences.
T-TEST
We use a statistical test called the t-test for
comparing and judging difference between two
group averages. The formula for t is the same as
for Z, but P-values are obtained from the t-
distribution instead of the Z-distribution.
The confidence interval
A 95% confidence interval is the range of values
we expect to contain the true difference between
the two group averages. It's based on the
"difference distribution of averages" not the
differences between individual observations. It
does not represent the range of values we expect
for the difference between individual growth
times; that range would be wider.
If there is no significant difference between the
group averages, the confidence level will contain 0
(that is, range from negative [ - ] to positive [ + ].
Summary of the t-test:
·
It is a test of hypothesis for comparing two
averages.
·
The hypothesis is that the two group
averages are the same.
·
Their difference = 0.
·
If P-value is low, <.05, reject the hypothesis.
Common notation:
Null hypothesis
H0: meanA = meanB
Alternative hypothesis
HA: meanA ¹ meanB
Paired T-Test
Matched or paired data
Two measurements are obtained for each
sampling unit (a transaction, phone call,
employee, deal, application, etc.) Measurements
in the second group are not independent from
those in the first group. They are matched or
paired. The second measurements are taken on
the same sampling units as the first
measurements.
Practical implications of Paired t-tests:
·
The Paired t-test is a powerful way to
compare two methods.
·
Requires a special matched data structure.
The sampling unit (machine, operation,
material, etc.) needs to have each method
applied to both; has little or no
carryover from the use of the first method
to the use of the second; requires
planning; applies in the analysis or
improvement stages of a project (example -
you're looking to find or demonstrate a
difference between two ways of running a
process).
ANOVA
Comparing two or more group averages.
ANOVA is a statistical test that uses variance to
compare multiple averages simultaneously.
Instead of comparing pair wise averages, it
compares the variance between groups to the
variance within groups. The between-group
variance is obtained from the variance, or S2, of
the group averages. The within-group variance is
obtained from the variance S2, among values
within each group, and then pooled (or averaged
with appropriate differences) across the groups. If
the variance between groups is the same as the
variance within groups, we say there is no
difference between the group averages.
S^{2} between
S^{2 }within
You need to:
·
Obtain the variance between groups.
·
Obtain the variance within groups.
·
If they are about the same, conclude there is
no significant difference between
groups.
·
The ratio of two variances = F-statistic.
·
We get a P-value from the F-distribution.
Assumptions for ANOVA
·
The samples are representative of the
population or process.
·
The process IS stable.
·
Only common causes of variation exist
within the process.
·
No shifts or drifts/trends over time (no time-
related special causes).
·
The variance for each group is the same.
·
Can be verified with the F-test
·
Violation of these assumptions can cause
incorrect conclusions in the ANOVA
analysis.
·
It is also assumed that the underlying
distribution of each group is Normal. This
can be checked with a Normal probability
plot of residuals (we will cover this in the
Regression section).
Does the variation differ between groups? How to
check: There is a statistical test called
Homogeneity of Variance to check this
assumption. "Homogeneity" means "the same" so
actually you're testing to determine if the
variances are the same.
Review of ANOVA:
·
Used to compare averages of two or more
groups.
·
Assumes variances of each group are the
same.
·
Also used to compare variances of two or
more groups.
·
Called the Homogeneity of Variance test.
·
Use this test to check the assumption that
variances are the same when comparing
averages.
CHI-SQUARE
This is the hypothesis test used to compare two or
more group proportions. It is used when both X
and Y are discrete. The counts are summarized in
a table known as a contingency table. The Chi-
Square measures the difference between the
observed and the expected counts in this way:
What Next?
Determine which group proportions are different.
Determine why the group proportions are
different.
Assumptions of the Chi-Square Test
·
The sample is representative of the
population or process.
·
We assume the underlying distribution is
binomial for discrete data used in a X2
test.
·
The expected count greater than or equal to
5 for each cell, or the test will not
perform properly.
·
If expected count is less than 5, collecting
additional data (bigger sample) is
probably needed.
Value of Chi-Square Test
·
Discrete data are commonly collected and
used to analyze process performance in
the service applications within
manufacturing.
·
Non-significant differences between two or
more groups keep you from chasing
ghosts.
·
There is little to gain by studying the best or
trying to motivate the worst
performers.
·
Significant differences between group
proportions can be detected.
·
A low P-value (less than .05) indicates that it
is appropriate to identify root causes
that night lead to significant differences
between groups.
·
Examine the chi-square values for each cell
to determine which groups are
different.
·
Remember to consider whether the size of
the "statistically significant" difference
in proportions is actually important to the
business.
CONTINUE TO PART THREE OF ANALYZE NEXT- -
PART THREE
© The Quality Web, authored by Frank E. Armstrong, Making Sense
Chronicles - 2003 - 2016