Stats

Continuous Improvement Toolkit

Descriptive ,S-tatistics

- Descriptive Statistics

Statistics is concerned with the describing, interpretation and
a na lyzi ng of data.

□ It is, therefor,e, an essentiil element in any improvement

process.

□ Statistics is often categorized into descri'ptive and inferential

statistics.
It uses analytical methods which provide
the math to model and predict variation.

□ It uses graphical m,ethods to help making
numbers visible for communication
purposes.

- Descriptive Statistics

Why do we Need Statistics?

□ To find why a process behaves the way it does.

□ To find why it produces defective goods or services.

□ To center our processes on 'Target' or 'Nominal'.

□ To check the accuracy and precision of the process.

□ To prevent problems caused by assignable causes
of variation.

□ To reduce variability and improve process capability.

□ To know the truth about the real world.

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- Descriptive Statistics

Descriptive Statistics:

□ Methods of describing the characteristics of a data set.

□ Useful because they allow you to make sense of the data.

□ Helps exploring and making conclusions about the data in order
to make rational decisions.

□ Includes ca lculating things such as the average of the data, its
spread and the shape it produces.

- Descriptive Statistics

□ For example, we may be concerned about describing:

• The weight of a product in a production line.
• The time taken to process an application.

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- Descriptive Statistics

□ Descriptive statistics involves describing, summarizing and

organizing the data so it can be easily understood.

□ Graphical displays are often used along with the quantitative
measures to enable clarity of communication.

- Descriptive Statistics

□ When analyzing a graphical display, you can draw conclusions

based on several characteristics of the graph.

□ You may ask questions such ask:

• Where is the approximate middle, or center, of the graph?
• How spread out are the data values on the graph?

• What is the overall shape of the graph?
• Does it have any interesting patterns?

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- Descriptive Statistics

Outlier:

□ A data point that is significantly greater or smaller than other
data points in a data set.

□ It is useful when analyzing data to identify outliers

□ They may affect the calculation of descriptive
statistics.

□ Outliers can occur in any given data set and in
any distribution.

- Descriptive Statistics

□ The easiest way to detect them is by graphing the data or using
graphical methods such as:

• Histograms.
• Boxplots.
• Normal probability plot s.

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- Descriptive Statistics

□ Outliers may indicate an experimental error or incorrect

recording of data.

□ They may also occur by chance.

• It may be normal to have high or low data points.

□ You need to decide whether to exclude them
before carrying out your analysis.

• An outlier should be excluded if it is due to

measurement or human error.

- Descriptive Statistics

□ This example is about the time taken to process a sample of
applicat ions.

2.8

8.7

0.7

4.9

3.4

2.1

4.0

Outlier

0 1 2 3 4 5 6 7 8 9

It is clear t hat one data point is far distant from the rest of the values.

This point is an 'outlier'

- Descriptive Statistics

The following measures are used to describe a data set:

□ Measures of position (also referred t o as central t endency or
locat ion measures).

□ Measures of spread (also referred to as variability or dispersion
measures).

□ Measures of shape.

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- Descriptive Statistics

□ If assignable causes of variation are affecting the process, we

will see changes in:

• Position.

• Spread.

• Shape.

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• Any combination of the three.
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- Descriptive Statistics

Measures of Position:

□ Position Statistics measure the data central tendency.

□ Central tendency refers to where the data is centered.

□ You may have calculated an average of some kind.

□ Despite the common use of average, there are different

statistics by which we can describe the average of a data set:

• Mean.

• Median.

• Mode.
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- Descriptive Statistics

Mean:

□ The total of all the values divided by the size of the data set.

□ It is the most commonly used statistic of position.

□ It is easy to understand and calculate.

□ It works well when the distribution is symmetric and there are

no outliers.

□ The mean of a sample is denoted by 'x-bar'.

□ The mean of a population is denoted by 'µ'.

Mean

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- Descriptive Statistics

Median:

□ The middle value where exactly half of the data values are
above it and half are below it.

□ Less widely used.

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□ A useful statistic due to its robustness.

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□ It can reduce the effect of outliers.

□ Often used when the data is nonsymmetrical.

□ Ensure that the values are ordered before calculation.

□ With an even number of values, the median is the mean of the

two middle values.

- Descriptive Statistics

Median Calculation:

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12

33

30

34

31
36

37

38

38
Median = 38 + 40 / 2 = 39
40

40

41

41

41

41

44

44
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- Descriptive Statistics

□ Why can the mean and median be different?

Median . . Mean

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- Descriptive Statistics

Mode:

□ The value that occurs the most often in a data set.

□ It is rarely used as a central tendency measure

□ It is more useful to distinguish between unimodal and

multimodal distributions

• When data has more than one peak.

- Descriptive Statistics

Measures of Spread:

□ The Spread refers to how the data deviates from the position
measure.

□ It gives an indication of the amount of variation in the process.

• An important indicator of quality.
• Used to control process variability and improve quality.

□ All manufacturing and transactional

processes are variable to some degree.

□ There are different statistics by which
we can describe the spread of a data set:

• Range.

• Standard deviation.

Spread

- Descriptive Statistics

Range:

□ The difference between the highest and the lowest values.

□ The simplest measure of variability.

□ Often denoted by 'R'.

□ It is good enough in many practical cases.

□ It does not make full use of the available data.

□ It can be misleading when the data is skewed or in the presence

of outliers.

• Just one outlier will increase

the range dramatically.
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Range

- Descriptive Statistics

Standard Deviation:

□ The average distance of the data points from their own mean.

□ A low standard deviation indicates that the data points are
clustered around the mean.

□ A large standard deviation indicates that they are widely
scattered around the mean.

□ The standard deviation of a sample is

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denoted by 's'.

□ The standard deviation of a population
is denoted by "µ".

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- Descriptive Statistics

o Perceived as difficult to understand because it is not easy to
picture what it is.

□ It is however a more robust measure of variability.

□ Standard deviation is computed as follows:

S=
L ( x - x )2

n-1

s = standard deviation

x = mean
x = values of the data set

n = size of the data set

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- Descriptive Statistics

Exercise:

□ This example is about the time taken to process a sample of
applications.

□ Find the mean, median, range and standard deviation for the

following set of data: 2.8, 8.7, 0.7, 4.9, 3.4, 2.1 & 4.0.

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- Descriptive Statistics

□ If someone hands you a sheet of data and asks you to find the
mean, median, range and standard deviation, what do you do?

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25 24 19 19 21 19 25 19

23 23 15 22 23 20 14 20

15 19 20 21 17 15 16 19

13 17 19 17 22 20 18 16

17 18 21 21 17 20 21 21

21 17 17 19 21 22 25 20

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- Descriptive Statistics

Measures of Shape:

□ Data can be plotted into a histogram to have a general idea of

its shape, or distribution.

□ The shape can reveal a lot of information about the data.

□ Data will always follow some know distribution.

- Descriptive Statistics

□ It may be symmetrical or nonsymmetrical.

□ In a symmetrica l distribution, the two sides of the distribution
are a mirror image of each other.

□ Examples of symmetrical distributions include:

• Uniform.
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• Normal.
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• Camel-back.

• Bow-tie shaped.

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- Descriptive Statistics

□ The shape helps identifying which descripfve statistic is more
appropriate to use in a given situation.

□ If the data is symmetrical, then we may use the mean or median
to ,measure the central tendency as they are a'lmost equal.
If the data is skewed, then the median will be a more
appropriate to measure the central tendency.

- Tw,o common statistics that 1m,easure the shap,e of the data:

•Skewness

•Kurtosis.

- Descriptive Statistics

Skewness:

□ Describes whether the data is distributed symmetrically around
the mean.

□ A skewness value of zero indicates perfect symmetry.

□ A negative value implies left-skewed data.

□ A positive value implies right-skewed data.

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- Descriptive Statistics

Further Information:
□ Variance is a measure of the variation around the mean.
□ It measures how far a set of data points are spread out from
their mean.

□ The units are the square of the units used for the original data.

•For example, a variable measured in meters will have a variance
measured in meters squared.

□ It is the square of the standard deviation.

Variance= s2

- Descriptive Statistics

□ The Inter Quartile Range is also used to measure
variability.

□ Quartiles divide an ordered data set into 4 parts.

□ Each contains 25% of the data.

□ The inter quartile range contains the middle
50% of the data (i.e. Q3-Ql).

□ It is often used when the data is not normally
distributed.

50%

Interquartile Range

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25%

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