Descriptive Statistics

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Descriptive Statistics

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 Statistics is concerned with the describing, interpretation and
analyzing of data.
 It is, therefore, an essential element in any improvement
process.
 Statistics is often categorized into descriptive and inferential
statistics.
 It uses analytical methods which provide
the math to model and predict variation.
 It uses graphical methods 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.

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 calculating things such as the average of the data, its
spread and the shape it produces.

 For example, we may be concerned about describing:
• The weight of a product in a production line.
• The time taken to process an application.

 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.

 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?

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.

Outlier:
 The easiest way to detect them is by graphing the data or using
graphical methods such as:
• Histograms.
• Boxplots.
• Normal probability plots.
*●

Outlier:
 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.

Outlier:
 This example is about the time taken to process a sample of
applications.
Outlier
0 1 2 3 4 5 6 7 8 9
2.8 8.7 0.7 4.9 3.4 2.1 4.0
It is clear that one data point is far distant from the rest of the values.
This point is an ‘outlier’

The following measures are used to describe a data set:
 Measures of position (also referred to as central tendency or
location measures).
 Measures of spread (also referred to as variability or dispersion
measures).
 Measures of shape.

 If assignable causes of variation are affecting the process, we
will see changes in:
• Position.
• Spread.
• Shape.
• Any combination of the three.

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.

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’.
0 1 2 3 4 5 6 7 8 9
Mean

Median:
 The middle value where exactly half of the data values are
above it and half are below it.
 Less widely used.
 A useful statistic due to its robustness.
 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.
0 1 2 3 4 5 6 7 8 9
MeanMedian

Median Calculation:
12
30
31
37
38
40
41
41
44
45
23
33
34
36
38
40
41
41
44
Median = 38 + 40 / 2 = 39

 Why can the mean and median be different?
0 1 2 3 4 5 6 7 8 9
MeanMedian

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.

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

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.
0 1 2 3 4 5 6 7 8 9
Range

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
denoted by ‘s’.
 The standard deviation of a population
is denoted by “μ”.

Standard Deviation:
 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:
Mean (x-bar)
s = standard deviation
x = mean
x = values of the data set
n = size of the data set
s =
∑ ( x – x )2
n - 1

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.
Time allowed: 10 minutes

 If someone hands you a sheet of data and asks you to find the
mean, median, range and standard deviation, what do you do?
21 19 20 24 23 21 26 23
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
19 20 24 28 26 26 25 24

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.

Measures of Shape:
 It may be symmetrical or nonsymmetrical.
 In a symmetrical distribution, the two sides of the distribution
are a mirror image of each other.
 Examples of symmetrical distributions include:
• Uniform.
• Normal.
• Camel-back.
• Bow-tie shaped.

Measures of Shape:
 The shape helps identifying which descriptive 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 almost equal.
 If the data is skewed, then the median will be a more
appropriate to measure the central tendency.
 Two common statistics that measure the shape of the data:
• Skewness.
• Kurtosis.

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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X
X
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XXX
X
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X
X
(+) – SK > 0 (-) – SK < 0

Kurtosis:
 Measures the degree of flatness (or peakness) of the shape.
 When the data values are clustered around the middle, then the
distribution is more peaked.
• A greater kurtosis value.
 When the data values are spread around more evenly, then the
distribution is more flatted.
• A smaller kurtosis values.
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(-) Platykurtic (0) Mesokurtic (+) Leptokurtic

 Skewness and kurtosis statistics can be evaluated visually via a
histogram.
 They can also be calculated by hand.
 This is generally unnecessary with modern statistical software
(such as Minitab).

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

Further Information:
 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-Q1).
 It is often used when the data is not normally
distributed.
25%
Interquartile Range
25%
25%
25%
50%

 Minitab is a statistical software that allows you to enter your
data to perform a wide range of statistical analyses.
 It can be used to calculate many types of descriptive statistics.
 It tells you a lot about your data in order to make more rational
decisions.
 Descriptive statistics summaries in Minitab
can be either quantitative or visual.
- Descriptive Statistics in Minitab

Example:
 A hospital is seeking to detect the presence of high glucose
levels in patients at admission.
 You may use the glucose_level_fasting worksheet or use data
that you have collected yourself.
 Remember to copy the data from
the excel sheet and paste it into
Minitab worksheet.
79 72 77 85 76 120 78 94
93 70 79 75 68 73 79 85
98 77 77 88 79 79 70 113
75 80 74 83 85 79 87 82
104 106 81 76 68 72 61 95
78 106 84 70 96 70 90 98
69 60 74 67 71 75 105 79
71 75 131 80 75 52 152 106
81 96

Example:
 To create a quantitative summary of your data:
• Select Stat > Basic Statistics > Display Descriptive Statistics.
• Select the variable to be analyzed, in this case ‘glucose level’.
• Click OK.
 Here is a screenshot of the various
descriptive statistics you may
choose when doing your analysis.

Example:
 Here is a screenshot of the example result:
Quantitative Summary

Example:
 To create a visual summary of your data:
• Select Stat > Basic Statistics > Graphical Summary.
• Select the variable to be analyzed, in this case ‘glucose level’.
• Click OK.
 Here is a screenshot
of the example result:

Example:
 By default, Minitab fits a normal distribution curve to the
histogram.
 A boxplot will also be shown to
display the four quartiles of the
data.
 The 95% confidence intervals are
also shown to illustrate where the
mean and median of the population
lie.

Example:
 Mean, standard deviation, sample size, and other descriptive
statistic values are shown in the adjacent data table.
 The skewed distribution shows the
differences that can occur between
the mean and median.
 The mean is pulled to the right by the
high value outliers.
 The positive value for skewness indicates
a positive skew of the data set.

Descriptive Statistics

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Descriptive Statistics