Descriptive Statistics
Uses data to provide descriptions of a population,
either through numerical calculations or graphical
displays.
Sampling
› When a population is very large, conducting a census can
be difficult, costly, and impractical.
› Instead, a subset of members selected from a population,
called a sample, is studied
How do I select a
representative sample
i.e. unbiased?
›
Simple random sampling: every combination of members of a given population has an equal likelihood being chosen
›
Systematic Sampling: members of the population are put into some order. A starting position is chosen at random and then every kth member in the
ordering is selected until a sample size is reached
›
Stratified Sampling: split the population into strata. Then randomly select the correct number of individuals from each strata so that their proportions
when combined together into a single sample would be representative of the population examined
›
Cluster Sampling: When the population is already organised into clusters that are each representative of the population, randomly selecting some
number of these clusters should provide a good sample
›
Sequence sampling: used frequently in quality control, involves randomly taking a sequence of successive units taken from a production line as the
sample
Identify the type of sampling used
A test to ensure the maximum capacity for a foot chair involves testing "to
destruction" every 100th stool from the assembly line (i.e., weight is added
to each stool until it collapses)
Systematic
The "Math:rand" function on the TI-83 calculator generates random decimal
values between 0 and 1. A group of 70 people are assigned numbers 00 to
69. A random value is generated on the TI-83 calculator using the
aforementioned function. If the first two digits of the random number
match someone's assigned number, then that person is selected to be part
of a sample. For example, if the number 0.35876397688 was generated,
person 35 would be selected to be in the sample. If the first two digits
don't match any of the assigned numbers, or if they correspond to
someone already selected, nothing happens. This process is repeated until
10 people have been selected to form the entire sample
Simple random
Numerical Summaries of Data
› Well-constructed data summaries and displays are
essential to good statistical thinking
› It is useful to describe data features numerically
› Characterizing the location (or central tendency) and
variability (spread or scatter) in the data is an example of
a numerical summary
› Data are often a sample of observations that have been
selected from some larger population of observations
– This type of population is called a conceptual or hypothetical
population because it does not physically exist
Sec 6.1 Numerical Summaries of Data
4
Sample Mean
The location or central tendency in the data can be characterized by the
arithmetic average or the sample mean.
For a finite population with 𝑁𝑁 equally likely values, the probability mass
function is 𝑓𝑓(𝑥𝑥𝑖𝑖 ) = 1/𝑁𝑁 and the mean is
Sec 6.1 Numerical Summaries of Data
5
Example 6.1 | Sample Mean
Consider 8 observations (𝑥𝑥𝑖𝑖 ) of pull-off force
from engine connectors as shown in the table.
8
∑x
i
=
x
=
average
=
8
=
104
= 13.0 pounds
8
i =1
12.6 + 12.9 + ... + 13.1
8
i
1
2
3
4
5
6
7
8
xi
12.6
12.9
13.4
12.3
13.6
13.5
12.6
13.1
13.00
= AVERAGE($B2:$B9)
Figure 6.1 The sample mean is the balance point.
Sec 6.1 Numerical Summaries of Data
6
Sample Variance and Standard Deviation
The variability or scatter in the data may be described by the sample
variance or the sample standard deviation.
The units of measurement for the sample variance are the square of the
original units of the variable, while the standard deviation measures
variability in the original units.
Sec 6.1 Numerical Summaries of Data
7
Example 6.2 | Sample Variance
=VAR.S($B2:$B9)
=STDEV.S($B2:$B9)
The table displays the quantities needed for
calculating the sample variance and sample
standard deviation.
The numerator of 𝑠𝑠 2 is
Sec 6.1 Numerical Summaries of Data
8
Computation of 𝒔𝒔𝟐𝟐
The prior calculation is definitional and tedious. A shortcut is
derived here and involves just 2 sums.
Sec 6.1 Numerical Summaries of Data
9
Example 6.3 | Shortcut Calculation for
For Example 6.2, we calculate the sample variance and standard deviation
using the shortcut method.
Sec 6.1 Numerical Summaries of Data
10
Exercise 6.1.6
› In Applied Life Data Analysis (Wiley, 1982), Wayne Nelson
presents the breakdown time of an insulating fluid
between electrodes at 34 kV. The times, in minutes, are
as follows:
 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01,
8.27, 12.06, 31.75, 32.52, 33.91, 36.71, and 72.89.
› Calculate the sample mean and sample standard
deviation.
› Check the values obtained with Excel (available through
Blackboard)
Exercise 6.1.7
› An article in Human Factors [“Visual Performance on CRT
Screens and Hard-Copy Displays” (1989, Vol. 31(3), pp.
247–257)] presented data on visual accommodation (a
function of eye movement) when recognizing a speckle
pattern on a high-resolution CRT screen. The data are as
follows:
 36.45, 67.90, 38.77, 42.18, 26.72, 50.77, 39.30, and 49.71.
› Calculate the sample mean and sample standard
deviation.
› Check the values obtained with Excel (available through
Blackboard)
Exercise 6.1.10
› Data from a second experiment using a low-resolution
screen were also reported in the article. They are:
 8.85, 35.80, 26.53, 64.63, 9.00, 15.38, 8.14, and 8.24.
› Compare it to the one for the first sample. What can you
conclude about CRT resolution in this situation?
Exercise 6.1.8
› Preventing fatigue crack propagation in aircraft structures
is an important element of aircraft safety. An engineering
study to investigate fatigue crack in n = 9 cyclically
loaded wing boxes reported the following crack lengths
(in mm):
 2.13, 2.96, 3.02, 1.82, 1.15, 1.37, 2.04, 2.47, 2.60.
› Calculate the sample mean and sample standard
deviation.
› Check the values obtained with Excel (available through
Blackboard)
Exercise 6.1.11
› The following data are the joint temperatures of the O-rings
(°F) for each test firing or actual launch of the Space Shuttle
rocket motor (from Presidential Commission on the Space
Shuttle Challenger Accident, Vol. 1, pp. 129–131):
 84, 49, 61, 40, 83, 67, 45, 66, 70, 69, 80, 58, 68, 60, 67, 72, 73, 70, 57,
63, 70, 78, 52, 67, 53, 67, 75, 61, 70, 81, 76, 79, 75, 76, 58, 31.
› Compute the sample mean and sample standard deviation
using Excel.
› Set aside the smallest observation (31°F) and recompute the
quantities. Comment on your findings. How “different” are the
other temperatures from this last value?
The meaning of 𝒏𝒏 − 𝟏𝟏 in the
denominator
› The population variance is calculated with 𝑁𝑁, the population
size. Why isn’t the sample variance calculated with 𝑛𝑛, the
sample size?
› The true variance is based on data deviations from the true
mean, 𝜇𝜇.
› The sample calculation is based on the data deviations from 𝑥𝑥,̅
not 𝜇𝜇.
› 𝑥𝑥̅ is an estimator of 𝜇𝜇; close but not the same.
› So the 𝑛𝑛 − 1 divisor is used to compensate for the error in the
mean estimation.
Sec 6.1 Numerical Summaries of Data
16
Degrees of Freedom
› When the sample variance is calculated with the quantity 𝑛𝑛 − 1 in
the denominator, the quantity 𝑛𝑛 − 1 is called the degrees of
freedom
› Origin of term:
– There are 𝑛𝑛 deviations from the 𝑥𝑥̅ in the sample
– The sum of the deviations is zero
– 𝑛𝑛 − 1 of the observations can be freely determined but the
𝑛𝑛𝑡𝑡𝑡 observation is fixed to maintain the zero sum
Sec 6.1 Numerical Summaries of Data
17
Sample Range
strongly affected by
extreme values
present in the
distribution
In addition to the sample variance and sample standard deviation,
the sample range is a useful measure of variability.
For Example 6.3 (pull-off force data), the sample range is 𝑟𝑟 =
13.6 − 12.3 = 1.3.
=MAX($B2:$B9) - MIN($B2:$B9)
Sec 6.1 Numerical Summaries of Data
18
Group Activity
› Compute the range of the previous data sets
› Compare the scatter of the range and the standard
deviation
› What would you conclude about the effect of the
measurements on each representation of the variability?
Quartiles
› The three quartiles partition the data into four equally sized counts
or segments.
– First or lower quartile : 25% of the data is less than q1.
– Second quartile : 50% of the data is less than q2, the median.
– Third or upper quartile : 75% of the data is less than q3.
› For the Table 6-2 data:
Usually not
affected by
the presence
of extreme
values in the
data set.
Denoted as 𝑥𝑥�
Descriptive Statistics
Mean
162.6625
Minimum
Q1
Median, Q2
Q3
Maximum
76
144.5
161.5
181
245
Midrange
84.5
(Sample) Variance
1140.631
(Sample) Standard Deviation 33.77324
Range
Mode
158
169
(76 to 245)
Interquartile Range (IQR)
36.5
(IQR Test) Lower Outlier
(IQR Test) Upper Outlier
89.75
181
SEC 6-2 STEM-AND-LEAF DIAGRAMS
20
Percentiles and Interquartile Range
› Percentiles are a special case of the quartiles.
› Percentiles partition the data into 100 segments.
› The interquartile range (IQR) is defined as:
IQR = q3 – q1.
› From the Quartiles example:
IQR = 181.00 – 143.25 = 37.75 = 37.8
› Impact of outlier data:
– IQR is not affected
– Range is directly affected.
SEC 6-2 STEM-AND-LEAF DIAGRAMS
21
Stem-and-Leaf Diagrams
Sec 6.2 Stem-and-Leaf Diagrams
22
Example 6.4a | Alloy Strength
› Consider the data in the table. We select as stem values the numbers 7, 8, 9, …,
24.
Sec 6.2 Stem-and-Leaf Diagrams
23
Example 6.4b | Alloy Strength
› The resulting stem-and-leaf diagram
is shown.
› Inspection of the diagram reveals
that most of the comprehensive
strengths lie between 110 and 200
psi and that a central value is
somewhere between 150 and 160
psi.
› The strengths are distributed
approximately symmetrically about
the central value
Sec 6.2 Stem-and-Leaf Diagrams
24
Group activity
› Please attempt one of the following exercises:
› 6.2.4
› 6.2.5
› 6.2.9
Frequency Distributions and Histograms
› A frequency distribution is a more compact summary of data than
a stem – and – leaf diagram
› To construct, we must divide the range of the data into intervals,
which are usually called class intervals, cells, or bins
› Choosing number of bins approximately equal to the square root of
the number of observations often works well in practice
Considerations:
Range = 245 – 76 = 169
Sqrt(80) = 8.9
Trial bin width = 169 / 8.9 = 18.9
Decisions:
Number of bins = 9
Bin width = 20
Range of bins = 20 * 9 = 180
Starting point = 70
Sec 6.3 Frequency Distributions and Histograms
26
Frequency Distribution Table
Sec 6.3 Frequency Distributions and Histograms
27
Histograms
• A histogram is a visual display of the frequency distribution
• Provides a visual impression of the shape and distribution of the
measurements and information about the central tendency and scatter
or dispersion in the data
bin frequency
• Unequal bin widths will be employed
Rectangle height =
bin width
Sec 6.3 Frequency Distributions and Histograms
28
Histograms
When the sample size is
large, the histogram can
provide a reasonably
reliable indicator of the
general shape of the
distribution or population of
measurements from which
the sample was drawn
the presence of skewness (or
outliers) can affect where the
measures of the middle are
located relative to one another
Sec 6.3 Frequency Distributions and Histograms
29
Examples of Distribution Shapes
Group activity
› Construct a Histogram on Excel using the data sets from
exercises:
› 6.2.4
› 6.2.5
› 6.2.9
› Adjust the number of bins for a smooth cumulative
frequency
Box Plots
› The box plot is a graphical display that simultaneously describes several
important features of a data set, such as center, spread, departure from
symmetry, and identification of unusual observations or outliers
› Sometimes called box – and – whisker plots
› Displays three quartiles
› A line, or whisker, extends from each end of the box
One might wonder where
the 1.5 in the above interval
comes from -- Paul Velleman,
a statistician at Cornell
University, was a student of
John Tukey, who invented this
test for outliers. He wondered
the same thing. When he
asked Tukey, "Why 1.5?",
Tukey answered, "Because 1 is
too small and 2 is too large."
Sec 6.4 Box Plots
32
Box Plots
Sec 6.4 Box Plots
33
Group activity
› Construct a box plot on Excel using the data sets from
exercises:
– 6.2.4
– 6.2.5
– 6.2.9
› How does the box plot compare in interpretive value to the
original stem-and-leaf diagram?
› Comment on the possible presence of outliers
› Repeat for the data without the extreme outlier and comment.
› Compare the distribution of the data with and without the
extreme outlier.
The most important functions in Excel (Data
Summaries)
•AVERAGE()
•MIN()
•MAX()
•QUARTILE.INC()
•MEDIAN()
•VAR.S() [or VAR.P()]
•STDEV.S() [or STDEV.P()]
•MODE.MULT()
Location (central tendency)
Variability (spread or
scatter)
Remarks
Mean (arithmetic average)
Standard deviation (or
variance)
Mean represents centre of
distribution and usually
affected by extreme values
and sample size. Likewise,
standard deviation is
exaggerated by values far
from the mean
Midrange
Range
Easy to compute and
strongly affected by extreme
values
Median
Inter-quartile range (IQR)
Can also be used for ordinal
values; usually not affected
by extreme values or sample
size
Mode
Number of modes
Most frequent data value in
the population/sample.
Modes can be used even for
nominal data.
Group Project
› It will be about an observational or retrospective study
› The assignment aims to collect a set of real data (the
number of observations should be more than 40) identify
statistical features, and perform relevant analysis assisted
by appropriate graphical plots.
› Refer to Section 1.2 to reflect on the limitations in your
chosen data sets
› YOU ARE REQUIRED TO SUBMIT AN EXCEL WORKBOOK
› More details on Blackboard
Specific Uses of Statistics
› Organize, summarize, and present data
› Generalize from knowledge of sample data to (probably
accurate) knowledge of a larger population
› Test hypotheses
› Determine relationships between variables
› Make predictions from existing data
Some Abuses
› Quoting statistics based on non-representative samples
› Choosing the “middle" value for a sample which most lends itself to your
position, when a different “middle" value would be more appropriate
› Speaking of changes in a variable in terms of actual values or
percentages to either inflate or deflate their importance psychologically.
(How happy would you be if your net worth increased by QAR 10 million.
What if that only represented a 0.3% increase?)
› Using detached statistics like "1/3 fewer carbs" (fewer than what?)
› Implying causal connections between variables without a well-designed
experiment to back it up (i.e., "Doctors say that taking lipotrim twice a day
''may reduce'' your weight by up to 30 lbs in the first 2 weeks!")
› Formatting graphs to mislead the eye
› Designing questions to be used on a survey that will bias the results