1. MAT 135 Introductory Statistics and Data Analysis Adjunct Instructor
Kenneth R. Martin
Lecture 7
October 12, 2016
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2. Confidential - Kenneth R. Martin
Agenda
Housekeeping
Readings
Exam #1 review
Chapter 1, 14, 10, 2, & 3
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Housekeeping
Read, Chapter 1.1 – 1.4
Read, Chapter 14.1 – 14.2
Read, Chapter 10.1
Read, Chapter 2
Read, Chapter 3
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Housekeeping
Exam #1 Review
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5. Statistics – Application to Research
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6. Confidential - Kenneth R. Martin
Statistics
Why collect samples ?
Often impractical to collect all the data from the entire population (i.e. U.S. census).
Some test methods are destructive – we wouldn’t have any products or services left to ship to a customer!
Too expensive to sample the entire population.
Don’t have to collect 100% of the population ! We can use inferential statistics to make sound conclusions about the population.
Population and Sample
Sampling
Scheme
POPULATION
SAMPLE
Measure
Data!
Use data from the SAMPLE to make conclusions about the POPULATION
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Statistics
Describing the Data
Two methods to summarize the data:
Graphical - Histogram
Analytical - Central Tendency
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Statistics
Central Tendency
A statistical measure which describes how the data is distributed around its central value: which includes the Mean, Median, and Mode.
However, Central Tendency does not tell about
data Variation / spread.
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Statistics
Relationship of Central Tendency
*** Normal distribution: Mean = Median = Mode
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Statistics
Frequency Distributions
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Statistics
Various curves (Different data spreads, common means)
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Statistics
Various curves (Different means, common data spreads)
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Statistics
Various Normal Curves
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Statistics
Measures of Variability - how the data is spread
from it’s central value
The central tendency does not indicate any levels of variability (dispersion) from the mean.
A = {100, 200, 300, 400, 500}
B = {50, 150, 300, 450, 550}
C = {250, 300, 300, 300, 350}
The mean & median of this data are all the same, but the variability of data is different in all data sets.
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Statistics
Measures of Variability:
Can be values from 0 to ∞ (infinity)
0 means no variability of data
A large value indicates lots of variability of data
Values can never be negative
As soon as one value in a data set differs from another, variability exists
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Statistics
Measures of Variability (Dispersion) - Range
Range (R) = Max. value – Min. value
= X H – X L
As data set size , the accuracy of using range .
Limit the usage of Range to ~ 10 readings.
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Statistics
Measures of Variability – Range Example
A = {100, 200, 300, 400, 500}
B = {50, 150, 300, 450, 550}
C = {250, 300, 300, 300, 350}
RA = ?
RB = ?
RC = ?
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Statistics
Measures of Variability
So what is the limitation of all three of these Range calculations ?
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Statistics
Measures of Variability (Dispersion) - Variance
Variance: a measure of the variability of the average squared distance that data points deviate from their mean.
Variance calculations include all data points.
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Statistics
Measures of Variability (Dispersion) - Variance
Sum of Squares (SS): the sum (addition) of the squared deviations of values from their mean. The SS is the numerator of the variance formula.
Variance, 2 , for the Population. μ is the population average
Variance, S2, for a Sample. M is the sample average.
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Statistics
Variance - Example
A = {100, 200, 300, 400, 500}
In this case, notice that the SS of both the population and the sample will be the same
Remember: PREMDAS
What is 2 ?
What is S2 ?
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Statistics
Measures of Variability (Dispersion) - Variance
What is a big limitation with Variance ?
What do you notice about the units of the mean, and the units of Variance ?
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Statistics
Measures of Dispersion – Standard Deviation
Also called the Root Mean Square deviation, it is a measure of the spread of the variability of the data; the average distance data deviate from their mean.
Calculated by taking the square root of the Variance
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Statistics
Measures of Dispersion – Standard Deviation
When the data comes from the “population”, we shall use “” (sigma) to denote the Standard Deviation.
The mean value will be represented by the Greek symbol  (mu)
The denominator does not have “uncertainty”, thus N
When the data comes from a “sample”, we shall use “SD” to denote the Standard Deviation.
The mean value will be represented by M or X ( X-bar)
The denominator shows “uncertainty”, thus n-1
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Statistics
Measures of Dispersion – Standard Deviation
We typically always want the standard deviation (variance) value to be as small as possible.
We typically want to minimize variability !
Standard deviation is always a better measure to precisely describe the data distribution versus range.
Other formulas exist for Standard Deviation, but will not be covered.
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Statistics
Standard Deviation - Example
A = {100, 200, 300, 400, 500}
What do we notice about the units of Standard Deviation and the units of the mean ?
 The Mean and Standard Deviation are typically reported together.
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Statistics
Standard Deviation - Example
B = {50, 150, 300, 450, 550}
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Statistics
Measures of Dispersion – Coefficient of Variation
CVar – Allows a comparison of standard deviations when the units of measure are not the same
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Statistics
Coefficient of Variation - Example
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Statistics
Box and Whisker Plot – Boxplot
Simple graphical tool to summarize data.
Need to determine 5 values (five-number summary) from data, to generate a boxplot:
Median (2nd Quartile)
Maximum data value
Minimum data value
1st Quartile (values below 1/4 observations)[whisker end]
3rd Quartile (values below 3/4 observations)[whisker end]
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Statistics
Process aim = 9.0 minutes
Spec = + / minutes
n = 125
R = 1.7
Box and Whisker Plot – Boxplot Example
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Statistics
Box and Whisker Plot - Boxplot Example
Inside box is the median value, and approximately 50% of observations
Whiskers extend from the box to extreme values
Example:
Median; n=125: Median = 63rd value = 9.8
Max = 10.7
Min = 9.0
1st Quartile = X 125 * 0.25 ~ X Avg 31 & 32 value = 9.6
3rd Quartile = X 125 * 0.75 ~ X Avg 94 & 95 value = 10.0
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Statistics
Box and Whisker Plot - Boxplot Example
Long Whiskers denote the existence of values much larger than other values.
For this example, mean  median.
Other variants exist, i.e. + / - 1.5*IQR [whisker ends], all other points are “outliers” as depicted as asterisks
IQR = Inner Quartile Range Q1 Q3 Q2
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Statistics
Box and Whisker Plot - Boxplot Example
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Statistics
Measures of Variability (Dispersion) - IQR
IQR – Interquartile Range
IQR = Q3 – Q1
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36. Confidential - Kenneth R. Martin
Statistics
Box and Whisker Plot - Boxplot Example
For this example,
IQR = ? Q1 Q3 Q2
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