Lecture 1

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INEN 270

• ENGINEERING STATISTICS
• Fall 2011

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Agenda
Purpose
Statistical Concepts
Descriptive Statistics and Some of Their Graphs
Inferential Statistics
Lecture Summary
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Lecture 1 Introduction to Statistics
Purpose
Statistical Concepts
Descriptive Statistics and Some of Their Graphs
Inferential Statistics
Lecture Summary
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Why Study Statistics?

• You need to know how to evaluate published
  numerical facts.
• Your profession or employment may require you to
  interpret the results of sampling or to employ
  statistical methods of analysis to make
  inferences in your work.

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What Is the Purpose of Statistics?

• One purpose of statistics is to make sense of
  your data.
• Statistics provide information about your data so
  you can answer questions and make informed
  business decisions.

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Lecture 1 Introduction to Statistics
Purpose
Statistical Concepts
Descriptive Statistics and Some of Their Graphs
Inferential Statistics
Lecture Summary
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Objectives

• Explain use of statistics.
• Define population and sample.
• Describe processes involved in statistical
  analysis.
• Compare descriptive and inferential statistics.
• Discuss the sampling plan.

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Defining the Problem

• Before you begin any analysis, you should
  complete certain tasks.
• 1. Outline the purpose of the study.
• 2. Document the study questions.
• 3. Define the population of interest.
• 4. Determine the need for sampling.
• 5. Define the data collection protocol.

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Example Speeding Data
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Population and Sample
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Basic Definition

• STATISTICS Area of science concerned with
  extraction of information from numerical data and
  its use in making inference about a population
  from data that are obtained from a sample.

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Extract Information
?
Population (set of all measurements)
Sample (set of measurements selected from the
population)
?
Make Inference
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Basic Definition

• Population and Parameter
• Population set representing all measurements of
  interest to the investigator.
• Parameters an unknown population characteristic
  of interest to the investigator.
• Sample and Statistic
• Sample subset of measurements selected from the
  population of interest.
• Statistic a sample characteristic of interest to
  the investigator.
• Descriptive Statistics
• Center of location mean, median, mode
• Variability variance, standard deviation
• Distribution

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Examples of Population and Sample

• Selecting the proper diet for shrimp or other sea
  animals is an important aspect of sea farming. A
  researcher wishes to estimate the average weight
  of shrimp maintained on a specific diet for a
  period of 6 months. One hundred shrimp are
  randomly selected from an artificial pond and
  each is weighed.
• Identify the population
• Identify the sample
• Identify the parameter
• Identify the statistic

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Simple Random Sampling
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Convenience Sampling
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Process of Statistical Data Analysis
Population
RandomSample
Make Inferences
Describe
SampleStatistics
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Sampling Plan
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Lecture 1 Introduction to Statistics
Purpose
Statistical Concepts
Descriptive Statistics and Some of Their Graphs
Inferential Statistics
Lecture Summary
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Objectives

• Compute and interpret statistics describing the
  location of a set of values, such as the mean and
  median and mode.
• Compute and interpret statistics describing the
  variability in a set of values, such as the range
  and standard deviation.
• Compute and interpret the measures of shape,
  skewness and kurtosis.
• Produce graphical displays of data.

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Some Frequently Used Statistics and Parameters
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Measure of Location

• Descriptive statistics that locate the center of
  your data are called measures of central tendency
  
• Sample Mean
• The sample mean of a set of n measurements (x1,
  x2,xn) is equal to the sum of the measurements
  divided by n.

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Measure of Location

• Sample Median
• Median the middle value (also known as the
  50th percentile)
• The median of a set of n measurements (x1,
  x2,xn) is the value that falls in the middle
  position when the measurements are ordered from
  the smallest to the largest.
• x1,xn are arranged in increasing order
  of magnitude

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RULE FOR CALCULATING THE MEDIAN

• 1. Order the measurements from the smallest to
  the largest.
• 2. A) If the sample size is odd, the median is
  the middle measurement.
• B) If the sample size is even, the median is
  the average of the two middle measurements.

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Percentiles
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Example

• A random sample of six values were
• taken from a population. These values were
• x17, x21, x310, x48, x54, and x612.
• What are the sample mean and
• sample median for these data?

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Sample Mean
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CALCULATIONS FOR THE SAMPLE MEDIAN
1. Order Sample
2.Median
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• 1. Order Sample

x21, x54, x17, x48, x310, x612
MEDIAN ( 7 8 ) / 2 7.5
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Example

• Given a set of data 1.7, 2.2, 3.11, 3.9, and
  14.7
• Sample mean
• Sample median

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Example
Consider the following sample 4 18 36
39 41 42 43 44 44 45 46 47
48 49 49 50 51 53 54 60
Which measure of central tendency best describes
the central location of the data THE SAMPLE
MEAN OR SAMPLE MEDIAN? Why?
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the median
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• Why?
• Because there is an outlier (extreme value),4 in
  the data set, the mean is heavily influenced by
  this single outlier.
• Solution
• Trimmed meandrops the highest and lowest extreme
  values and averages the rest.
• e.g. 5 trimmed mean drops the highest and lowest
  5 and averages the rest.

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Sample Mode

• Sample Mode
• What is the mode for the previous example?
• 44 (occurs twice)
• 49 (occurs twice)

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Measures of Central Tendency (Mode, mean and
median)

• How are they related to a given data set?
• Depending on the skewness of the population

(a) A bell-shaped distribution
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(b) A distribution skewed to the left
(c) A distribution skewed to the right
A mean B median C mode
A mode B median C mean
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• Suppose IRS wants to measure the central tendency
  of the income of the American population, which
  measure will you recommend and why?
• Hint Bill Gates
• Skewed to the right

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Other Measures of Locations

• Trimmed means
• Computed by trimming away a certain percent of
  both the largest and smallest set of values.
• Less sensitive to outliers than the mean but
  more-so than the median.
• What is the relationship between trimmed mean and
  the median?
• Example 0.32 0.53 0.28 0.37
  0.47 0.43 0.36 0.42 0.38 0.43

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0.32 0.53 0.28 0.37 0.47 0.43 0.36 0.42 0.38 0
.43
0.28 0.32 0.36 0.37 0.38 0.42 0.43
0.43 0.47 0.53
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The Spread of a Distribution Variation
Measure Definition
range the difference between the maximum and minimum data values
interquartile range the difference between the 25th and 75th percentiles (IR or IQR)
variance a measure of dispersion of the data around the mean
standard deviation a measure of dispersion expressed in the same units of measurement as your data (the square root of the variance)
coefficient of variation standard deviation as a percentage of of the mean
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Typical Variation Standard Deviation

• The variance is a measure of variation. The
  square root of the variance, or standard
  deviation, is a measure of variation in terms of
  the original linear scale.
• is the population standard
  deviation
• is an estimate of the population standard
  deviation.

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Typical Variation Average Squared Deviation

• Consider the data 3, 4, 8

Obs Data Deviation (Deviation)2
1 3 -2 4
2 4 -1 1
3 8 3 9
Sum 15 0 14
Average 5 0 14/3
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Measures of Variability

• Sample Range
• XMax-XMin
• Sample Variance
• Sample Standard Deviation

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Obs.
Obs.
1 7 49 2 1
1 3 10 100 4 8 64 5
4 16 6 12 144
1 7 0 0 2 1 -6
36 3 10 3 9 4 8 1
1 5 4 -3 9 6
12 5 25
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374
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Sample Variance
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Unbiased Estimate of Population Variance

• Calculate the unbiased estimate of population
  variance by averaging with n-1 instead of n.
• This estimator is unbiased because, on average,
  it equals the population variance.

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Discrete and Continuous Data

• Discrete Data
• Counted of defective items, of accidents
• Continuous Data
• Measured all possible heights, weights,
  distance,etc.

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Distributions

• When you examine the distribution of values for
  speed, you can determine
• the range of possible data values
• the frequency of data values
• whether the data values accumulate in the middle
  of the distribution or at one end.

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Graphical Methods and Data Description

• Stem and Leaf Plot
• Relative Frequency distribution
• Relative Frequency Histogram

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Construction of a Stem-Leaf Display

• List the stem values, in order, in a vertical
  column
• Draw a vertical line to the right of the stem
  values
• For each observation, record the leaf portion of
  the observation in the row corresponding to the
  appropriate stem
• Reorder the leaves from the lowest to highest
  within each stem row
• If the number of leaves appearing in each stem is
  too large, divide the stems into two groups, the
  first corresponding to leaves 0 through 4, and
  the second corresponding to leaves 5 through 9.
  (This subdivision can be increased to five groups
  if necessary).

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Car Battery Life
2.2 4.1 3.5 4.5 3.2 3.7 3.0 2.6
3.4 1.6 3.1 3.3 3.8 3.1 4.7 3.7
2.5 4.3 3.4 3.6 2.9 3.3 3.9 3.1
3.3 3.1 3.7 4.4 3.2 4.1 1.9 3.4
4.7 3.8 3.2 2.6 3.9 3.0 4.2 3.5
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Stem and Leaf Plot of Battery Life
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Double-Stem and Leaf Plot of Battery Life
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Relative Frequency Distribution

• Group data into different classes or intervals
• Counting leaves belonging to each stem
• Each stem defines a class interval
• Divide each class frequency by the total number
  of observations, we obtain the proportion of the
  set of observations in each of the classes.

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Relative Frequency Distribution of Battery Life
Class Interval Class midpoint Frequency, f Relative frequency
1.5-1.9 1.7 2 0.05
2.0-2.4 2.2 1 0.025
2.5-2.9 2.7 4 0.100
3.0-3.4 3.2 15 0.375
3.5-3.9 ? ? ?
4.0-4.4 ? ? ?
4.5-4.9 ? ? ?
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Class Interval Class midpoint Frequency, f Relative frequency
1.5-1.9 1.7 2 0.05
2.0-2.4 2.2 1 0.025
2.5-2.9 2.7 4 0.100
3.0-3.4 3.2 15 0.375
3.5-3.9 3.7 10 0.250
4.0-4.4 4.2 5 0.125
4.5-4.9 4.7 3 0.075
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Relative Frequency Histogram of Battery Life
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Picturing Distributions Histogram

• Each bar in the histogram represents a group of
  values (a bin).
• The height of the bar is the percent of values in
  the bin.

PERCENT
Bins
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Measures of Shape Skewness
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Measures of Shape Kurtosis
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Data Displays and Graphical Methods

• Box and Whisker Plot or Boxplot
• Pth Percentile
• The Pth Percentile is the value Xp such that p
  of the measurements will fall below that value
  and (100-p) of the measurements will fall above
  the value.
• Quartile
• Quartiles divide the measurements into four parts
  such that 25 of the measurements are contained
  in each part. The first quartile (Lower
  Quartile) is denoted by Q1, the second by Q2, and
  the third (Upper Quartile) by Q3.

P
(100-P)
Xp
Q1
Q2
Q3
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• InterQuartile Range (IQR)
• IQRQ3-Q1
• Outlier
• Observations that are considered to be unusually
  far removed from the bulk of the data.
• We label the observations as outliers when the
  distance from the box exceeds 1.5 times the
  interquartile range (in either direction).
• Box encloses the interquartile range of the data
• Whiskers show the extreme observations in the
  sample.

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Box and Whiskers Plot or Boxplot

• Calculating Fence Values
• Lower Inner Fence
• Q1-1.5(IQR)
• Upper Inner Fence
• Q31.5(IQR)
• Lower Outer Fence
• Q1-3(IQR)
• Upper Outer Fence
• Q33(IQR)

Maximum
Upper Quartile
Median
Lower Quartile
Minimum
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A Quick Method

• 1. Order the data from smallest to largest value.
• 2. Divide the ordered data set into two data sets
  using the median as the dividing value.
• 3. Let the lower quartile be the median of the
  set of values consisting the smaller values.
• 4. Let the upper quartile be the median of the
  set of values consisting of the larger values.

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Example

• Nicotine content was measured in a random sample
  of 40 cigarettes. The data is displayed below.

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1.Order the data from the smallest to the
largest 2.Divide the ordered data set into two
data sets using the median as the dividing value
0.72 0.85 1.09 1.24 1.37
1.40 1.47 1.51 1.58 1.63
1.64 1.64 1.67 1.68 1.69
1.69 1.70 1.74 1.75 1.75
1.79 1.79 1.82 1.85 1.86
1.88 1.90 1.92 1.93 1.97
2.03 2.08 2.09 2.11 2.17
2.28 2.31 2.37 2.46 2.55
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• Q2?
• Q1?
• Q3?
• IQRQ3-Q1?
• Q1(1.631.64)/21.635
• Q2(1.751.79)/21.77
• Q3(1.972.03)/22.000
• IQRQ3-Q10.365

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Box-whisker Plot
Outlier
Outlier
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Information Drawn from Boxplot

• The center of the distribution is indicated by
  the median line in the box.
• A measure of the variability is given by the
  interquartile range, the length of the box.
• The relative position of the median line
  indicates the symmetry of the middle 50 of the
  data.
• The skewness can be obtained by the length of the
  whiskers.
• The presence of outliers can be examined.

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Quantile Plot
A quantile plot simply plots the data values on
the vertical axis against an empirical assessment
of the fraction of observations exceeded by the
data value.
Where i is the order of observations when they
are ranked from low to high.
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Quantile Plot for paint data (table 8.2 page 238)
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Normal Quantile Plots

• The normal quantile-quantile plot is a plot of
  y(i) (ordered observations)
• against

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Normal Quantile Plots
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Lecture 1 Introduction to Statistics
Purpose
Statistical Concepts
Descriptive Statistics and Some of Their Graphs
Inferential Statistics
Lecture Summary
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Objectives

• Understand the importance of making inference.
• Understand the steps conducting a statistical
  study.

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Statistical Inference

• making an "INFORMED GUESS" about a parameter
  based on a statistic.
• (This is the main objective of statistics.)

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STATISTICAL INFERENCE
GATHER DATA
MAKE INFERENCES
SAMPLE STATISTICS
PARAMETERS
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Variable

• A VARIABLE is a characteristic of an individual
  or object that may vary for different
  observations.
• A QUANTITATIVE VARIABLE measures a variable on
  some sort of scale.
• A QUALITATIVE VARIABLE categorizes the values of
  the variable.

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RAISIN BRAN EXAMPLE

• A cereal company claims that the average amount
  of raisins in its boxes of raisin bran is two
  scoops.
• A random sample of five boxes was taken off the
  production line, and an analysis revealed an
  average of 1.9 scoops per box.

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Components of the Problem

• Identify the population
• Identify the sample
• Identify the symbol for the parameter
• Identify the symbol for the statistic

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Five Steps in a Statistical Study

• 1. Stating the problem
• 2. Gathering the data
• 3. Summarizing the data
• 4. Analyzing the data
• 5. Reporting the results

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Stating the Problem

• Specifically identifying the population to be
  sampled
• Identifying the parameter (s) being studied

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Gathering the Data

• SURVEYS
• Random Sampling
• Stratified Sampling
• Cluster Sampling
• Systematic sampling
• EXPERIMENTS
• Completely Randomized Design
• Randomized Block Design
• Factorial Design

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Lecture 1 Introduction to Statistics
Purpose
Statistical Concepts
Descriptive Statistics and Some of Their Graphs
Inferential Statistics
Lecture Summary
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Summary

• Basics of statistics
• Descriptive statistics and graphs
• Inferential statistics
• Textbook
• Chapter 1 (page 1-28)
• Chapter 8 (page 229-243)

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