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Transcript of Educational Statistics
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Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Educational StatisticsEducational Statistics
GURU K MOORTHYGURU K MOORTHYGURU K MOORTHYGURU K MOORTHY
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OutlineOutline
Introduction Frequency Distribution Measures of Central Tendency Measures of Dispersion
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Outline-ContinuedOutline-Continued
Other Measures Concept of a Population and Sample The Normal Curve Tests for Normality
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Learning ObjectivesLearning Objectives
When you have completed this chapter you should be able to:
Know the difference between a variable and an attribute.
Perform mathematical calculations to the correct number of significant figures.
Construct histograms for simple and complex data.
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Learning Objectives-cont’d.Learning Objectives-cont’d.
When you have completed this chapter you should be able to:
Calculate and effectively use the different measures of central tendency, dispersion, and interrelationship.
Understand the concept of a universe and a sample.
Understand the concept of a normal curve and the relationship to the mean and standard deviation.
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Learning Objectives-cont’d.Learning Objectives-cont’d.
When you have completed this chapter you should be able to:
Calculate the percent of items below a value, above a value, or between two values for data that are normally distributed.
Calculate the process center given the percent of items below a value
Perform the different tests of normality Construct a scatter diagram and perform
the necessary related calculations.
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Definition of Statistics:
1. A collection of quantitative data pertaining to a subject or group. Examples are blood pressure statistics etc.
2. The science that deals with the collection, tabulation, analysis, interpretation, and presentation of quantitative data
IntroductionIntroduction
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Two phases of statistics:Descriptive Statistics:
Describes the characteristics of a product or process using information collected on it.
Inferential Statistics (Inductive):Draws conclusions on unknown process
parameters based on information contained in a sample.
Uses probability
IntroductionIntroduction
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Types of Data:
Attribute:
Discrete data. Data values can only be integers. Counted data or attribute data. Examples include: How many of the products are
defective? How often are the machines repaired? How many people are absent each day?
Collection of DataCollection of Data
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Types of Data:
Attribute:
Discrete data. Data values can only be integers. Counted data or attribute data. Examples include: How many days did it rain last month? What kind of performance was
achieved? Number of defects, defectives
Collection of Data – Cont’d.Collection of Data – Cont’d.
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Types of Data:
Variable:
Continuous data. Data values can be any real number. Measured data.
Examples include: How long is each item? How long did it take to complete the
task? What is the weight of the product? Length, volume, time
Collection of DataCollection of Data
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Collection of DataCollection of Data
Significant Figures Rounding
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Significant Figures = Measured numbers When you measure something there is
always room for a little bit of error How tall are you 5 ft 9 inches or 5 ft 9.1
inches? Counted numbers and defined numbers ( 12
ins. = 1 ft, there are 6 people in my family)
Significant FiguresSignificant Figures
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Significant figures are used to indicate the amount of variation which is allowed in a number.
It is believed to be closer to the actual value than any other digit.
Significant figures:3.69 – 3 significant digits.36.900 – 5 significant digits.
Significant FiguresSignificant Figures
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Use Scientific Notation3x10^2 (1 significant digit)3.0x10^2 (2 significant digits)
Significant Figures – Cont’d.Significant Figures – Cont’d.
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Rules for Multiplying and Dividing Number of sig. = the same as the number
with the least number of significant digits.6.59 x 2.3 = 1532.65/24 = 1.4 (where 24 is not a
counting number)32.64/24=1.360(24 is a counting
number i.e. 24.00)
Significant FiguresSignificant Figures
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Rules for Adding and Subtracting Result can have no more sig. fig. after the
decimal point than the number with the fewest sig. fig. after the decimal point.38.26 – 6 = 32 (6 is not a counting
number)38.2 -6 = 32.2 (6 is a counting number)38.26 – 6.1 = 32.2 (rounded from 32.16) If the last digit >=5 then round up, else
round down
Significant FiguresSignificant Figures
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Precision
The precision of a measurement is determined by how reproducible that measurement value is.
For example if a sample is weighed by a student to be 42.58 g, and then measured by another student five different times with the resulting data: 42.09 g, 42.15 g, 42.1 g, 42.16 g, 42.12 g Then the original measurement is not very precise since it cannot be reproduced.
Precision and AccuracyPrecision and Accuracy
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Accuracy The accuracy of a measurement is
determined by how close a measured value is to its “true” value.
For example, if a sample is known to weigh 3.182 g, then weighed five different times by a student with the resulting data: 3.200 g, 3.180 g, 3.152 g, 3.168 g, 3.189 g
The most accurate measurement would be 3.180 g, because it is closest to the true “weight” of the sample.
Precision and AccuracyPrecision and Accuracy
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Precision and AccuracyPrecision and Accuracy
Figure 4-1 Difference between accuracy and precision
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Frequency Distribution Measures of Central Tendency Measures of Dispersion
DescribingDescribing DataData
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Ungrouped Data Grouped Data
Frequency DistributionFrequency Distribution
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2-72-7There are three types of frequency distributions
Categorical frequency distributions Ungrouped frequency distributions Grouped frequency distributions
Frequency DistributionFrequency Distribution
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2-72-7Categorical frequency distributions Can be used for data that can be placed
in specific categories, such as nominal- or ordinal-level data.
Examples - political affiliation, religious affiliation, blood type etc.
CategoricalCategorical
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2-82-8 Example :Blood Type Frequency Example :Blood Type Frequency DistributionDistribution
Class Frequency Percent
A 5 20
B 7 28
O 9 36
AB 4 16
CategoricalCategorical
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2-92-9Ungrouped frequency distributions Ungrouped frequency distributions - can
be used for data that can be enumerated and when the range of values in the data set is not large.
Examples - number of miles your instructors have to travel from home to campus, number of girls in a 4-child family etc.
UngroupedUngrouped
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2-102-10 Example :Number of Miles TraveledExample :Number of Miles Traveled
Class Frequency
5 24
10 16
15 10
UngroupedUngrouped
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2-112-11 Grouped frequency distributions Can be used when the range of values
in the data set is very large. The data must be grouped into classes that are more than one unit in width.
Examples - the life of boat batteries in hours.
GroupedGrouped
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2-122-12Example: Lifetimes of Boat BatteriesExample: Lifetimes of Boat Batteries
Classlimits
ClassBoundaries
Cumulative
24 - 30 23.5 - 37.5 4 4
38 - 51 37.5 - 51.5 14 18
52 - 65 51.5 - 65.5 7 25
frequencyFrequency
GroupedGrouped
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Number non conforming
Frequency Relative Frequency
Cumulative Frequency
RelativeFrequency
0 15 0.29 15 0.29
1 20 0.38 35 0.67
2 8 0.15 43 0.83
3 5 0.10 48 0.92
4 3 0.06 51 0.98
5 1 0.02 52 1.00
Table 4-3 Different Frequency Distributions of Data Given in Table 4-1
Frequency DistributionsFrequency Distributions
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Frequency Histogram
0
5
10
15
20
25
0 1 2 3 4 5
Number Nonconforming
Freq
uenc
y
Frequency HistogramFrequency Histogram
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Relative Frequency Histogram
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0 1 2 3 4 5
Number Nonconforming
Rela
tive F
req
uen
cy
Relative Frequency Relative Frequency HistogramHistogram
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Cumulative Frequency Histogram
0
10
20
30
40
50
60
0 1 2 3 4 5
Number Nonconforming
Cu
mu
lati
ve F
req
uen
cy
Cumulative Frequency Cumulative Frequency HistogramHistogram
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The histogram is the most important graphical tool for exploring the shape of data distributions.
Check: http://quarknet.fnal.gov/toolkits/ati/histograms.html for the construction ,analysis and understanding of histograms
The HistogramThe Histogram
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The Fast WayStep 1: Find range of distribution, largest - smallest valuesStep 2: Choose number of classes, 5 to 20Step 3: Determine width of classes, one decimal place more than the data, class width = range/number of classesStep 4: Determine class boundariesStep 5: Draw frequency histogram
#classes n
Constructing a HistogramConstructing a Histogram
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Number of groups or cellsIf no. of observations < 100 – 5 to 9
cellsBetween 100-500 – 8 to 17 cellsGreater than 500 – 15 to 20 cells
Constructing a HistogramConstructing a Histogram
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For a more accurate way of drawing a histogram see the section on grouped data in your textbook
Constructing a HistogramConstructing a Histogram
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Bar Graph Polygon of Data Cumulative Frequency Distribution or
Ogive
Other Types of Other Types of Frequency Distribution Frequency Distribution
GraphsGraphs
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Bar Graph and Polygon of Bar Graph and Polygon of DataData
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Cumulative FrequencyCumulative Frequency
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Figure 4-6 Characteristics of frequency distributions
Characteristics of FrequencyCharacteristics of FrequencyDistribution GraphsDistribution Graphs
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Analysis of HistogramsAnalysis of Histograms
Figure 4-7 Differences due to location, spread, and shape
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Analysis of HistogramsAnalysis of Histograms
Figure 4-8 Histogram of Wash Concentration
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The three measures in common use are the: Average Median Mode
Measures of Central Measures of Central TendencyTendency
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There are three different techniques available for calculating the average three measures in common use are the:
Ungrouped data Grouped data Weighted average
AverageAverage
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1
ni
i
XX
n
Average-Ungrouped DataAverage-Ungrouped Data
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1
1 1 2 2
1 2
... .
...
hi i
i
h h
h
f XX
n
f X f X f X
f f f
h = number of cellsh = number of cellsfi=frequencyfi=frequencyXi=midpointXi=midpoint
Average-Grouped DataAverage-Grouped Data
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1
1
n
i iiw n
ii
w XX
w
Used when a number of averages are combined with different frequencies
Average-Weighted AverageAverage-Weighted Average
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2m
d mm
ncf
M L if
Lm=lower boundary of the cell with the medianN=total number of observationsCfm=cumulative frequency of all cells below mFm=frequency of median celli=cell interval
Median-Grouped DataMedian-Grouped Data
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Boundaries Midpoint Frequency Computation
23.6-26.5 25.0 4 100
26.6-29.5 28.0 36 1008
29.6-32.5 31.0 51 1581
32.6-35.5 34.0 63 2142
35.6-38.5 37.0 58 2146
38.6-41.5 40.0 52 2080
41.6-44.5 43.0 34 1462
44.6-47.5 46.0 16 736
47.6-50.5 49.0 6 294
Total 320 11549
Table 4-7 Frequency Distribution of the Life of 320 tires in 1000 km
Example ProblemExample Problem
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2m
d mm
ncf
M L if
320154
235.6 3 35.958
Md
Median-Grouped DataMedian-Grouped Data
Using data from Table 4-7
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ModeMode
The Mode is the value that occurs with the greatest frequency.
It is possible to have no modes in a series or numbers or to have more than one mode.
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Figure 4-9 Relationship among average, median and mode
Relationship Among theRelationship Among theMeasures of Central Measures of Central
TendencyTendency
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Range Standard Deviation Variance
Measures of DispersionMeasures of Dispersion
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The range is the simplest and easiest to calculate of the measures of dispersion.
Range = R = Xh - Xl Largest value - Smallest value in
data set
MeasuresMeasures of Dispersion- of Dispersion-RangeRange
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Sample Standard Deviation:
2
1( )
1
n
iXi X
Sn
2
2
11
/
1
nn
ii
Xi Xi n
Sn
Measures of Dispersion-Measures of Dispersion-Standard DeviationStandard Deviation
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Ungrouped Technique
2 2
1 1( )
( 1)
n n
i in Xi Xi
Sn n
Standard DeviationStandard Deviation
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2 2
11
( ) ( )
( 1)
hh
i i i iii
n f X f Xs
n n
Standard DeviationStandard Deviation
Grouped Technique
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Relationship Between the Relationship Between the Measures of DispersionMeasures of Dispersion
As n increases, accuracy of R decreases Use R when there is small amount of data
or data is too scattered If n> 10 use standard deviation A smaller standard deviation means better
quality
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Relationship Between the Relationship Between the Measures of DispersionMeasures of Dispersion
Figure 4-10 Comparison of two distributions with equal average and range
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Other MeasuresOther Measures
There are three other measures that are frequently used to analyze a collection of data: Skewness Kurtosis Coefficient of Variation
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Skewness is the lack of symmetry of the data.
For grouped data:3
13 3
( ) /h
i iif X X n
as
SkewnessSkewness
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SkewnessSkewness
Figure 4-11 Left (negative) and right (positive) skewness distributions
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Kurtosis provides information regrading the shape of the population distribution (the peakedness or heaviness of the tails of a distribution).
For grouped data:4
14 4
( ) /h
i iif X X n
as
KurtosisKurtosis
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KurtosisKurtosis
Figure 4-11 Leptokurtic and Platykurtic distributions
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Correlation variation (CV) is a measure of how much variation exists in relation to the mean.
Coefficient of VariationCoefficient of Variation
(100%)sCV
X
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Population Set of all items that possess a characteristic of interest
Sample Subset of a population
Population and SamplePopulation and Sample
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Parameter is a characteristic of a population, i.o.w. it describes a population Example: average weight of the population, e.g. 50,000 cans made in a month.Statistic is a characteristic of a sample, used to make inferences on the population parameters that are typically unknown, called an estimator Example: average weight of a sample of 500 cans from that month’s output, an estimate of the average weight of the 50,000 cans.
Parameter and StatisticParameter and Statistic
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Characteristics of the normal curve: It is symmetrical -- Half the cases are to
one side of the center; the other half is on the other side.
The distribution is single peaked, not bimodal or multi-modal
Also known as the Gaussian distribution
The Normal CurveThe Normal Curve
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Characteristics:
Most of the cases will fall in the center portion of the curve and as values of the variable become more extreme they become less frequent, with "outliers" at the "tail" of the distribution few in number. It is one of many frequency distributions.
The Normal CurveThe Normal Curve
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The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula:
iXZ
Standard Normal DistributionStandard Normal Distribution
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Relationship between the Relationship between the
Mean and Standard Mean and Standard DeviationDeviation
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Mean and Standard Mean and Standard DeviationDeviation
Same mean but different standard deviation
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Mean and Standard Mean and Standard DeviationDeviation
Same mean but different standard deviation
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IF THE DISTRIBUTION IS NORMAL
Then the mean is the best measure of central tendencyMost scores “bunched up” in
middleExtreme scores are less frequent,
therefore less probable
Normal DistributionNormal Distribution
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Percent of items included between certain values of the std. deviation
Normal DistributionNormal Distribution
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Histogram Skewness Kurtosis
Tests for NormalityTests for Normality
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Histogram:
ShapeSymmetrical
The larger the sampler size, the better the judgment of normality. A minimum sample size of 50 is recommended
Tests for NormalityTests for Normality
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Skewness (a3) and Kurtosis (a4)” Skewed to the left or to the right (a3=0 for
a normal distribution) The data are peaked as the normal
distribution (a4=3 for a normal distribution)
The larger the sample size, the better the judgment of normality (sample size of 100 is recommended)
Tests for NormalityTests for Normality
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Probability Plots Order the data from the smallest to the
largest Rank the observations (starting from 1 for
the lowest observation) Calculate the plotting position
100( 0.5)iPP
n
Where i = rank PP=plotting position n=sample size
Tests for NormalityTests for Normality
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Procedure: Order the data Rank the observations Calculate the plotting position
Probability PlotsProbability Plots
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Procedure cont’d: Label the data scale Plot the points Attempt to fit by eye a “best
line” Determine normality
Probability PlotsProbability Plots
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Procedure cont’d: Order the data Rank the observations Calculate the plotting position Label the data scale Plot the points Attempt to fit by eye a “best line” Determine normality
Probability PlotsProbability Plots
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Chi-Square Test
2 Chi-squared
Observed value in a cell
Expected value for a cell
i
i
O
E
Where
22
1
( )ik
i
ii
O E
E
Chi-Square Goodness of Fit Chi-Square Goodness of Fit TestTest
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The simplest way to determine if a The simplest way to determine if a cause and-effect relationship exists cause and-effect relationship exists between two variablesbetween two variables
Scatter DiagramScatter Diagram
Figure 4-19 Scatter Diagram
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Supplies the data to confirm a Supplies the data to confirm a hypothesis that two variables are hypothesis that two variables are relatedrelated
Provides both a visual and statistical Provides both a visual and statistical means to test the strength of a means to test the strength of a relationshiprelationship
Provides a good follow-up to cause and Provides a good follow-up to cause and effect diagramseffect diagrams
Scatter DiagramScatter Diagram
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Straight Line FitStraight Line Fit
2 2
[( )( ) /
[( ) / ]
/ ( / )
xy x y nm
x x n
a y n m x n
y a mx
Where m=slope of the line and a is the intercept on the y axis