Introductory Statistics
Learning Objectives
lDistinguish between different data types
lEvaluate the central tendency of realisticbusiness data
lEvaluate the dispersion of data
lEvaluate test statistics
lUse a test statistic to formulate a businessdecisions using regression analysis
After the session the students should be able to:
Types of data
•Discrete (A variable controlled by a fixed set of values)
•Continuous data (A variable measured on a continuous scale )
•These data may be collected (ungrouped) and then groupedtogether in particular form so that can be easily inspected
•But how would we collect data?
Simple random sampling
Stratified sampling
Cluster sampling
Quota sampling
Systematic sampling
Mechanical sampling
Convenience sampling
Sampling Techniques
Frequency distributions
The following are data of ages of a sample of agesmanagers
How could we represent these data effectively?
Scattering the data
Scatter Diagrams
Bar Diagrams
The histogram
We could group the datainto convenient classintervals thus
What measures of the centraltendency do we have
Measures of the central tendency
•Mode
•The maximum value of the distribution e.g. the mostoccurring value (in reality this can be evaluated using astandard formula
• Median
•The central value of a set of data or a distribution. Can beevaluated using a standard method of using the CDF
•Arithmetic mean
•The central value assuming the data are distributed inaccordance to an arithmetic progression
•Geometric mean
•The central value assuming the data are distributedaccording to a geometric progression
The mode
•For our data this occursbetween 30-39 (the modalrange)
•The construction showncan be employed to homein on the exact value
•Or the formula: whereL=lower boundary, l=lowerfreq diff, u=upper freq diff& c=the class boundarywidth
The mode
•Here, for our data
•L=29.5,
•l=5,
•u=1and
•the class boundarywidth c=10
The Median
For our data we couldevaluate this quantity twofold
•Via logical inference
Measures of Dispersion
•The range
•Largest value minus Smallestvalue
•Mean Square variation fromthe mean
•Square root of the variance
NOTE:
Use of Computer packages
Example:
Given the following data use a spreadsheet toproduce a grouped histogram using 9 binsalso produce a CFD. Hence or otherwiseevaluate:
a)Three measures of the central tendency and,
b)Three measures of the dispersion
Decision Processes
This is all very welland good however,how does this allowus to make researchand managerial &research decisions?
To answer this weneed to consider thepattern of the data,thus:
•Many sets of data adhere to thenormal distribution.
•The most important distribution ofthem all
•It is pretty much this property thatallows us to obtain (research)management decisions
•The normal distribution is usuallywritten N(μ,σ2); with μ thepopulation mean and σ2 thevariance
The Normal distribution
Properties of N(μ,σ2)
•For any normal curve withmean mu and standarddeviation sigma:
•68 percent of the observationsfall within one standarddeviation sigma of the mean.
•95 percent of observation fallwithin 2 standard deviations.
•99.7 percent of observations fallwithin 3 standard deviations ofthe mean.
The Z-Score
This is formula thatallows us to evaluatethe probability of anevent if we know thata particularpopulation isnormally distributed
N(48,12), find theprobability that some value of X<20.Example: If a population is N(48,12), find theprobability that some value of X<20.
Solution Protocol
-2.15
p
Spreadsheet SolutionProtocol
Exercise
•Example: Using a z score If a population is N(111,33.82),find the probability that some value of 100 <X<150.
Exercise
•Using a z score and given that the population isN(37,4.352), find the probability that some value ofX>150.
Samples
If we are using a sample of values as aconsequence of the central limit theorem the zscore will change, thus
The mean expenditure per customer at a tire store is £60and the sd £6. It is known that the nominal customer perday is 40. A new product costs £64, what is theprobability of selling such a product per customer
Example
Try one
In a store, the average number of shoppers is448, with an sd of 21. What is the probability that49 shopping hours have a mean between 441and446.
Regression & Correlationanalysis
Correlation analysis is used to measure strength ofthe association (linear relationship) between twovariables
Correlation is only concerned with strength of therelationship
No causal effect is implied with correlation
Scatter diagrams were presented in the last sessions
As was Correlation
Regression & Correlationanalysis
A scatter diagram can be used to show therelationship between two variables
Correlation analysis is used to measure strength ofthe association (linear relationship) between twovariables
Correlation is only concerned with strength of therelationship
No causal effect is implied with correlation
Scatter diagrams were presented in the last sessions
As was Correlation
Introduction toRegression Analysis
oRegression analysis is used to:
Predict the value of a dependent variable based on thevalue of at least one independent variable
Explain the impact of changes in an independentvariable on the dependent variable
oDependent variable: the variable we wish to predictor explain
oIndependent variable: the variable used to explainthe dependent variable
Simple Linear RegressionModel
oOnly one independent variable, X
oRelationship between X and Y isdescribed by a linear function
oChanges in Y are assumed to becaused by changes in X
Types of Relationships
Y
X
Y
X
Y
Y
X
X
Linear relationships
Curvilinear relationships
Types of relationshipscont…
Y
X
Y
X
Y
Y
X
X
Strong relationships
Weak relationships
Types of Relationships
Y
X
Y
X
No relationship
The regression model
Linear component
PopulationY intercept
PopulationSlopeCoefficient
RandomErrorterm
DependentVariable
IndependentVariable
Random Error
component
The regression model
Random Errorfor this Xi value
Y
X
Observed Valueof Y for Xi
Predicted Valueof Y for Xi
Xi
Slope = β1
Intercept = β0
εi
The Least Squaresapproach
b0 and b1 are obtained by finding thevalues of b0 and b1 that minimize the sumof the squared differences between Y and:
Rendering:
Regression Formulae
Thus the formulae canbe summarized as:
Where:
Regression Example
An estate agent wishes to find the relationshipbetween the house prices and size, it is suspectedthat a linear relationship exists between the houseprice (the dependent variable Y) and the house sizein square metres (the independent variable X). Usinglinear regression, find the relationship and make aprediction of a house price measuring 200m2. Thefollowing data have been collected by the estateagent.
Regression data
House Price in £k
(Y)
Area in m sqr
(X)
123
156
156
178
140
189
154
208
100
122
110
172
203
261
162
272
160
158
128
189
Regression Solution Cont…
Regression SolutionCont…
Then we evaluate the regression constant
There are various computer methods availablewhich do these calculations for you these aredetailed in the handout
Regression computersolution
There a three methods to evaluate theRegression coefficient and constantusing an Excel spreadsheet. These being:
•Graphical
•Calculation
•Functions
Regression computersolution Cont…
This is an example of the graphical method,which is required for a pass grade in theforthcoming assignment! If you want highergrades however you will have to check theseanswers using the other two methods shownin the handout
Summary
lDistinguish between different data types
lEvaluate the central tendency of realistic businessdata
lEvaluate the dispersion of data
lEvaluate test statistics
lUse a test statistic to formulate a businessdecisions using regression analysis
Have we met out learning objectives? Specifically areyou able to: