92465757 statistical techniques for analyzing quantitative data

7/29/2019 92465757 Statistical Techniques for Analyzing Quantitative Data

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Statistical Techniques for

Analyzing Quantitative Data

Maryam RamezaniValues in Computer Technology

CSC 426


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Outline

Statistics in Research

Exploring and Organizing a Data Set

Nature of the Data , Nominal , Ordinal, Interval, Ratio

Normal and Non-Normal Distributions

Descriptive Statistics

Inferential Statistics

Statistical Software Packages


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Role of Statistics in Research

With Statistics , we can summarize large bodies

of data, make predictions about future trends

,and determine when different experimentaltreatments have led to significantly different

outcomes.

Statistics are among the most powerful tools in

the research's toolbox.


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How statistics come to research?

In quantitative research we use numbers to

represent physical or nonphysical

phenomena

We use statistics to summarize and interpret

numbers


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Look at your data and find the ways of organizingthem

example: Scores of test for 11 children:

What do you see?

Ruth 96, Robert 60, chuck 68, Margaret 88

Tom 56, Mary 92,Ralph 64, Bill 72,Alice 80

Adam 76,Kathy 84


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Student Score

Ruth 96

Robert 60

Chuck 68

Margaret 88

Tom 56

Mary 92

Ralph 64

Bill 72

Alice 80Adam 76

Kathy 84

Student Score

Adam 76

Alice 80

Bill 72

Chuck 68

Kathy 84

Margaret 88

Mary 92

Ralph 64

Robert 60

Ruth 96

Tom 56

Student Score

Alice 80

Kathy 84

Margaret 88

Mary 92

Ruth 96

Alphabetical

Order

Adam 76

Bill 72

Chuck 68

Ralph 64

Robert 60

Tom 56

0

20

40

60

80

100

120

0 5 10 15

Series1


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Using Computer Spreadsheets to Organize

and Analyze Data

Sorting

Graphing

Formulas What Ifs

Save, Store, recall, update information


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Functions of Statistics

Descriptive Statistics:

describes what the data look like

Inferential Statistics :

inference about a large population by collecting

small samples.


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Considering the Nature of the Data

Continuous or discrete

Nominal, ordinal, interval or ratio scale

Normal or non-normal distribution


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Continuous versus Discrete Variables

Continuous Data :takes on any value within a finite or infinite interval.You can count, order and measure continuous data.

Example :height, weight, temperature, the amount of sugar in an orange, the

time required to run a mile.

Discrete Data : values / observations belong are distinct and separate,i.e. they can be counted (1,2,3,....).

Example: the number of kittens in a litter; the number of patients in a doctors

surgery; the number of flaws in one metre of cloth; gender (male, female); blood

group (O, A, B, AB).


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Nominal Data

the numbers are simply labels. You can count but not order or

measure nominal data

Example: males could be coded as 0, females as 1; marital status of anindividual could be coded as Y if married, N if single.

classification data, e.g. m/f

no ordering, e.g. it makes no sense to state that M > F

arbitrary labels, e.g., m/f, 0/1, etc


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Ordinal Data

ordered but differences between values are not

important e.g., Like scales, rank on a scale of 1..5 your degree of satisfaction

rating of 2 rather than 1 might be much less than the difference inenjoyment expressed by giving a rating of 4 rather than 3.

You can count and order, but not measure, ordinal

data.


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Interval Data

ordered, constant scale, but no natural zero

differences make sense, but ratios do not

e.g.: 30-20=20-10, but 20/10 is not twice as hot!

e.g.: Dates: the time interval between the starts of years 1981 and 1982

is the same as that between 1983 and 1984, namely 365 days. The

zero point, year 1 AD, is arbitrary; time did not begin then


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Ratio Data

Like interval data but has true zero

Ordered, Constant scale, natural zero

e.g., height, weight, age, length


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Normal and Non-Normal Distributions


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Normal Distribution


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Non-Normal Distributions

Skewed to the Left

(Negatively Skewed)Skewed to the Right

(Positevely Skewed)


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Leptokurtic and Platykurtic

Distributions


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Descriptive Statistics

Descriptive Statistics describes data

Points of Central Tendency

Amount of Variability

Relation of different variables to eachother


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Measuring center: If the n observations are x1, x2,,

xn, arithmetic mean is

n

xxx

xn

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Points Of Central Tendency: Mean

Geometric Mean

e.x.: Biological growth, Population growth


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Measure of Central Tendency

Mode The Most frequently occurringscore is identified.

Data on nominal, ordinal,

interval and ratio

Median The midpoint of the data Ordinal, interval, and ratio

Arithmetic

mean

All scores are added and the sum

is divided by the number of scores

Interval and ration

Geometric

mean

All scores are multiplied together,

and the nth root of their product is

computed.

Ratio scales


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Measures of Variability

How great is the Spread?

Range=Highest Score-Lowest score

the quartiles: The pth percentile of a distribution is the value

such that p percent of the observations fall at or below it.

The 50th percentile = median, M

The 25th percentile = first quartile, Q1

The 75th percentile = third quartile, Q3

Interquartile: Quartile 3- Quartile 1

Example:

13 13 16 19 21 21 23 23 24 26 26 27 27 27 28 28 30 30

M=?, Q1=?, Q3=?


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Measures of Variability

2

11

1

n

i

i xx

n

s

Standard Devastation

standardized score

xz


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Measure of Relationship: Correlation

correlation indicates the strength and direction of a linearrelationship between two variables.

See page 266 for other examples or correlation statistics


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Notes about Correlation

Substantial correlations between two

characteristics needs reasonable Validity and

Reliability in measuring

Correlation does not indicate causation


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Examples of using Statistics in

Computer Science Conceptual Representation of User Transactions or Sessions

A B C D E F

user0 15 5 0 0 0 185

user1 0 0 32 4 0 0

user2 12 0 0 56 236 0

user3 9 47 0 0 0 134

user4 0 0 23 15 0 0user5 17 0 0 157 69 0

user6 24 89 0 0 0 354

user7 0 0 78 27 0 0

user8 7 0 45 20 127 0

user9 0 38 57 0 0 15

Session/user

data

Pageview/objects


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We use the samples as estimate of population parameter.

The quality of all statistical analysis depends on the quality of

the sample data

Sample

Population

Random Sampling: every unit in the

population has an equal chance to be

Chosen

A random sample should represent thepopulation well, so sample statistics

from a random sample should provide

reasonable estimates of population parameters


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Some definitions

Parameter: describes a population

Statistic: describes a sample

Sample statistics Population parameter

Sample mean x

Sample proportion p P

Sample variance s2 2

Sample number n N

A parameter is a characteristic or quality of a population that in concept is

constant ,however, its value is variable.

example: radius is a parameter in a circle


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Estimate a population parameter from a

random sample

Test statistically hypotheses


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Inferential Statistics: Estimate a

Population Parameter from Sample

All sample statistics have some error in estimating population parameters

Example: estimate mean height of 10 year old boys in Chicago, Sample:200 boys

How close the sample mean is to the population mean?

we dont know but we know:

The mean from an infinite number of samples form a normal distribution.

The population mean equals the average (mean) of all samples.

The Standard deviation of sample distribution ( standard error) is directly

related to the std of the characteristic in question for the overallpopulation.


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Standard Error

Standard error tell us how much the particular mean vary from one

sample to another when all samples are the same size and drawn

randomly from the sample population.

Standard Error:

n is size of all samples and is the population std which we dont have!

We use the std of sample:

nM

1

n

s

M


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Accuracy of the Estimator

As in many problems, there

is a trade off between

accuracy and dollars.

What we will get from

our money if we invest

dollars in obtaining a largersize?

n = 100?

n = 200?


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Point versus Interval Estimate

A point estimate is a single value--a point--taken from a sample and used to estimate thecorresponding parameter of a population

, s, s2 and r estimate , , 2, respectively

An interval estimate is a range of values--an interval within whose limits apopulation parameter probably lies.

we say that we are 95% confident that the unknown population mean lies in the interval

X

(x -2/(n1/2),x+2 /(n1/2))

95% confidence interval for .

In only 5% of all samples,

the sample mean x is not in the above interval,

that is 5% of all samples give inaccurate results.


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Testing Hypothesis

Confidence intervals are used when the goal of our analysis is toestimate an unknown parameter in the population.

A second goal of a statistical analysis is to verify some claim aboutthe population on the basis of the data.

Research Hypothesis =/=Statistical hypothesis

A test of significance is a procedure to assess the truth about ahypothesis using the observed data. The results of the test areexpressed in terms of a probability that measures how well the datasupport the hypothesis.


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Sample values: The sample average of nicotine = 1.51 mlg

The standard deviation = 1.016.

The estimated amount of nicotine is 1.51mlg, based on the sample values.

The standard error of the sample average is

S.E.=s.d./sqrt(n-1)=0.045

Is there an actual difference between the sample value (1.51mlg) and theadvertised value (1.4 mlg)? Or is it just due to sampling error?

To answer this question we need aTest of Significance:

Example

To determine whether the mean nicotine content of a brand of cigarettes is

greater than the advertised value of 1.4 milligrams, a health advocacy grouptakes a sample of 500 cigarettes and measures the amount of nicotine in the

sample.


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Stating an hypotheses

The null hypothesis H0 expresses the idea that the observed difference is

due to chance. It is a statement of no effect or no difference, and is

expressed in terms of the population parameter.

Let denote the true average amount of nicotine.H0 : =1.4mlg

The alternative hypothesisHarepresents the idea that the difference is real. It

is expressed as the statement we hope or suspect is true instead of the null

hypothesis.

The alternative hypothesis states that the cigarettes contain a higher

amount of nicotine, that is: Ha : > 14mlg


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General comments on stating hypotheses

It is not easy to state the null and the alternative hypothesis!

The hypotheses are statements on the population values.

The alternative hypothesis Ha is often called researcher hypothesis,

because it is the hypothesis we are interested about.

A significance test is a test against the null hypothesis

Often we set Ha first and then Ho is defined as the opposite

statement!


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Errors in Hypothesis testing

Type I Error: the null hypothesis is rejected when it is in fact true; that is,

H0 is wrongly rejected.

Type II Error:the null hypothesis H0, is not rejected when it is in fact false


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Meta- Analysis

Meta-analysis refers to the analysis of analyses...the statistical

analysis of a large collection of analysis results from individual

studies for the purpose of integrating the findings. (Glass, 1976, p. 3)

Conduct a fairly extensive search for relevant studies

Identify appropriate studies to include in meta-analysis

Convert each studys results to a common statistical index


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Using Statistical Software Packages

SPSS

SAS

Matlab Statistics toolbox

SYSTAT, Minitab, Stat View, Statistica


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Interpreting the Data

Relating the findings to the original research problem and to the

specific research questions and hypothesis

Relating the findings to preexisting literature, concepts, theories and

research results.

Determining whether the findings have practical significance as well

as statistical significance

Identifying limitations of the study

92465757 statistical techniques for analyzing quantitative data

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