Statistics Class 2

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Statistics Class 2

• Descriptive Statistics
• Central Tendency, Variability, and Standard
  Deviation
• Probability and Sampling
• Frequency Distributions

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Where we have been?

• Looked at definition of statistics
• Looked at key terms
• The role of statistical thinking in science and
  society at large
• Distinction inferential and descriptive
  statistics
• Distinction quantitative and qualitative data
• Reliability

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Where we are going?

• Talk about descriptive statistics
• Graphs
• Numerical methods
• Aim of descriptive statistics

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Describing Qualitative Data

• Definition of qualitative data
• Types of qualitative data
• nominal (yes/no 0, 1)
• ordinal (never, rarely, once a month, once a
  week, daily) ordered or ranked data

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Key Terms

• Class is one of several categories in which
  observations can be classified
• Class frequency is the number of observations in
  a particular class
• Class relative frequency is the class frequency
  in relation to all observations ie divided by
  the total number of observations

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Aphasia Example

• Consider a study of aphasia published in the
  Journal of Communication Disorders (Mar. 1995).
  Aphasia is the "impariment of loss of the faculty
  of using or understaind spoken or written
  languages." Three types of aphasia have been
  identified by researchers Broca's, conduction,
  and anomic. They wanted to determine whether one
  type of aphasia occurs more often than nay other,
  an, if so, how often. Consequently, they measured
  apahsia types for a sample of 22 adult
  aphasiac's. Table 2.1 gives the type of aphasia
  diagnosed for each aphasiac in the sample.

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Summary Table for Data on 22 Aphasiacs
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Calculating Relative Frequencies
Type Aphasia Relative Frequency Cumulative
Frequency Brocas 5 5/22 .227 .227 Conduct
ion 7 7/22 .318 .645 Anomic 10 10/22
.455 1.00 Totals 22 1.00 1.00
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Example Problem Dimensions

• Information systems design can progress toward
  meting the needs of the population of
  decision-makers, managers, policy-makers, and
  interdisciplinary workers by attention to
  specifications obtained from the users
  situation. The user situation is represented by
  problems and their dimensions. Problem dimensions
  are discussed to propose a new orientation for
  the design of information systems. (MacMullin
  Taylor, The Information Society, 3, 91-111).

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Bar Graph

• Visual depiction of the relative frequency of a
  variable. Shows how many times each case appears
  in the sample.
• Can also depict accumulative frequency or
  absolute frequency.

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Figure 1. Types of Aphasia in 22 Adult Aphasiacs.
Source Journal of Communication Disorders.
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Graphical Methods for Describing Quantitative Data

• Dot Plots
• Stem-and-Leaf Display
• Histograms

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• break

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Dot Plots

• In a dot plot the numerical value of each case is
  located on the horizontal axis.
• When data values repeat, dots are placed one on
  top of the other.
• Example

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Stem-and-Leaf

• A very easy and simple way of depicting data. The
  stem represents the full number, whereas the leaf
  represents the decimal points.
• Example

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Relative Frequency Histogram

• A relative frequency histogram has a vertical and
  a horizontal axes.
• The vertical axis indicates the proportion or
  relative frequency of the data.
• The horizontal axis represents the possible
  numerical values appearing in the data.
• Example.

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Measurement Classes

• The horizontal axis is divided into intervals
  called measurement classes.
• The intervals are of equal size.
• To each interval a frequency and relative
  frequency can be assigned on the vertical axis.

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Types of Histograms

• Absolute frequency
• Relative frequency
• Cumulative frequency
• Percentage
• Cumulative percentage
• Absolute frequency including cumulative percentage

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Numerical Measures of Central Tendency

• Central tendency is the numerical indictor of
  where the data tends to cluster.
• Variability indicates what the spread of the data
  is.
• Example height (general).

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Mean

• The mean is the most commonly used measure of
  central tendency.
• It is the sum of all the data points divided by
  the number of points in the data set.
• Example height.

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Calculating the Mean

• Mean sum of scores/ number of observations
• X ?xi/n average
• Example 2, 4, 6, 8,
• X 20/45

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Mean is an estimator of the population center

• The goodness of the inference depends on
• the size of the sample
• the spread of the data

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Center and Spread

• Insert picture p. 41

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Median

• When the data is arranged from the largest to the
  smallest number,
• the median is the number in the middle if the
  population size is odd, and
• the median is the number in the middle once the
  smallest number is eliminated if the population
  size is even.

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Location of Media

• Insert picture p. 43
• the median cuts the data in two 50 below and 50
  above.

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Mode

• Is the most frequent data point in the data set.
• Thus, has the largest relative frequency.

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Symbols

• X sample mean
• µ population mean
• M median
• n population size

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Comparing central tendency measures mean, and
medianinsert diagram p. 45
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Numerical Measures of Variability

• Range is the largest point in the data set minus
  the smallest.
• Sample variance sum of squared distance form the
  mean divided by n-1.
• Sample standard deviation is the positive square
  root of the sample variance.

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Why range not adequate?
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Symbols

• S2 sample variance
• s sample standard deviation
• sigma square (?2) population variance
• sigma (?) population standard deviation

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Interpreting the Standard Deviation

• The meaning of the mean is very much dependent on
  the size of the standard deviation.
• It provides information about the spread or the
  homogeneity of the sample.

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Calculating Variance and SD

• Example variance 2, 4, 6, 8,
• Mean 20/45
• Distance from mean 2-5-3
• 4-5-1 6-51 8-53
• Add all up 0
• What is the problem?
• Mean is in the middle!

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Solution

• Square the differences!
• Distance from mean 2-5-3
• 4-5-1 6-51 8-53
• (-3 )2 (-1) 2 (1) 2 (3) 2 9119 20
• Variance is 20.
• SD 20/n-120/36.66

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Two hypothetical data sets

• Sample 1 1,2,3,4,5
• Sample 2 2,3,3,3,4

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Solution
3
1
5

• X13
• X23
• -2-1012 square 4101410
• -10001 square 10001 2

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SD

• Take the square root of the variance
• Sample 1 SD3.16
• Sample 2 SD1.41

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Numerical Measures of Relative Standing

• These are a series of descriptive measures of the
  relationship of a measurement to the rest of the
  data.
• For example the pth percentile indicates where
  the rest of the points are located. P of the
  measures fall below the pth percentile and
  (100-p) fall above.

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• Insert figure 2.23

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Z-scores as numerical measures of relative
standing

• Z-distribution indicates where the measure stands
  in relation to the other measures.
• The sample z score is calculated by zx - x / s.

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Interpretation of z-scores for bell-shaped
variables

• 1. Approximately 68 of the measurements will
  have z-scores between -1 and 1SD.
• 2. Approximately 68 of the measurements will
  have z-scores between -2 and 2SD.
• 3. Approximately 99.7 of the measurements will
  have z-scores between -3 and 3 SD.

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• Picture 2.25

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Probability
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Probability

• Is the basis for inferential statistics
• Take dices and roll them 10 times
• What are the possible outcomes?
• There are 6 possible sample points 1, 2, 3, 4,
  5, and 6
• Each result is called an observation
• In this case 10 observations

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Coin Example

• Take a coin and toss it head or tale?
• If you do this with 2 coins, what are the
  possible outcomes or sample points
• Process of observing is called experiment

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All Possible Sample Points for 2 Coins

• 1. Observe HH
• 2. Observe TT
• 3. Observe TH
• 4. Observe HT
• All possible sample points are referred to as
  sample space.

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Key Terms

• Experiment is an act of observation that leads to
  a single outcome that cannot be predicted with
  certainty.
• Sample point is the most basic outcome of an
  experiment.
• Sample space is the collection of all its sample
  points.
• Event is a specific collection of sample points.

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Experiments and Their Sample Spaces

• Experiment Observe the up face on a coin.
• Sample Space
• 1. Observe a head.
• 2. Observe a tale.
• S H, T

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• Experiment Observe the up face on a die.
• Sample Space
• 1. Observe a 1
• 2. Observe a 2
• 3. Observe a 3
• 4. Observe a 4
• 5. Observe a 5
• 6. Observe a 6
• S 1, 2, 3, 4, 5, and 6

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• Experiment Observe the up faces on two coins
• Sample Space
• 1. Observe HH
• 2. Observe HT
• 3. Observe TH
• 4. Observe TT
• S HH, HT, TH, TT

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Venn Diagrams
H T
1 2 3 4 5 6
HH HT TH TT
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Probability Rules for Sample Points

• 1. All sample point probabilities must lie
  between 0 and 1
• 2. The probabilities of all the sample points
  within a sample space must sum to 1.

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Probability of an Event

• The probability of an event A is calculated by
  summing the probabilities of the sample points in
  the sample space for A.

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Steps for Calculating Probabilities of Events

• 1. Define the experiment , that is, describe the
  process used to make an observation and the type
  of observation that will be recorded.
• 2. List the sample points.
• 3. Assign probabilities to the sample points.
• 4. Determine the collection of sample points
  contained in the event of interest.
• 5. Sum the sample point probabilities to get the
  event probability.

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Lecture on Sampling

• Statistics Course
• Faculty of Information Studies

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Inferential statistics

• Hypothesis testing set a null hypothesis and an
  alternative hypothesis
• Parameter estimation determine the magnitude of
  a characteristic in a population

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Two central topics are

• 1. Random sampling
• 2. Probability

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Relationship population and sample

• The sample is a part of the population
• The sample is taken from the population
• Why is it that we do not use the population?
• Why would it be more advantageous to use the
  population?
• What is then the relationship between sample and
  population

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Representative sample

• This is a key issue in guaranteeing that we can
  make inferences from our sample to the population
  of interest
• Why important concept?
• Can you imagine what happens if a sample is
  non-representative?
• What other concepts are relevant in
  experimentation?

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Validity-Generalizability

• These are two central concepts in methodology
  (mainly in testing)
• Definition of validity to measure what we intend
  to measure
• Definition of generalizability to be able to
  generalize the results to the population of
  interest to us
• Why relevant concepts?

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Random Sample

• Definition A random sample is defined as a
  sample selected from the population by a process
  that assures the following
• 1) each possible sample of a given size has an
  equal chance of being selected and
• 2) all the members of the population have an
  equal chance of being selected into the sample

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Sampling with replacement

• Take members from population to include them in
  the sample and then put them back in the
  population, so that they can be drawn again
• Why important?
• If not, then sampling without replacement (often
  done in practice) why more often used?

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Why important (random sampling)?

• To be able to generalize from a sample to a
  population
• The sample has to be representative of the
  population
• Potential problem biased
  sample!!!

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Biased sample

• A sample that is not representative of the
  population we intend to study
• Example student population selected on the
  basis of email
• why not representative social aspect of
  technology

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Techniques for random sampling

• 1) Computer programs e.g. excel
• 2) Table of random numbers (good method)
• 3) Intuition
• 4) Stratification

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Tricks for evaluating papers

• Look at article in critical manner
• Look at the methodology section
• Focus on sample-population relationship
• Focus on how sample was selected
• Focus on generalizability
• Focus on validity
• Focus on tests used

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• Focus on type of statistical analysis
• Focus on level of significance
• Focus on rationale
• Think about relationship results and
  interpretation
• What is your overall feeling?