Statistics : Describing Data

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Statistics Describing Data

• Week 18 to 19

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For weeks 18 and 19

• Summarising data
• Graphic representations
• The shape of distributions
• Measures of central tendency
• Measures of dispersion
• Percentiles, quartiles, interquartile range

3
Notice / Caveat

• This stats course is a very quick
• and short version of parts of
• a full stats course that would normally
• take 1 full year to cover
• However, because we only have 8 weeks
• there are many issues of the topics
• we will study that we will not have time
• to explain/cover in class

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Notice / Caveat

• There are therefore certain question
• of why and/or how which we will not
• have time to cover or explain.
• However, do not let this stop you from
• doing your extra reading if you want to.
• So, dont let the constraint of the syllabus
• hold you back if you want to learn more

5
Booklist

• Suggested books for you to study from include
  most of those found in the SOAS library under
  class mark number 519.4
• Scan through them to find those appropriate to
  the stats we will do

6
Booklist

• But it is up to you to decide which is your most
  preferred texts to use in support of the topic of
  statistics
• Basically all you need to look for are books with
  headings the same as or similar to those in
  these, and future, slides

7
The Nature of Statistics

• Some examples to highlight the application of
  stats
• Someone taking a poll during an election and
  predicting the outcome of the ellection based on
  the data they get
• To get an idea of the economic status of a town a
  researcher collects data on salaries of people in
  that town and calculates the average salary

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The Nature of Statistics

• Some examples to highlight the application of
  stats
• Police forces collect crime figures from
  different parts of the country and produce
  summaries of the data for different types of
  crime in different areas
• Production of goods in the thousands by a factory
  usually results in some of the goods being
  defective. The company tries to estimate the
  number of defective goods produced.

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The Nature of Statistics

• Basic material of statistics is data or
  observation gathered from experiments
• Data can be
• Numerical age, weight, number of people
• Non-numerical colour, who to vote for in an
  election,

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The Nature of Statistics

• Definition
• Descriptive stats
• Statistics is the science of collecting,
  simplifying and describing/presenting data
• Inferential stats
• It is also the science of making inferences
  (drawing conclusions) based on the analysis of
  data

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The Nature of Statistics

• When collecting data we cannot collect the total
  data possible
• We cannot interview all the people who voted
• We cannot collect all the salaries of the people
  in a town
• etc
• so a judgement has to be made about the larger
  body of data by studying the info from some of
  the data

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The Nature of Statistics

• Definition
• The entire collection of all data we are
  interested in is called a population
• A collection of some of the elements obtained
  from the population is called a sample of the
  population

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The Nature of Statistics

• Example
• In studying voter preferences the population is
  everybody who votes in an election. The votes are
  the data values of interest.
• The sample is those people who are interviewed
  after voting

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The Nature of Statistics

• Example
• In studying average salaries of a town the
  population is everybody in the town. The salaries
  are the data values of interest.
• The sample is the salaries of those people we
  interview

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The Nature of Statistics

• In the salaries example above
• the average salary of the population is called a
  parameter of the population
• the average salary of the sample is called a
  statistic of the population
• Definition
• A numerical property of a population is called a
  parameter. A numerical property of a sample is
  called a statistic.

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Sampling

• Statistics uses samples instead of population
• we therefore need our samples to be
  representative of the population as a whole.
• We therefore need to be careful about the way we
  collect sample data

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Sampling

• Random Samples
• These are simply samples chosen at random,
  without using any criteria or prejudices
• It also means that however many different
  samples we choose from a population they will all
  represent characteristics and properties similar
  to that of the population

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Sampling

• Two specific methods of sampling are prejudices
  lottery and random number method.
• Lottery method elements of a population are
  written on separate tags, placed in a container,
  and mixed. Tags are then chosen at random
• Random number suppose we have 1000 employees
  numbered 0001 to 1000. Use a table of random
  numbers to choose 100 employees

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Sampling

• Stratified Samples
• If we divide a population into sub-populations
  this is known as stratifying the population.
• Samples taken from sub-populations are known as
  stratified samples

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Sampling

• Stratified Samples
• An examples of this might be to subdivide
  student population into 1st year, 2nd year and
  3rd year students.
• Each year is then a strata of the population,
  and we may take random samples from each strata.

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Sampling

• Stratified Samples
• Suppose, for example, that we had 40 1st years,
  25 2nd years and 35 3rd years.
• To obtain a stratified sample of size 100 we
  would therefore take 40 1st years, 25 2nd years
  and 35 3rd years

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Sampling

• Cluster Samples
• Stratified sampling is practical only when we
  have a small number of sub-populations / strata.
• However, if there are a large number of strata
  then it is impractical to do so and we have
  therefore to choose a random number of strata
• This is cluster sampling

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Sampling

• Cluster Samples
• As an example consider voting district. A
  particular county will have many voting districts
  too many to sample from all of them.
• So, each voting district is a stratum of the
  county as a whole and we take a random sample of
  the strata, i.e. a random number of voting
  district (from which we sample)

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Some Problems with Sampling

• Non-response Bias
• Consider the case of the 1936 USA presidential
  campaign between Franklin Roosevelt and Alf
  Landon (a true story)
• a survey was conducted of 2.4 million people
• the survey predicted that Alf Landon would win
  by a landslide
• but in fact Franklin Roosevelt won by a
  landslide

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Some Problems with Sampling

• Non-response Bias
• why ?
• It was found that a lot of the participants of
  the survey were people whose names were taken
  from the telephone book
• in 1936 people who had phones were well off
  (not poor) and the politics of such people was
  that of Alf Landon
• but this was not representative of the politics
  of the population as a whole

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Some Problems with Sampling

• Non-response Bias
• thus the sample was not representative of the
  population as a whole
• Such a situation is known as bias. Particularly
  this situation is known as non-response bias
  since a segment of the population was not
  represented in the sample.

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Some Problems with Sampling

• Non-response bias can range from
• People not answering specific questions
• to people refusing to participate at all
• to whole groups of people being excluded from
  the sample
• E.g. internet surveys suffer major non-response
  bias. Why ?
• E.g. surveys of home owners suffer major
  non-response bias. Why ?

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Some Problems with Sampling

• To prevent Non-response bias some companies
• 1) Call back people if they could not be
  contacted 1st time
• 2) Or, offer incentives such as complete this
  survey and you can choose 1 of the following free
  products.
• What problems can occur with doing 2) ?

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Some Problems with Sampling

• Response Bias
• Response bias relates to participants not giving
  true, honest answers or forgetting what their
  answer might have been.
• This can also occur in questionnaires when
  questions are written in a way to elicit a
  certain answer

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Some Problems with Sampling

• Response Bias
• E.g. Given that the congestion charge has
  produced no net reduction in traffic congestion,
  would you favour an increase in congestion
  charges next year ?
• Hence the design of questions is a questionnaires
  is very important to make sure response bias does
  not occur or is minimised.

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Some Problems with Sampling

• Lying is more difficult to prevent have you
  ever cheated when filling in your tax return form
  ?

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Descriptive Statistics

• (to do
• raw data, frequency distribution
• Histogram, frequency polygon, comparison of
  latter two. See p81-96 of Saunders white stats
  book
• )-----

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Graphic Representation

• -----

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The Shape of Distribution

• -----
• see p273 chase and bown
• For descrip of normal distribution

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Measures of Central Tendencies

• There exists three different types of "averages"
  to a set of data.
• Such averages are generally called central
  tendencies because averages tell us the value
  of the data lying in the middle
• hence averages are called measures of central
  tendencies.

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Measures of Central Tendencies

• The Mean
• This is the usual average as we know it, i.e.
• add up all the data and
• divide by the number of data values
• x , ? (? x) / n

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Measures of Central Tendencies

• Notation
• ? population mean
• x sample mean
• Example
• See lecture
• Note extreme values can affect the value of the
  average.

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Measures of Central Tendencies

• The larger any one value in the data the more it
  will drag the mean away from the central area of
  the data
• These extreme values are called outliers,
• Because of the effect of outliers on the average,
  the mean is said to be not a resistant measure.

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Measures of Central Tendencies

• The Median
• Arrange the set of data in order of increasing
  magnitude. The median of that set is the data
  value which lies in the middle.
• The symbol for this median is x (x tilde).

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Measures of Central Tendencies

• The Median
• If we have an even number of data then there
  will be only one value in the middle and that is
  the median.
• What do we do if we have an odd number of values
  in our data ?

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Measures of Central Tendencies

• Examples
• See lecture
• For the mean we saw that there was an outlier,
  but that the median calculation was not affected
  by this.
• Thus the median is a resistant measure of central
  tendency.

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Measures of Central Tendencies

• Q when to use Mean and when to use Median as
  you measure of central tendency
• 1) generally (but not always) use the median as
  measure if you have outliers.
• 2) otherwise, use the mean to calculate your
  measure.
• but
• 3) always investigate the data to know which one
  to use.

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Dispersion

• When data is
• plotted on a graph it can either look compact or
  spread out.

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Dispersion

• Such spreading out is generally called dispersion
  
• Then is the data tightly packed around the mean
  or is it loosely spread out around the mean.
• Looking at the two diagrams above we see that the
  1st is more loosely spread out than the 2nd one.

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Dispersion

• Knowing such info can be crucial.
• Example
• A manufacturer produces items of a certain
  strength.
• You would expect that wherever you used the
  product it would be of the strength you expected.

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Dispersion

• You don't want there to be a great difference
  between the strength of the product in Leeds
  compared to Luton.
• Such unreliability could be very dangerous if
  the product was a bridge.
• Thus we would want a very low difference in
  strength, i.e. a very small measure of dispersion
  in strength between products.

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Measures of Dispersion

• Closely related to dispersion is the position of
  a piece of data w.r.t. all the other data
• and the way to describe this position is called
  measure of dispersion.
• Q How do we calculate or measure dispersion ?

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Measures of Dispersion

• Range R
• This is the simplest way but the least useful.
• R highest data values lowest data value
• Example
• See lecture

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Measures of Dispersion

• Range does not
• Measure dispersion well enough.
• In the diag both sets of data have range 10

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Measures of Dispersion

• but the data is spread out very differently
• range does not measure variability within
  data.
• Hence we need another measure
• one which measure variability away from the
  mean.

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Measures of Dispersion

• Average deviation
• So, to calculate the deviation of each data
  point from the mean we can find the average
  distance of each data from the mean
• A. D. 1 ? x mean
• N
• where N is number of values of a population

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Measures of Dispersion

• Examples
• See lecture
• However, calculating absolute distances in
  average deviation does not statistically/
  naturally represent the most appropriate spread
  of data.
• The more commonly used measure of spread
  is -----gt

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Measures of Dispersion

• Standard Deviation
• So, we now calculate standard deviation of each
  data point from the mean
• S.D. v 1/N ? (x - ?)2
• where N is number of values of a population, and
  ? is the population mean

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Measures of Dispersion

• Intuitively it makes sense to use the average
  deviation
• but this type of measure is not the most
  representative of the way naturally occurring
  data is spread out
• standard deviation is more representative of
  the spread of data which is normally distributed
• -----gt

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Measures of Dispersion
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Measures of Dispersion

• Standard deviation can be thought of as a typical
  distance from the mean.
• Populations are generally too big for us to
  calculate means and S.D.s
• So we need to calculate means and S.D.s of
  samples taken from populations

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Measures of Dispersion

• Hence we use sample standard deviation
• s v (1/(n-1) . ? (x - x)2)
• where n is sample size and x is sample mean

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Measures of Dispersion

• For sample S.D. we divide by n-1 since samples
  tend to be small in size and dividing by n biases
  the result of S.D. compared to population S.D.
• Example
• See lecture

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Percentiles, quartiles, interquartile range

• Another way of measuring dispersion is by a
  (percentage distance from the mean ?)
• (see all 3 stats books I have)

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Descriptive Statistics

• The End