Course Lecture: Analytic Techniques

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Course Lecture Analytic Techniques
Descriptive Statistics
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Terminology

• Population complete collection of elements to be
  studied
• Sample sub-collection of elements drawn from the
  population
• Parameters population characteristics,
• eg. Mean, Median, Variance
• Estimates sample statistics that estimate the
  population parameters, eg. Sample mean, sample
  median, sample variance.

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Terminology Levels of measurement

• Nominal
• names, labels or categories only
• Eg. Marital status, Gender,ethnic groups
• Ordinal
• can be arranged an some order
• (but the difference between data values is
  meaningless)
• Eg. Level of education, qualitative evaluation
  good, very good, excellent
• Interval
• like ordinal but the difference between values
  is meaningful
• (however there is no zero starting point)
• Eg. Dates
• Ratio
• like interval but there is a natural zero
  starting point
• (meaning that, say, doubling the number in
  meaningful)
• Eg. Distance, speed, income

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MEASURES OF LOCATION

• A measure of central tendency is probably the
  most common number used to describe data sets.
  It gives us some idea of what the average or
  middle or most occurring number in the data
  set is.
• There are many different measures, in this unit
  we will look at only the three most important
  measures

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MEASURES OF CENTRAL TENDENCY
1. Mean - µ or 2. Mode - most common 3.
Median - middle value i.e. the (N1)th value
when data arranged in order 2 They can be
calculated even when data has been grouped Note
Sample and population means are different.
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The arithmetic mean

• Usually referred to as simply the mean
• in a sample is called x bar, symbol
• in a population is called mu (pronounced Mew as
  in dew), symbol

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The arithmetic mean (Cont.)

• The mean is the sum of all the values, divided by
  the number of values. IE for a sample with n
  observations, the mean would be

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The arithmetic mean (Cont.)

• The sample mean is often used as an estimate of
  the population mean ?
• Because the mean is calculated by summing every
  observation, it is greatly affected by any
  extreme values, and can as such present a
  distorted representation of the data.

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The Median

• When the data are not unimodal and symmetrical
  (ie skewed), the median is preferred
• The median is the middle value when the data are
  arranged in order. IE there are an equal number
  of observations above and below the median

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The Median (Cont.)

• If there are an odd number of values, the median
  is the value of the middle observation.
  Otherwise it is somewhere between the two middle
  values, and generally calculated as the average
  of these two numbers

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The Median (Cont.)

• arrange the data in order (decreasing or
  increasing)
• locate the middle value using the formula
• middle value th value

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The Mode

• The mode is the value(s) that occurs most often
• Useful on Nominal scale data, where it is not
  possible to calculate the mean or median
• A distribution can have more than one mode (eg
  two modes bimodal)

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Measures of Central tendency Example

• Consider the following data
• 12 34 56 34 21 23 1 19 17 12 34 53
• Calculate the mean, median and mode

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Measures of Central tendency Example (Cont.)

• The mean.
• (12345634212311917123453)
  12
• 26.33

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Measures of Central tendency Example (Cont.)

• The Median
• middle value th value
• th 6.5th value
• average of 6th and 7th values
• in order
• 1 12 12 17 19 21 23 34 34 34 53 56
• Median (2123)/2 22

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Measures of Central tendency Example (Cont.)

• The mode
• number 1 12 17 19 21 23 34 53 56
• frequency 1 2 1 1 1 1 3 1 1
• Therefore, the mode is 34

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Mean, Median and Mode

• If the distribution is exactly symmetric, the
  mean, the median and the mode are exactly the
  same.
• If the distribution is skewed, the three measures
  differ.

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Which one to use?

• Different by definition
• Mean and median are unique, and only for
  quantitative variables.
• Mode is not unique.
• Mode is defined for categorical variables also.
• The choice depends on the shape of the
  distribution, the type of data and the purpose of
  your study
• Skewed
• Categorical
• Total quantity

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Measures of Dispersion

• A second important property of a distribution is
  a measure of dispersion. IE how variable the
  data are
• The four most commonly used measures are the
  range, variance, standard deviation and
  coefficient of variation
• We will also look at the Inter Quartile Range

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The Range

• The range is simply the difference between the
  highest and lowest values in a data set
  Range xmax - xmin

• The range however gives no indication of the
  dispersion of values between these two extreme
  values. IE there may be a lot of values clumped
  at either end of the distribution

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The Variance

• The two most commonly used measures which take
  into account all the data values are the variance
  and the standard deviation
• A data set that is more variable will have a
  larger variance than a data set that which is
  relatively homogeneous
• The variance is the sum of the squared deviations
  divided by the number of observations

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The variance (Cont.)

• Consider these data 5, 17, 12, 10
• The mean of the data is
• (5171210)/4 11
• A deviation is the distance of each observation
  from the mean

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The variance (Cont.)
Deviations

• For these data, the deviations are
• 5 - 11 -6
• 10 - 11 -1
• 12 - 11 1
• 17 - 11 6

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The variance (Cont.)

• We are interested in the squared deviations, so
  the numbers are squared
• Number Deviation Squared deviation
• 5 -6 36
• 10 -1 1
• 12 1 1
• 17 6 36

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The variance (Cont.)

• The squared deviations are then summed and
  divided by the number of observations to give the
  variance
• Variance (36 1 1 36) / 4
• 18.5
• The variance is hence the average squared
  deviation of the data

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The variance (Cont.)

• For a population, the variance is notated by and
  the formula

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The variance (Cont.)

• For a sample, the Variance is notated by s2 and
  given by the formula
• Note the subtle difference in these two formulas.
  Your calculator can calculate both these numbers
  in a matter of seconds

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The Standard Deviation

• The standard deviation is simple the ve square
  root of the variance. Hence for a population the
  standard deviation is and for s a sample,
  IE
• The standard deviation is in the same units as
  the mean.

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The coefficient of variation

• The coefficient of variation (CV) is a relative
  measure of variability which has no units and is
  generally expressed in terms of a percentage
• It is used for comparing data that are not
  measured using the same units, or when comparing
  data with quite different means
• It is simply the standard deviation divided by
  the mean

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The coefficient of variation (Cont.)

• The CV can only be calculated on data collected
  at the ratio level

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The Quartiles

• We can improve the description by also looking at
  the middle half of the data
• Recall that the Median is the middle value of the
  data set. IE the value that 50 of observations
  are greater than and 50 of observations are less
  than
• The quartiles are calculated in a similar fashion

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The Quartiles (Cont.)

• The first quartile lies one quarter of the way
  through the data. IE One quarter of the data
  values are less than the first quartile
• The third quartile lies three quarters of the way
  through the data. IE Three quarters of the data
  values are less than the third quartile

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The Quartiles (Cont.)

• EG Consider the following data (ordered)
• 2 3 5 9 12 17 23 29 31 32
  35
• There are 11 values, so the median is the 6th
  value, in this case 17. The first quartile is
  the middle value of the observations below the
  median,
• 2 3 5 9 12

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The Quartiles (Cont.)

• The third quartile is the middle value of the
  observations above the median,
• 23 29 31 32 35
• So, the data with Q1, M and Q3 are
• 2 3 5 9 12 17 23 29 31 32
  35
• And the middle 50 of data lie between Q1 and Q3.
  In this case, between 5 and 31

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The Quartiles (Cont.)

• The difference between the 1st and 3rd quartiles
  is called the Inter Quartile Range

Inter Quartile Range Q3-Q1
31-5 26
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Approximate statistics for grouped data

• When the data are given in a frequency
  distribution table, we cannot calculate the exact
  mean and standard deviation
• We can however calculate the approximate values

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Statistics for grouped data

• For the mean
• and for the variance

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Statistics for grouped data eg.

• Consider the following frequency table
• Class Interval Frequency
• 1 2 - 5 3
• 2 5 - 8 6
• 3 8 - 11 8
• 4 11 - 14 7
• 5 14 - 17 4
• 6 17 - 20 2

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Grouped data example (Cont.)

• Class Interval Frequency Midpoint fimi
  fimi2
• 1 2 - 5 3 3.5 10.5 36.75
• 2 5 - 8 6 6.5 39 253.5
• 3 8 - 11 8 9.5 76 722
• 4 11 - 14 7 12.5 87.5 1093.75
• 5 14 - 17 4 15.5 62 961
• 6 17 - 20 2 18.5 37 684.5
• totals 30 312 3751.5

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Grouped data example (Cont.)

• 17.47
• s 4.18