Describing and Exploring Data

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Chapter 2

• Describing and Exploring Data

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Describing and Exploring Data

• Once a bunch of data has been collected, the raw
  numbers must be manipulated in some fashion to
  make them more informative.
• Several options are available including plotting
  the data or calculating descriptive statistics.

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Plotting Data

• Often, the first thing one does with a set of raw
  data is to plot frequency distributions.
• Usually this is done by first creating a table of
  the frequencies broken down by values of the
  relevant variable, then the frequencies in the
  table are plotted in a histogram.

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TABLE Example Your age as
estimated by the questionnaire from the first
class.

• Note The frequencies in the adjacent table were
  calculated by simply counting the number of
  subjects having the specified value for the age
  variable.

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Histogram
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Grouping Data

• Plotting is easy when the variable of interest
  has a relatively small number of values (like our
  age variable did).
• However, the values of a variable are sometimes
  more continuous, resulting in uninformative
  frequency plots if done in the above manner.

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Grouping Data

• For example, our weight variable ranges from 100
  lb. to 200 lb. If we used the previously
  described technique, we would end up with 100
  bars, most of which with a frequency less than 2
  or 3 (and many with a frequency of zero).
• We can get around this problem by grouping our
  values into bins. Try for around 10 bins with
  natural splits.

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TableExample Binning our weight variable.
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Histogram
Check out this demo which clearly shows how the
width of the bin that you select can clearly
affect the look of the data Here is another
similar demonstration of the effects of bin width

• See section in text on cumulative frequency
  distributions

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Stem Leaf Plots

• If values of a variable must be grouped prior to
  creating a frequency plot, then the information
  related to the specific values becomes lost in
  the process (i.e., the resulting graph depicts
  only the frequency values associated with the
  grouped values).
• However, it is possible to obtain the graphical
  advantage of grouping and still keep all of the
  information if stem leaf plots are used.

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Stem Leaf Plots

• These plots are created by splitting a data point
  into that part associated with the group and
  that associated with the individual point.
• For example, the numbers 180, 180, 181, 182, 185,
  186, 187, 187, 189 could be represented as
• 18 001256779

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Thus, we could represent our weight data in the
following stem leaf plot
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Stem leaf plots are especially nice for
comparing distributions
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Terminology Related to Distributions

• Often, frequency histograms tend to have a
  roughly symmetrical bell-shape and such
  distributions are called normal or gaussion.

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Terminology Related to Distributions

• Sometimes, the bell shape is not symmetrical.
• The term positive skew refers to the situation
  where the tail of the distribution is to the
  right, negative skew is when the tail is to the
  left.

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Example Pizza Data.
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Notation Variables

• When we describe a set of data corresponding to
  the values of some variable, we will refer to
  that set using an uppercase letter such as X or
  Y.
• When we want to talk about specific data points
  within that set, we specify those points by
  adding a subscript to the uppercase letter like
  X1.

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For Example

• 5, 8, 12, 3, 6, 8, 7
• X1, X2, X3, X4, X5, X6, X7

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Summation

• The Greek letter sigma, which looks like ?, means
  add up or sum whatever follows it.
• Thus, ?Xi, means add up all the Xis.
• If we use the Xis from the previous example, ?Xi
  49 (or just ?X).

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Summation

• Note, that sometimes the ? has number above and
  below if. These numbers specify the range over
  which to sum.
• For example, if we again use the the Xis from the
  previous example, but now limit the summation
• ?Xi 34

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Nasty Example
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Nasty Example . . .continued

• ?X
• ?Y
• ?(X-Y)
• ?X2
• (?X)2

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Your turn

• ?(XY)
• (?(X-Y))2
• ?(X2-Y2)

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Double Subscripts

• Sometimes things are made more complicated
  because capital letters (e.g., X) are sometimes
  used to refer to entire data sets (as opposed to
  single variables) and multiple subscripts are
  used to specify specific data points.

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X24 3??X or ??Xij 61
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Measures of Central Tendency

• While distributions provide an overall picture of
  some data set, it is sometimes desirable to
  represent the entire data set using descriptive
  statistics.
• The first descriptive statistics we will discuss,
  are those used to indicate where the centre of
  the distribution lies.

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The Mode

• There are, in fact, three different measures of
  central tendency.
• The first of these is called the mode.
• The mode is simply the value of the relevant
  variable that occurs most often (i.e., has the
  highest frequency) in the sample.

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The Mode

• Note that if you have done a frequency histogram,
  you can often identify the mode simply by finding
  the value with the highest bar.
• However, that will not work when grouping was
  performed prior to plotting the histogram
  (although you can still use the histogram to
  identify the modal group, just not the modal
  value).

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Finding the mode

• Create a non-grouped frequency table as described
  previously, then identify the value with the
  greatest frequency.
• Example Class height.

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The Median

• A second measure of central tendency is called
  the median.
• The median is the point corresponding to the
  score that lies in the middle of the distribution
  (i.e., there are as many data points above the
  median as there are below the median).

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The Median

• To find the median, the data points must first be
  sorted into either ascending or descending
  numerical order.
• The position of the median value can then be
  calculated using the following formula

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Examples

• 1) If there are an odd number of data points
• (1, 3, 3, 4, 4, 5, 6, 7, 12)
• 2) If there are an even number of data points
• The median is the item in the fifth position of
  the
• ordered data set, therefore the median is 4.

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The Mean

• Finally, the most commonly used measure of
  central tendency is called the mean (denoted for
  a sample, and for a population).
• The mean is the same of what most of us call the
  average, and it is calculated in the following
  manner

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The Mean

• For example, given the data set that we used to
  calculate the median (odd number example), the
  corresponding mean would be
• Similarly, the mean height of our class,
• as indicated by our sample, is

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Mode vs. Median vs. Mean

• In our height example, the mode and median were
  the same, and the mean was fairly close to the
  mode and median.
• This was the case because the height distribution
  was fairly symmetrical.
• However, when the underlying distribution is not
  symmetrical, the three measures of central
  tendency can be quite different.

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• This raises the issue of which measure is best.
• Note that if you were calculating these values,
  you would show all your steps (its good to be
  prof!).

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Some Visual Demos
Here is a demonstration that allows you to change
a frequency histogram while simultaneously noting
the effects of those changes on the mean versus
the median. As you use the demo, you should
easily be able to think about how these changes
are also affecting the mode, right?
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Measures of Variability

• In addition to knowing where the centre of the
  distribution is, it is often helpful to know the
  degree to which individual values cluster around
  the centre.
• This is known as variability.

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Range

• There are various measures of variability, the
  most straightforward being the range of the
  sample
• Highest value minus lowest value
• While range provides a good first pass at
  variance, it is not the best measure because of
  its sensitivity to extreme scores (see text).

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The Average Deviation

• Another approach to estimating variance is to
  directly measure the degree to which individual
  data points differ from the mean and then average
  those deviations.
• That is

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The Average Deviation

• However, if we try to do this with real data, the
  result will always be zero
• Example (2,3,4,4,6,6,12)

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The Mean Absolute Deviation (MAD)

• One way to get around the problem with the
  average deviation is to use the absolute value of
  the differences, instead of the differences
  themselves.
• The absolute value of some number is just the
  number without any sign
• For Example -3 3

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The Mean Absolute Deviation (MAD)

• Thus, we could re-write and solve our average
  deviation question as follows
• The data set in question has a mean of 5 and a
  mean absolute deviation of 2.

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The Variance

• Although the MAD is an acceptable measure of
  variability, the most commonly used measure is
  variance (denoted s2 for a sample and ?2 for a
  population) and its square root termed the
  standard deviation (denoted s for a sample and ?
  for a population).

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The Variance

• The computation of variance is also based on the
  basic notion of the average deviation however,
  instead of getting around the zero problem by
  using absolute deviations (as in MAD), the zero
  problem is eliminating by squaring the
  differences from the mean.
• Specifically

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Alternate formula for s2 and s

• The definitional formula of variance just
  presented was
• An equivalent formula that is easier to work with
  when calculating variances by hand is
• Although this second formula may
  look more intimidating, a few examples
  will show you that it is actually easier to
  work with (as youll see in assignment 2).

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Visualizing Means and Standard Deviations
This demonstration allows you to play with the
mean and standard deviation of a distribution.
Note that changing the mean of the distribution
simply moves the entire distribution to the left
or right without changing its shape. In
contrast, changing the standard deviation alters
the spread of the data but does not affect where
the distribution is centered Run demo
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Estimating Population Parameters

• So, the mean (X) and variance (s2) are the
  descriptive statistics that are most commonly
  used to represent the data points of some sample.
• The real reason that they are the preferred
  measures of central tendency and variance is
  because of certain properties they have as
  estimators of their corresponding population
  parameters and ?2.

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Estimating Population Parameters

• Four properties are considered desirable in a
  population estimator sufficiency, unbiasedness,
  efficiency, resistance.
• Both the mean and the variance are the best
  estimators in their class in terms of the first
  three of these four properties.
• To understand these properties, you first need to
  understand a concept in statistics called the
  sampling distribution

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Sampling Distribution Demo
We will discuss sampling distributions off and on
throughout the course, and I only want to touch
on the notion now. Basically, the idea is this
in order to exam the properties of a statistic we
often want to take repeated samples from some
population of data and calculate the relevant
statistic on each sample. We can then look at
the distribution of the statistic across these
samples and ask a variety of questions about
it. Check out this demonstration which I hope
makes the concept of sampling distributions more
clear.
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Properties of a Statistic

• 1) Sufficiency
• A sufficient statistic is one that makes use of
  all of the information in the sample to estimate
  its corresponding parameter.

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Estimating Population Parameters

• 2) Unbiasedness
• A statistic is said to be an unbiased estimator
  if its expected value (i.e., the mean of a number
  of sample means) is equal to the population
  parameter it is estimating.
• Explanation of N-1 in s2 formula.

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Assessing the Bias of an Estimator

• Using the procedure, the mean can be shown to be
  an unbiased estimator (see p 47).
• However, if the more intuitive formula for s2 is
  used
• it turns out to underestimate ?2

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Assessing the Bias of an Estimator

• This bias to underestimate is caused by the act
  of sampling and it can be shown that this bias
  can be eliminated if N-1 is used in the
  denominator instead of N.
• Note that this is only true when calculating s2,
  if you have a measurable population and you want
  to calculate ?2, you use N in the denominator,
  not N-1.

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Degrees of Freedom

• The mean of 6, 8, 10 is 8.
• If I allow you to change as many of these numbers
  as you want BUT the mean must stay 8, how many of
  the numbers are you free to vary?

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Degrees of Freedom

• The point of this exercise is that when the mean
  is fixed, it removes a degree of freedom from
  your sample -- this is like actually subtracting
  1 from the number of observations in your sample.
• It is for exactly this reason that we use N-1 in
  the denominator when we calculate s2 (i.e., the
  calculation requires that the mean be fixed first
  which effectively removes -- fixes -- one of the
  data points).

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Estimating Population Parameters

• 3) Efficiency
• The efficiency of a statistic is reflected in the
  variance that is observed when one examines the
  means of a bunch of independently chosen samples.
  The smaller the variance, the more efficient the
  statistic is said to be.

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Estimating Population Parameters

• 4) Resistance
• The resistance of an estimator refers to the
  degree to which that estimate is effected by
  extreme values.
• As mentioned previously, both X and s2 are
  highly sensitive to extreme values.

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Estimating Population Parameters

• 4) Resistance
• Despite this, they are still the most commonly
  used estimates of the corresponding population
  parameters, mostly because of their superiority
  over other measures in terms sufficiency,
  unbiasedness, efficiency.