CSEM03 REPLI

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CSEM03REPLI

• Research and the use of statistical tools

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Objectives

• At the end of this lecture you will be able to
• Be able to plot graphs and know your way around
  SPSS
• Describe the shape of normal and non-normal
  distributions
• Describe the characteristics of a normal and
  non-normal distribution
• Mode, median, mean, standard deviation

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The SPSS statistical tool

• Recommended text
• Andy Fields Field, A (2005) Discovering
  Statistics using SPSS, 2nd Edition, Sage
  Publishing, London (ISBN 0-7619-4452-4)
• You can get SPSS for your own machine for 5 from
  the LRC

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Simple statistical models mean, sum of squares,
variance and standard deviation

• Mean
• We can consider the mean of a sample as one of
  the simplest statistical models because it
  represents a summary of the data.
• For example
• How may CDs does a group of students own?
• If we take 5 students then the numbers of CDs
  owned respectively are 1,2,3,4,6.
• The mean is the sum of these (?) divided by the
  number of students.
• Mean 16 3.2
• 5
• This is a theoretical mean, you cannot have .2 of
  a CD.
• How well does mean represent the data?
• What are the differences between the observed and
  the mean?

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Differences between the mean and the observed data
Figure 1 Differences between observed no of CDs
and the mean
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Cancelling out the errors in the data (deviation
from the mean)

• xi - 1-3.2 -2.2
• where xi first data point x1
• so x2 - 2 3.2 -1.2 etc
• The deviances are -2.2, -1.2, -0.2, 0.8 and
  2.8
• Total error ? xi - (sum of deviances) 0

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Sum of Squared errors (SS)

• So there is no total between our model and the
  observed data. Some errors are negative, some are
  positive but they cancel each other out. To avoid
  the problem of knowing the direction of error (eg
  in a large dataset) we square each error (a
  negative squared becomes positive).
• This is called the Sum of Squared errors (SS) and
  is a good measure of the accuracy of our model.
  This however depends on the amount of data
  collected and the more data points the higher the
  SS. To overcome this we average the error by
  dividing SS by the number of observations N.

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• A more useful statistic is to use the error in
  the sample to estimate the error in the
  population this is done by dividing SS by the
  no of observations 1.
• This measure is known as the variance

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Variance and Standard Deviation

• Variance s2 SS ? (xi - )2
• N-1
  N-1
• ? ( xi - )2 4.84 1.44 0.04 0.64 7.84
  14.8
• ? (xi - )2 14.8 3.7
• N-1 4
• From this statistic we can derive a very useful
  measure called Standard Deviation (SD)
• SD ?Variance ? 3.7 1.92

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Levels of data

• The type of data you collect will depend on the
  design of your study. Data can be measured on
  different scales
• Interval data
• These data are measured on a scale along which
  intervals are equal. For example if you record
  ratings of a pop video on a range of 1 to 5 the
  change between each number should be equal.
• Categorical data
• These are any variables that are made up of
  objects/ entities. For example the UK degree
  classification system comprises 1, 21, 22, 3,
  pass or fail. The interval between each class is
  not equal.
• Nominal data
• This is where numbers can represent names e.g.
  the numbers on the back of football shirts
  denotes the type of player ( 1 goalkeeper).
• .
• Measurement data
• The objects being studied are measured on a
  quantitative scale. With discrete measurement
  data only certain values are possible The data
  can be discrete or continuous. Examples of
  continuous measurement data are age, height,
  cholesterol level.
• Ordinal data
• A type of categorical data where the order is
  important e.g Degree classification, seriousness
  of illness.

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Median

• The median is the "Middle value" of a list. The
  smallest number such that at least half the
  numbers in the list are no greater than it. If
  the list has an odd number of entries, the median
  is the middle entry in the list after sorting the
  list into increasing order.
• If the list has an even number of entries, the
  median is equal to the sum of the two middle
  (after sorting) numbers divided by two.
• The median can be estimated from a histogram by
  finding the smallest number such that the area
  under the histogram to the left of that number is
  50

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Mode

• For lists, the mode is the most common (frequent)
  value. A list can have more than one mode. In a
  histogram, a mode is the most frequently
  occurring interval (seen as a bump).

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Normal distribution example (scores in a test)
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Positive skew
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Standard deviation ?
Mean ?
34.1
34.1
13.6
13.6
0.1
0.1
2.1
2.1
-3?
-2?
-1?
1?
2?
3?
?
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Find the mean, median, mode, and range of these
data.

• ExampleThree dice are rolled 12 times. The sum
  of the numbers after each roll is recorded
  below
• Numbers rolled
• 12, 11, 3 , 7, 4, 4, 17, 13, 12, 5, 8, 12
• Step 1 Rearrange the data elements.3, 4, 4, 5,
  7, 8, 11, 12 ,12, 12, 13, 17Step 2 Find the
  mean.Add all the numbers and divide by 12
• 116/12 9.7Step 3 Find the median.The
  sample size is even, for there are 12 data
  elements.The median is the average value of the
  sixth and the seventh elements.median
  (811)/2 9.5

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• Step 4 Find the mode.The number 3 occurs
  once.The number 4 occurs twice.The number 5
  occurs once.The number 7 occurs once.The number
  8 occurs once.The number 11 occurs once.The
  number 12 occurs three times.The number 13
  occurs once.The number 17 occurs once.mode
  12Step 5 Find the range.The highest value is
  17.The lowest value is 3.range 17-3 14

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Examples for practice

• The weekly salaries of six employees at McDonalds
  are70, 100, 90, 80, 70, 100. For these
  six salaries, find (a) the mean (b) the median
  (c) the mode
• List the data in order 70,70,80,90,100,100
                                                    
                               Mean 60 70 80
  90 90 120 510 85                      
                6                         6     
  Median    60, 70, 80, 90 , 90 , 120 The two
  numbers that fall in the middle need to
  beaveraged.     80 90 85
•                            2 Mode   The number
  that appears the most is 90