Basic Quantitative Methods in the Social Sciences (AKA Intro Stats)

1
Basic Quantitative Methods in the Social
Sciences(AKA Intro Stats)

• 02-250-01
• Lecture 3

2
Variation

• Variability The extent numbers in a data set are
  dissimilar (different) from each other
• When all elements measured receive the same
  scores (e.g., everyone in the data set is the
  same age, in years), there is no variability in
  the data set
• As the scores in a data set become more
  dissimilar, variability increases

3
Variation Range

• The range tells us the span over which the data
  are distributed, and is only a very rough measure
  of variability
• Range The difference between the maximum and
  minimum scores
• Example The youngest student in a class is 19
  and the oldest is 46. Therefore, the age range of
  the class is 46 19 27 years.

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Variation

• X
• 5 0.00 This is an example of data
• 5 0.00 with NO variability
• 5 0.00
• 5 0.00
• 5 0.00
• 25 n 5 5

5
Variation

• X
• 6 1.00 This is an example of data
• 4 -1.00 with low variability
• 6 1.00
• 5 0.00
• 4 -1.00
• 25 n 5 5

6
Variation

• X
• 8 3.00 This is an example of data
• 1 -4.00 with higher variability
• 9 4.00
• 5 0.00
• 2 -3.00
• 25 n 5 5

7
Note

• Lets say we wanted to figure out the average
  deviation from the mean. Normally, we would want
  to sum all deviations from the mean and then
  divide by n, i.e.,
• (recall look at your formula for the mean from
  last lecture)
• BUT We have a problem. will
  always add up to zero

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Variation

• However, if we square each of the deviations from
  the mean, we obtain a sum that is not equal to
  zero
• This is the basis for the measures of variance
  and standard deviation, the two most common
  measures of variability of data

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Variation

• X
  
• 8 3.00
  9.00
• 1 -4.00
  16.00
• 9 4.00
  16.00
• 5 0.00
  0.00
• 2 -3.00
  9.00
• 25 0.00
  50.00
• Note The is called the Sum of
  Squares

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Variance of a Population

• VARIANCE OF A POPULATION the sum of squared
  deviations from the mean divided by the number of
  scores (sigma squared)

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Population Standard Deviation

• Square root of the variance

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Sample Variance

• the sum of squared deviations from the mean
  divided by the number of degrees of freedom (an
  estimate of the population variance, n-1)

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Sample Standard Deviation

• Square root of the variance s2

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Why use Standard Deviation and not Variance!??!

• Normally, you will only calculate variance in
  order to calculate standard deviation, as
  standard deviation is what we typically want
• Why? Because standard deviation expresses
  variability in the same units as the data
• Example Standard deviation of ages in a class is
  3.7 years

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Variance

• The above formulae are definitional - they are
  the mathematical representation of the concepts
  of variance and standard deviation
• When calculating variance and standard deviation
  (especially when doing so by hand) the following
  computational formulae are easiest to use (trust
  us, they really are easier to use. You should
  however have a good understanding of the
  definitional formulae)

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Population Variance

• Computational Formula

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Population Standard Deviation

• Computational Formula

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Sample Variance

• Computational Formula

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Sample Standard Deviation

• Computational Formula

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Sample Standard Deviation Example

• Data
• X X2
• 8 64 n 5, 5
• 1 1
• 9 81
• 5 25 s2 175 (25)2/5
• 2 4 4
• X25 175 s2 12.50
• s s 3.54

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Computing Standard Deviation

• When calculating standard deviation, create a
  table that looks like this

X X2
X1 X12
X2 X22
X3 X32
X4 X42
2
X X2
2 4
4 16
7 49
9 81
22 2 150
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Computing Standard Deviation

• The values are then entered into the formula as
  follows
• 150
  222 484
• n
  4
• 
  n-1 3

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Computing Standard Deviation

• The values are then entered into the formula as
  follows
• 150
  222 484
• n
  4
• 
  n-1 3

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Computing Standard Deviation

• The values are then entered into the formula as
  follows

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Degrees of Freedom

• Degrees of Freedom The number of independent
  observations, or, the number of observations that
  are free to vary
• In our data example above, there are 5 numbers
  that total 25 ( 25, n 5)

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Degrees of Freedom

• Many combinations of numbers can total 25, but
  only the first 4 can be any value
• The 5th number cannot vary if 25
• This example has 4 degrees of freedom, as four of
  the five numbers are free to vary
• Sample standard deviation usually underestimates
  population standard deviation. Using n-1 in the
  denominator corrects for this and gives us a
  better estimate of the population standard
  deviation.

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Degrees of Freedom

• Degrees of freedom are usually
• n-1
• (the total of data points minus one)

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Time for an example

• Seven people were asked to rate the taste of
  McDonalds french fries on a scale of 1 to 10.
  Their ratings are as follows
• 8, 4, 6, 2, 5, 7, 7
• Calculate the population standard deviation
• Calculate the sample variance
• Class discussion When would this be a
  population, and when would it be a sample?

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Why is Standard Deviation so Important?

• What does the standard deviation really tell us?
• Why would a samples standard deviation be small?
• Why would a samples standard deviation be large?

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An Example

• Youre sitting in the CAW Student Centre with 4
  of your friends. A member of the opposite sex
  walks by, and you and your friends rate this
  persons attractiveness on a scale from 1 to 10
  (where 1very unattractive and 10drop dead
  gorgeous)

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Food for thought

• 1) What would it mean if all five of you rated
  this person a 9 on 10?
• 2) What would it mean if all five of you rated
  this person a 5 on 10?
• 3) What would it mean if the five of you produced
  the following ratings 1, 10, 2, 9, and 3 (note
  that the mean rating would be 5)?
• Why would scenario 3 happen instead of scenario
  2? What factors would lead to these different
  ratings?
• These questions form the basis of why
  statisticians like to explain variability

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An In-Depth Look at Scenario 3

• So if the five of you produced the following
  ratings 1, 10, 2, 9, and 3, what is the standard
  deviation of these ratings?
• Calculate!
• What is the standard deviation in Scenario 2?
  Calculate!

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Normal Distribution

• The normal distribution is a theoretical
  distribution
• Normal does not mean typical or average, it is
  a technical term given to this mathematical
  function
• The normal distribution is unimodal and
  symmetrical, and is often referred to as the Bell
  Curve

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Normal Distribution
Mean Median Mode
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Normal Distribution

• We study the normal distribution because many
  naturally occurring events yield a distribution
  that approximates the normal distribution

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Properties of Area Under the Normal Distribution

• One of the properties of the Normal Distribution
  is the fixed area under the curve
• If we split the distribution in half, 50 of the
  scores of the sample lie to the left of the mean
  (or median, or mode), and 50 of the scores lie
  to the right of the mean (or median, or mode)

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Properties of Area Under the Normal Distribution

• The mean, median, and mode always cut the Normal
  Distribution in half, and are equal since the
  Normal Distribution is unimodal and symmetrical

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Properties of Area Under the Normal Distribution
50 of scores
50 of scores
Mean, Median, Mode
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Properties of Area Under the Normal Distribution

• The entire area under the normal curve can be
  considered to be a proportion of 1.0000
• Thus, half, or .5000 of the scores lie in the
  bottom half (i.e., left of the mean) of the
  distribution, and half, or .5000 of the scores
  lie in the top half (i.e., right of the mean)

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Properties of Area Under the Normal Distribution
.5000 of scores
.5000 of scores
Mean, Median, Mode
41
Z-scores

• Z-Scores (or standard scores) are a way of
  expressing a raw scores place in a distribution

• Z-score formula

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Z-scores

• The mean and standard deviation are
  always notated in Greek letters
• Z-scores only reflect the data points position
  relative to the overall data set (so youre now
  considering the data as a population, as youre
  not looking to infer to a greater population)
• This means use the population formula for
  standard deviation rather than the sample formula
  whenever you calculate Z

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Z-scores

• A z-score is a better indicator of where your
  score falls in a distribution than a raw score
• A student could get a 75/100 on a test (75) and
  consider this to be a very high score

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Z-scores

• If the average of the class marks is 89 and the
  (population) standard deviation is 5.2, then the
  z-score for a mark of 75 would be
• 89 5.2
• z (75-89)/5.2
• z (-14)/5.2
• z -2.69

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Z-scores

• This means that a mark of 75 is actually 2.69
  standard deviations BELOW the mean
• The student would have done poorly on this test,
  as compared to the rest of the class

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Z-scores

• z 0 represents the mean score (which would be
  89 in this example)
• z lt 0 represents a score less than the mean
  (which would be less than 89)
• z gt 0 represents a score greater than the mean
  (which would be greater than 89)

47
Z-scores

• For any set of scores
• the sum of z-scores will equal zero
• ( 0.00)
• have a mean equal to zero
• ( 0.00)
• and a standard deviation equal to one
• ( 1.00)

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Z-scores

• A z-score expresses the position of the raw score
  above or below the mean in standard deviation
  sized units
• E.g.,
• z 1.50 means that the raw score is 1 and
  one-half standard deviations above the mean
• z -2.00 means that the raw score is 2 standard
  deviations below the mean

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Z-score Example

• If you write two exams, in Math and English, and
  get the following scores
• Math 70 (class 55, 10)
• English 60 (class 50, 5)
• Which test mark represents the better performance
  (relative to the class)?

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Z-score Example cont.

• Math mark
• z (70-55)/10
• z 1.50
• English mark
• z (60-50)/5
• z 2.00

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Z-score Example Illustration
Mean Z0.00
Z1.50
Z2.00
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The Answer

• Because Z 2.00 is greater than Z 1.50, the
  English class mark of 60 reflects a better
  performance relative to that class than does the
  Math class mark of 70

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Z-score Solving for X

• The z-score formula can be rearranged to solve
  for X

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Z-scores Solving for X

• This formula is used when you know the z-score of
  a data point, and want to solve for the raw score.

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Example

• E.g., if a class midterm exam has 65 and
  5, what exam mark has a z-score value of 1.25?
• X (1.25)(5) 65
• 6.25 65
• 71.25
• So, a person whose test is 1.25 standard
  deviations above the mean obtained a score of
  71.25

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Z-scores

• Z-score problems ask you to solve for X or solve
  for z
• Review both types of problems!