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 2

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Major Points Today

• Types of Measurement
• Summation Notation
• Organizing Data
• Stem and Leaf Displays
• Graphs
• Measures of Central Tendency

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Types of Measurement

• There are 4 types of measurement most often used
  in statistics
• Nominal
• Ordinal
• Interval
• Ratio

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Nominal Measurement

• Nominal Measurement the classification of
  measurements into a set of categories
• The numbers produced by nominal measurement are
  frequencies of occurrence in the categories
  (e.g., 22 ducks, 12 chickens, 2 geese, etc)

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Nominal Measurement cont.

• A second example is gender 2 categories, male
  and female
• Nominal measurement applies to qualitative
  variables - elements are assigned to a category
  because they possess one characteristic or
  another
• Nominal data is also termed qualitative data

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Ordinal Measurement

• Ordinal Measurement the rank ordering of
  elements on a continuum
• Ordinal measurement does not measure the amount
  of the variable - it represents the individuals
  placement in a continuum (or ranking e.g., the
  winner of a race is in first place)

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Ordinal Measurement cont.

• It is important to note that the amount of
  variable difference between rank position is not
  constant - the difference in amount of talent
  between the 1st and 2nd place finishers in a race
  cannot be assumed to be the same as the
  difference in amount of talent between the 5th
  and 6th place finishers
• Ordinal data can tell you that the person in 1st
  place finished before the person in 3rd place,
  but not by how much

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Interval Measurement

• Interval Measurement the assignment of numerical
  quantity to the variable in a way that
• the number assigned reflects the amount of the
  variable
• the size of the measurement unit remains constant
  
• and the zero point is defined arbitrarily and
  does not represent an absence of the property
  being measured

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Interval Measurement cont.

• The best example is temperature
• 40C represents how hot something is (the amount
  of heat it has)
• The unit of measurement (1C) represents the same
  amount of heat regardless of where it occurs in
  the range of measurement (the amount of change in
  temperature is the same between 25C - 26C and
  32C - 33C)
• The zero point (0C) is arbitrary - it represents
  the point at which water freezes, not the absence
  of temperature

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Interval Measurement cont.

• Interval measurement can contain negative
  numbers, whereas Nominal and Ordinal Measurement
  do not

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Ratio Measurement

• Ratio Measurement The assignment of numerical
  quantity to the variable in such a way that
• the number assigned reflects the amount of the
  variable
• the size of the measurement unit remains constant
• and the zero point represents an absence of the
  property being measured

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Ratio Measurement

• Good examples are time and length
• A ratio scale cannot produce negative numbers
• Interval and ratio measurement are equivalent
  for statistical purposes and are often referred
  to as one thing (interval/ratio data)

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Summation Notation

• We commonly use the letters X and Y to
  represent the variables we have measured
• Upper case Greek letter sigma (?) is known as
  the summation operator it means the sum of

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? Example

• Suppose we keep a record for 6 days of every time
  someone slips in the CAW Student Centre Cafeteria
  (represented by X), the data may look like this

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Data Example

• Day X
• Mon 10
• Tues 5
• Weds 12
• Thurs 11
• Fri 21
• Sat 28

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?X

• ?X means the sum of all the X scores, so that
• ?X X1 X2 X3 ... XN
• 10 5 12 11 21 28
• 87
• Note X1 means the first X score XN means
  the last X score

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(?X)2

• (?X)2 means the square of the sum (total all
  numbers within parentheses and then square), so
  that
• (?X)2 (X1 X2 X3 ... XN)2
• (10 5 12 11 21 28)2
• (87)(87)
• 7569

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?X 2

• ?X 2 means the sum of the squares (square each
  number and then sum), so that
• ?X 2 X1 2 X2 2 X3 2 ... XN 2
• 10 2 5 2 12 2 11 2 21 2 28 2
• 100 25 144 121 441 784
• 1615

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More Summation Notation

• Suppose you also keep track of the number of
  pieces of garbage dropped on the floor of the CAW
  Student Centre for the same days as above
  (variable Y) and the data were as follows

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Example Data

• Day X Y
• Mon 10 210
• Tues 5 160
• Weds 12 245
• Thurs 11 240
• Fri 21 340
• Sat 28 415

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?XY

• ?XY means the sum of the products
• ?XY (X1)(Y1) (X2)(Y2) (X3)(Y3) ...
  (XN)(YN)
• (10)(210) (5)(160) (12)(245)
  (11)(240) (21)(340) (28)(415)
• 2100 800 2940 2640 7140 11620
• 27240

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Organizing Data

• Frequency Distributions A frequency distribution
  is a table which shows the number of individuals
  or events that occurred at each measurement value
• this is the most common form of organizing data

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Frequency Distributions

• The following hypothetical frequency distribution
  shows the number of women in different majors at
  the University of Windsor
• Major of Women
• Art 15
• Biology 35
• Chemistry 34
• Music 85
• Psychology 97

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Frequency Distributions

• This frequency distribution organizes the data
  into nominal categories (by major)
• Frequency distributions can also organize data by
  points of measurement on a continuous variable,
  as follows

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Frequency Distributions

• Age of Students in 02-250
• Age Frequency
• 18 14
• 19 85
• 20 58
• 21 40
• 22 35
• 23 16
• 24 10
• 25 6
• 26 4

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Frequency Distributions

• Frequency distributions should not exceed 15 to
  20 lines, as the point is to summarize the data
  in a way that represents all the information
  concisely
• When there are more data than can be classified
  in 20 lines, the data can be grouped into score
  ranges known as class intervals, as in this
  example

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Class Interval Example

• Canada Population Estimates for the Year 2016 (in
  millions)
• Age Pop Age
  Pop
• 0 - 4 2.05 50 - 54 2.79
• 5 - 9 2.07 55 - 59 2.69
• 10 - 14 2.12 60 - 64 2.31
• 15 - 19 2.19 65 - 69 1.97
• 20 - 24 2.38 70 - 74 1.42
• 25 - 29 2.48 75 - 79 0.99
• 30 - 34 2.54 80 - 84 0.71
• 35 - 39 2.53 85 - 89 0.47
• 40 - 44 2.51 90 0.33
• 45 - 49 2.57
  

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Frequency Distributions cont.

• Looking at the frequency distribution tells us
• The most frequently occurring age is expected to
  be in the 50-54 age range (b/c this is the
  largest population estimate, 2.79 million).
• The age frequencies are expected to be fairly
  evenly distributed from 0 to 70 years old and
  then fall off
• The expected distributions of ages is not
  symmetrical very low (young) and high (old) ages
  do not occur with equal likelihood

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Frequency Distributions cont.

• Dividing the data into class intervals makes the
  data more accessible
• Data which has been divided into class intervals
  is sometimes referred to as grouped data

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Cumulative Frequency Distributions

• Frequency distributions can be made to contain
  more information, as when a column of cumulative
  frequencies is added
• Cumulative Frequency Distribution A table in
  which the frequency of individuals or events at
  each measurement value is added to previous
  frequencies so that each line reads as the total
  frequency of that and lower measurement values

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Cumulative Frequency Ex.

• Age of Students in 02-250
• Age Frequency Cumulative Frequency
• 18 14 14
• 19 85 99
• 20 58 157
• 21 40 197
• 22 35 232
• 23 16 248
• 24 10 258
• 25 6 264
• 26 4 268

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More Frequency Distributions

• Frequency distributions can also contain
  information about the percentages and cumulative
  percentages of observations at the various scores

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More Frequency Distributions

• Age of Students in 02-250
• Age Frequency Cumulative Cumulative
• Frequency
• 18 14 14 5.22 5.22
• 19 85 99 31.72 36.94
• 20 58 157 21.64 58.58
• 21 40 197 14.93 73.51
• 22 35 232 13.06 86.57
• 23 16 248 5.97 92.54
• 24 10 258 3.93 96.27
• 25 6 264 2.24 98.51
• 26 4 268 1.49 100.00
  

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Exact Limits

• All measurements are expressed in discrete units,
  such as seconds or centimeters
• No matter how small the unit of measurement, it
  is always possible to imagine finer measurement
• 1 cm 10 mm

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Exact Limits

• So, for continuous variables, any measure should
  be viewed as representing a range of values
• This range has a width equal to the unit of
  measurement used, and the boundaries of this
  range are the exact limits of the measure

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Exact Limits

• E.g., If we say an event lasted 12 seconds, we
  mean it is closer to 12 seconds than to 11 or 13
  seconds. A score of 12 represents a range of
  values. This range is one second wide (one unit
  of the measurement) and extends between 11.5 and
  12.5 seconds

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Exact Limits

• Exact limits identify the upper and lower ends of
  the range represented by the raw score and are
  the real boundaries of the measure in question

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Exact Limits

• Exact Limits Values one-half unit of measurement
  above and below the score or class interval.
  Exact limits are the boundaries of the range of
  values represented by the measure
• Some authors refer to exact limits as real limits

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Exact Limits Examples

• Measure Exact Limits
• 52 51.5 - 52.5
• 51 50.5 - 51.5
• 52.2 52.15 - 52.25
• 52.1 52.05 - 52.15

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Exact Limits Examples

• Measure Exact Limits
• 50.02 50.015 - 50.025
• 50.01 50.005 - 50.015
• Class Interval Exact Limits
• 50 - 54 49.5 - 54.5
• 55 - 59 54.5 - 59.5

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Stem-and-Leaf Displays

• Stem-and-Leaf Display partitions each score into
  a stem and a leaf and groups the scores
  according to common stems
• The Leaf is the rightmost digit
• The Stem is the digit (or digits) to the left
  of the leaf (the stem is 0 for 1 digit numbers)

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Stem-and-Leaf

• E.g.,
• Stem Leaf
• 4 0 4
• 54 5 4
• 123 12 3
• 123 4
• The numbers 24 and 26 have different leaves(4
  and
• 6) but the same stem (2)

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Stem-and-Leaf

• Consider this raw data and their stem-and-leaf
  plot

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Stem-and-Leaf

• stem leaf
• 3 6
• 4 477
• 5 05899
• 6 01225788
• 7 24559
• 8 578
• 9 2

Data 36, 44, 47, 47, 50, 55, 58, 59, 59, 60, 61,
62, 62, 65, 67, 68, 68, 72, 74, 75, 75, 79, 85,
87, 88, 92
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Stem-and-Leaf

• Or this example
• Data 102, 104, 115, 116, 116, 125, 127, 128,
  129, 129, 131, 136, 137, 145, 145
• stem leaf
• 10 24
• 11 566
• 12 57899
• 13 167
• 14 55

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Stem-and-Leaf

• Unlike frequency distributions, stem-and-leaf
  plots give an indication of the overall
  distribution of the scores (e.g., evenly spread
  or bunched, symmetrical or nonsymmetrical)
• Note Make sure you include every instance of a
  given value, e.g., if 57 occurs 3 times in the
  data set, this should be represented in the stem
  and leaf display with a stem of 5 and three 7s in
  the leaf.

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Graphs

• Graph refers to all manner of pictorial, or
  graphic, representation of data
• We will consider histograms and frequency
  polygons

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Graphs

• The horizontal axis (X axis) is labeled with
  units representing points of measurement and the
  vertical axis (Y axis) is labeled with values
  representing frequency of occurrence
• Histograms and frequency polygons are like
  2-dimensional representations of frequency
  distributions

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Histogram

• Histogram A graphic in which the horizontal axis
  identifies points of measurement, and the
  vertical axis represents frequency of occurrence
• Solid bars are used to represent the frequency at
  each point of measurement (a histogram is a bar
  graph)

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Age Data Histogram Example
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Frequency Polygon

• Frequency Polygon A graphic in which the
  horizontal axis identifies points of measurement,
  and the vertical axis represents frequency of
  occurrence (a frequency polygon is a line graph)

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Age Data Frequency Polygon
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Graphs cont.

• Both histograms and frequency polygons can be
  embellished by the simultaneous plotting of more
  than one variable, as shown next

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Graphs cont.
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Graphs cont.
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Describing Data

• Averages an average is a numerical value that
  indicates the middle point or central region of
  the raw data
• Averages are sometimes referred to as measures of
  central tendency

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Averages

• 3 statistics are commonly termed averages
• Mode
• Median
• Mean

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Mode

• Mode The most frequently occurring score
• A distribution with a single most frequently
  occurring score (one hump) is termed a unimodal
  (single mode) distribution
• A distribution with 2 values that share the
  quality of being most frequently occurring (2
  humps) is termed bimodal (2 modes)

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Mode Example

• Age of Students in 02-250
• Age Frequency
• 18 14
• 19 85 ? In this example, the Mode is
• 20 58 19 as it has the highest
• 21 40 frequency
• 22 35
• 23 16
• 24 10
• 25 6
• 26 4

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A la Mode

• The mode does not take into account all of the
  data - only the one most frequently occurring
  score
• The mode is the score with the highest bar in a
  histogram, or the highest point in a frequency
  polygon
• When the data are combined into class intervals,
  the mode is the mid-point of the class interval
  that contains the most scores

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Median

• Median The middle point of the distribution, or
  the score which bisects the distribution (divides
  it into upper and lower halves)

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Median

• If there are an ODD number of scores, the median
  is the middle score
• 1, 3, 6, 7, 8, 13, 15, 17, 18, 21, 23
• ?
• Median 13
• There are 5 scores above the median,
• and 5 below

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Median

• If there are an EVEN number of scores, the median
  is the midpoint between the two middle scores
• 1, 3, 6, 7, 8, 13, 15, 17, 18, 23
• ?
• Median (8 13)/2 10.5

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Median Notes

• NOTE!
• When determining the median, you must arrange the
  scores in ascending or descending order first!

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Steps to Finding the Median

1. Arrange data in ascending or descending order
2. Count the number of scores (N)
3. If there are an odd number of scores, find the
   middle point (the score where there are the same
   number of scores above and below it) - this is
   the median

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Steps to Finding the Median

• 4. If there are an even number of scores,
• find the 2 middle scores - add them,
• and divide by 2 - this is the median

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More Median

• When a distribution is viewed as area, the median
  divides the total area in half

50
50
Median
69
Median cont.

• The median is based on the value of one or two
  scores, and does not take into account all of the
  data
• When the data are grouped into class intervals,
  the median can be viewed as the midpoint of the
  class interval which contains the middle score
  (50th frequency). This is only a rough estimate

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Arithmetic Mean

• Arithmetic Mean the sum of the scores divided by
  the number of scores (what is generally thought
  of as the average)

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Mean

• The mean of a sample of X scores is symbolized as
  ? , which is said as X bar
• The mean of a population of X scores is
  symbolized by the Greek letter mu (µ)
• Greek letters tend to be used for parameters,
  while conventional letters are used for statistics

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Mean

• The algebraic definition of the population mean
  is as follows
• N is used to refer to the number of scores
• in the data set (termed population size)

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Sample Mean

• The algebraic definition of the sample mean is as
  follows
• n is used to refer to the number of scores
• in the data set (termed sample size)

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Mean cont.

• The algebraic formula for the sample and
  population mean is the same, (although some terms
  have different formulae for samples and
  populations)

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Mean cont.

• The mean is used as the measure of average almost
  exclusively (rather than the mode or median)
  because it is defined algebraically and considers
  all the raw scores in the data set

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Mean cont.

• In any group of scores, the sum of the deviations
  from the mean equals zero
• X X- ? n 6
• 3 3 - 5.50 -2.50 ? ?X/n
• 5 5 - 5.50 -0.50 ? 33/6
• 9 9 - 5.50 3.50 ? 5.50
• 2 2 - 5.50 -3.50
• 8 8 - 5.50 2.50
• 6 6 - 5.50 0.50
• ?X 33 ?(X- ?) 0.00

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Relative Characteristics of Averages

• If the distribution is symmetrical, the mean,
  median, and mode have the same value
• The longer tail of a non-symmetrical distribution
  pulls the mean more than the mode and median
• Therefore the mean is more effected by outliers
  (very large or very small data points) than are
  the mode and median

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Relative Characteristics of Averages

• Relative positions of the mean and median
• Note The mode is the highest point in the
  distribution