Chapter 2 online slides

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Chapter 2 online slides
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Chapter 2

• Frequency Distributions, Stem-and-leaf displays,
  and Histograms

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Where have we been?
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To calculate SS, the variance, and the standard
deviation find the deviations from ?, square and
sum them (SS), divide by N (?2) and take a square
root(?).
Example Scores on a Psychology quiz
Student John Jennifer Arthur Patrick Marie
X 7 8 3 5 7
X - ? 1.00 2.00 -3.00 -1.00 1.00
(X - ?)2 1.00 4.00 9.00 1.00 1.00
?2 SS/N 3.20
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Ways of showing how scores are distributed around
the mean

• Frequency Distributions,
• Stem-and-leaf displays
• Histograms

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Some definitions

• Frequency Distribution - a tabular display of the
  way scores are distributed across all the
  possible values of a variable
• Absolute Frequency Distribution - displays the
  count (how many there are) of each score.
• Cumulative Frequency Distribution - displays the
  total number of scores at and below each score.
• Relative Frequency Distribution - displays the
  proportion of each score.
• Relative Cumulative Frequency Distribution -
  displays the proportion of scores at and below
  each score.

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Example Data

• Traffic accidents by bus drivers
• Studied 708 bus drivers, all of whom had worked
  for the company for the past 5 years or more.
• Recorded all accidents for the last 4 years.
• Data looks like3, 0, 6, 0, 0, 2, 1, 4, 1, 6,
  0, 2

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Frequency Distributions

• of
• accidents
• 0
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11

Calculate relative frequency.
Divide each absolute frequency by the N.
For example, 117/708 .165
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What pops out of such a display

• 18 drivers (about 2.5 of the drivers) had 7 or
  more accidents during the 4 years just before the
  study.
• Those 18 drivers caused 147 of the 708 accidents
  or slightly over 20 (20.76) of the accidents.
• Maybe they should be given eye/reflex exams?
• Maybe they should be given desk jobs?

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Frequency Distributions

• of
• accidents
• 0
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11

Calculate relative frequency.
Divide each absolute frequency by the N.
For example, 117/708 .165
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What can you answer?

• of
• accidents
• 0
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11

Percent with at most 1 accident?
.165 .222 .387
.387 100 38.7
Proportion with 8 or more accidents?
.008 .001 .004 .001 .014
Percent with between 4 and 7 accidents?
.110 .062 .030 .010 .212 21.2
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Cumulative Frequencies
Cumulative Relative Frequency .165 .387 .610 .773
.883 .945 .975 .983 .993 .994 .999 1.000

• of acdnts
• 0
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11

Cumulative Frequency 117 274 432 547 625 669 690 6
97 703 704 707 708
Cumulative frequencies show number of scores at
or below each point.
Calculate by adding all scores below each point.
Cumulative relative frequencies show the
proportion of scores at or below each point.
Calculate by dividing cumulative frequencies by N
at each point.
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Grouped Frequencies

• Needed when
• number of values is large OR
• values are continuous.

• To calculate group intervals
• First find the range.
• Determine a good interval based on
• on number of resulting intervals,
• meaning of data, and
• common, regular numbers.
• List intervals from largest to smallest.

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Grouped Frequency Example
100 High school students average time in seconds
to read ambiguous sentences. Values range
between 2.50 seconds and 2.99 seconds.
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Determining i (the size of the interval)

• WHAT IS THE RULE FOR DETERMINING THE SIZE OF
  INTERVALS TO USE IN WHICH TO GROUP DATA?
• Whatever intervals seems appropriate to most
  informatively present the data. It is a matter of
  judgement. Usually we use 6 12 same size
  intervals each of which use intuitively obvious
  endpoints (e.g., 5s and 0s).

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Grouped Frequencies
Range 2.99 - 2.50 .49 .50
i .1 i 5
i .05 i 10
Reading Time 2.95-2.99 2.90-2.94 2.85-2.89 2.80-2.
84 2.75-2.79 2.70-2.74 2.65-2.69 2.60-2.64 2.55-2.
59 2.50-2.54
Frequency 9 7 20 11 10 10 4 8 10 11
Reading Time 2.90-2.99 2.80-2.89 2.70-2.79 2.60-2.
69 2.50-2.59
Frequency 16 31 20 12 21
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Either is acceptable.

• Use whichever display seems most informative.
• In this case, the smaller intervals and 10
  category table seems more informative.
• Sometimes it goes the other way and less detailed
  presentation is necessary to prevent the reader
  from missing the forest for the trees.

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How you organize the data is up to you.

• When engaged in this kind of thing, there is
  often more that one way to organize the data.
• You should organize the data so that people can
  easily understand what is going on.
• Thus, the point is to use the grouped frequency
  distribution to provide a simplified description
  of the data.

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Stem and Leaf Displays

• Used when seeing all of the values is important.
• Shows
• data grouped
• all values
• visual summary

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Stem and Leaf Display

• Reading time data

Reading Time 2.9 2.9 2.8 2.8 2.7 2.7 2.6 2.6 2.5 2
.5
Leaves 5,5,6,6,6,6,8,8,9 0,0,1,2,3,3,3 5,5,5,5,5,
6,6,6,7,7,7,7,7,7,7,8,9,9,9,9 0,0,1,2,3,3,3,3,4,4,
4 5,5,5,5,6,6,6,8,9,9 0,0,0,1,2,3,3,3,4,4 5,6,6,6
0,1,1,1,2,3,3,4 6,6,8,8,8,8,8,9,9,9 0,1,1,1,2,2,2,
4,4,4,4
i .05 i 10
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Stem and Leaf Display

• Reading time data

Reading Time 2.9 2.8 2.7 2.6 2.5
Leaves 0,0,1,2,3,3,3,5,5,6,6,6,6,8,8,9 0,0,1,2,3,
3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,7,7,7,7,8,9,9,9,
9 0,0,0,1,2,3,3,3,4,4,5,5,5,5,6,6,6,8,9,9 0,1,1,1,
2,3,3,4,5,6,6,6 0,1,1,1,2,2,2,4,4,4,4,6,6,8,8,8,8,
8,9,9,9
i .1 i 5
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Figural displays of frequency data
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Bar graphs

• Bar graphs are used to show frequency of scores
  when you have a discrete variable.
• Discrete data can only take on a limited number
  of values.
• Numbers between adjoining values of a discrete
  variable are impossible or meaningless.
• Bar graphs show the frequency of specific scores
  or ranges of scores of a discrete variable.
• The proportion of the total area of the figure
  taken by a specific bar equals the proportion of
  that kind of score.
• Note, in this context proportion and relative
  frequency are synonymous.

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The results of rolling a six-sided die 120 times
100 75 50 25 0
120 rolls and it came out 20 ones, 20 twos,
etc..
1 2 3 4 5
6
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Bar graphs and Histograms

• Use bar graphs, not histograms, for discrete
  data. (The bars dont touch in a bar graph, they
  do in a histogram.)
• You rarely see data that is really discrete.
• Discrete data are almost always categories or
  rankings.ANYTHING ELSE IS ALMOST CERTAINLY A
  CONTINUOUS VARIABLE.
• Use histograms for continuous variables.
• AGAIN, almost every score you will obtain
  reflects the measurement of a continuous variable.

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A stem and leaf display turned on its side shows
the transition to purely figural displays of a
continuous variable
9 9 9 9 7 7 7 7 7 7 7 6 6 6 5 5 5 5
4 4 4 3 3 3 3 2 1 0 0
4 4 4 4 2 2 2 1 1 1 0
9 9 8 6 6 6 5 5 5 5
4 4 3 3 3 2 1 0 0 0
9 9 9 8 8 8 8 8 6 6
9 8 8 6 6 6 6 5 5
4 3 3 2 1 1 1 0
3 3 3 2 1 0 0
6 6 6 5
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Histogram of reading times notice how the bars
touch at the real limits of each class!
20 18 16 14 12 10 8 6 4 2 0
F r e q u e n c y
Reading Time (seconds)
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Histogram concepts - 1

• Histograms must be used to display continuous
  data.
• Most scores obtained by psychologists are
  continuous, even if the scores are integers.
• WHAT COUNTS IS WHAT YOU ARE MEASURING, NOT THE
  PRECISION OF MEASUREMENT.
• INTEGER SCORES IN PSYCHOLOGY ARE USUALLY ROUGH
  MEASUREMENTS OF CONTINUOUS VARIABLES.

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An Example

• For example, while scores on a ten question
  multiple choice intro psych quiz ( 1, 2, 10) are
  integers, you are measuring knowledge, which is a
  continuous variable that could be measured with
  10,000 questions, each counting .001 points. Or
  1,000,000 questions each worth .00001 points.
• You measure at a specific level of precision,
  because thats all you need or can afford.
  Logistics, not the nature of the variable,
  constrains the measurement of a continuous
  variable.

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Histogram concepts - 2

• If you have continuous data, you can use
  histograms, but remember real class limits.
• Histograms can be used for relative frequencies
  as well.
• Histograms can be used to describe theoretical
  distributions as well as actual distributions.

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What are the real limits of the fifth class? The
highest class?
20 18 16 14 12 10 8 6 4 2 0
F r e q u e n c y
Real limits of the fifth class are ???? - ????
Real limits of the highest class are ???? - ????.

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Real limits of the fifth class are 2.695-2.745
Real limits of the highest class are 2.945 -
2.995
20 18 16 14 12 10 8 6 4 2 0
F r e q u e n c y
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Displaying theoretical distributions is the most
important function of histograms.

• Theoretical distributions show how scores can be
  expected to be distributed around the mean.

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TYPES OF THEORETICAL DISTRIBUTIONS

• Distributions are named after the shapes of their
  histograms. For psychologists, the most important
  are
• Rectangular
• J-shaped
• Bell (Normal)
• t distributions - Close to Bell shaped, but a
  little flatter

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Rectangular Distribution of scores
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The rectangular distribution is the know
nothing distribution

• Our best prediction is that everyone will score
  at the mean.
• But in a rectangular distribution, scores far
  from the mean occur as often as do scores close
  to the mean.
• So the mean tells us nothing about where the next
  score will fall (or how the next person will
  behave).
• We know nothing in that case.

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Flipping a coin Rectangular distributions are
frequently seen in games of chance, but rarely
elsewhere.
100 75 50 25 0
100 flips - how many heads and tails do you
expect?
Heads Tails
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Rolling a die
100 75 50 25 0
120 rolls - how many of each number do you expect?
1 2 3 4 5
6
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What happens when you sample two scores at a time?

• All of a sudden things change.
• The distribution of scores begins to resemble a
  normal curve!!!!
• The normal curve is the we know something
  distribution, because most scores are close to
  the mean.

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Rolling 2 dice
Dice Total 1 2 3 4 5 6 7 8 9 10 11 12
Look at the histogram to see how this resembles a
bell shaped curve.
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Rolling 2 dice
100 90 80 70 60 50 40 30 20 10 0
360 rolls
1 2 3 4 5 6 7 8 9 10
11 12
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Normal Curve
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J Curve
Occurs when socially normative behaviors are
measured.
Most people follow the norm, but there are
always a few outliers.
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What does the J shaped distribution represent?

• The J shaped distribution represents situations
  in which most everyone does about the same thing.
  These are unusual social situations with very
  clear contingencies.
• For example, how long do cars without handicapped
  plates park in a handicapped spot when there is a
  cop standing next to the spot.
• Answer Zero minutes!
• So, the J shaped distribution is the we know
  almost everything distribution, because we can
  predict how a large majority of people will
  behave.

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When do you get a J shaped distribution?
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When do you get a J shaped distribution?
Occurs when socially normative behaviors are
measured.
Most people follow the norm, but there are
always a few outliers.
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Principles of Theoretical Curves

• Expected frequency Theoretical relative
  frequency X N
• Expected frequencies are your best estimates
  because they are closer, on the average, than any
  other estimate when we square the difference
  between observed and predicted frequencies.
• Law of Large Numbers - The more observations that
  we have, the closer the relative frequencies we
  actually observe should come to the theoretical
  relative frequency distribution.