Types of Distributions Frequency Distribution

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Types of DistributionsFrequency Distribution

• Frequency distribution
• Showing what you have
• A way to illustrate how many of each thing.

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Types of DistributionsFrequency Distribution
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Types of DistributionsNormal Distribution

• Normal distribution
• Also known as the bell-shaped curve
• An illustration of the expectation of what most
  types of data will look like
• A few data points at each extreme
• Most data points in the middle area

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Types of DistributionsNormal Distribution
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Types of DistributionsPositively Skewed
Distribution

• Not all data are created equal
• Positive skew
• Many data points near the origin of the graph

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Types of DistributionsNegatively Skewed
Distribution

• Negative skew
• Many data points away from the origin of the
  graph

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Types of DistributionsBimodal Distribution

• Bimodal
• Two areas under the curve with many data points

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Types of DistributionsNon-normal Distributions

• Nonnormal distributions
• But not abnormal
• Platykurtic flat like a plate

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Types of DistributionsNon-normal Distributions

• Leptokurtic up down (like leaping)
• Bimodal lumpy

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Grouping data

• A way of organizing data so that they are
  manageable.
• Which is easier to understand?
• 3, 1, 7, 4, 1, 2, 3, 5, 4, 9
• or
• 1, 1, 2, 3, 3, 4, 4, 5, 7, 9

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Grouping dataTips for grouping data

• Tips for grouping lots of data
• Choose interval widths that reduce your data to 5
  to 10 intervals.

5
10
15
20
25
30
35
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Grouping dataTips for grouping data

• Choose meaningful intervals.
• Which is easier to understand at a glance?

5
10
15
20
25
30
35
or
4
7
10
13
16
19
22
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Grouping dataTips for grouping data

• Interval widths must be the same.

5
10
15
20
25
30
35
NOT
5
10
20
22
30
33
35
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Grouping dataTips for grouping data

• Intervals cannot overlap.

5-10
11-15
16-20
21-25
26-30
31-35
36-40
NOT
5-10
10-15
14-20
20-26
25-30
30-35
35
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Grouping dataAn example

• The data are displayed using
• A frequency table of individual data points
• A frequency table by intervals
• Graph of data by intervals

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Grouping dataAn example
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Grouping dataAn example
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Grouping dataAn example
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Measures of Central TendencyChoosing the right
measure

• Normal distribution
• Mean median/mode
• Median mean/mode
• Mode mean/median
• They all work.
• Pick the one that fits the need.

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Measures of Central TendencyChoosing the right
measure

• Positively skewed
• Mean little high
• Median middle score
• Mode little low
• Median works best

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Measures of Central TendencyChoosing the right
measure

• Negatively skewed
• Mean too low
• Median middle score
• Mode little high
• Median works best

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Measures of Central TendencyChoosing the right
measure

• What does a community make?

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50
100
500
750
30
35
40
45
20
25
70
Mode
Median
Mean
one family
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Klinkers and outliers

• Klinkers
• Something is wrong
• Broken equipment, language problem, etc.
• The scores/values do not legitimately represent
  anything.
• 0 because the equipment broke.
• Extra low score b/c participant cannot read
  English.
• Dont use the value

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Klinkers and outliers

• Outliers
• A way out there score, but nothing is broke
• Extra high score of intelligence
• Bill Gatess salary
• Extra low heart rate
• Conservative statisticians say keep it.

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Measures of variation

• Measures of variation
• How far the numbers are scattered around a center
  point.
• Range
• Standard deviation
• Variance

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Measures of variation Standard deviation

• Standard deviation
• An approximate picture of the average amount that
  each number in a set of number differs from the
  mean.

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Measures of variation Standard deviation

• What is it good for?
• The empirical rule If the data are normally
  distributed, then . . .
• Approximately 68 of all numbers in a set will
  fall within one standard deviation of the mean.
• Approximately 95 of all numbers in a set will
  fall within two standard deviations of the
  mean.
• Approximately 98 of all numbers in a set will
  fall within three standard deviations of the
  mean.

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Measures of variation Variance

• Variance
• Average of the square of the distances
  (deviations) of a set of numbers from their mean.
• In other words, the standard deviation without
  taking the square root.

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Sampling Distribution of Means

• Theoretical frequency distribution
• If one repeatedly drew equal-sized samples from
  the population, measured them on some trait or
  characteristic, and took the means of each of
  those samples, one could plot a distribution of
  those means.
• Resulting distribution is the sampling
  distribution of means.

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Sampling Distribution of Means

• Standard error of the mean
• Standard deviation of the sampling distribution.
• As sample size increases, standard error of the
  mean becomes smaller.

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Sampling Distribution of Means

• Whats it used for?
• Central Limit Theorem
• If a population is normally distributed on some
  trait or characteristic, then the sampling
  distribution of the means is also normally
  distributed.
• Even if scores are not normally distributed in
  the population, the sampling distribution of
  means will be normally distributed if the sample
  sizes are large.

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Use of the Sampling Distribution

• The sampling distribution of the means serves as
  the basis for many of the inferential statistics
  to be examined later in this course.