Basic Statistical Concepts

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Basic Statistical Concepts
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So, you have collected your data

• Now what?
• We use statistical analysis to
• test our hypotheses
• make claims about the population
• This type of analyses are called inferential
  statistics

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But, first we must

• Organize, simplify, and describe our body of
  data (distribution).
• These statistical techniques are called
  descriptive statistics

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Distributions

• Recall a variable is a characteristic that can
  take different values
• A distribution of a variable is a summary of all
  the different values of a variable
• Both type (each value) and token (each instance)

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Distribution
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7 Types 1,2,3,4,5,6,7
9 Tokens 1,2,2,3,4,4,5,6,7
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Distribution
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N 9
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Properties of a Distribution

• Shape
• symmetric vs. skewed
• unimodal vs. multimodal
• Central Tendency
• where most of the data are
• mean, median, and mode
• Variability (spread)
• how similar the scores are
• range, variance, and standard deviation

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Representing a Distribution

• Often it is helpful to visually represent
  distributions in various ways
• Graphs
• continuous variables (histogram, line graph)
• categorical variables (pie chart, bar chart)
• Tables
• frequency distribution table

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Distribution

• What if we collected 200 observations instead of
  only 9?

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Distribution
N 200
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Continuous Variables
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Categorical Variables
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Frequency Distribution Table
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Shape of a Distribution

• Symmetrical (normal)
• scores are evenly distributed about the central
  tendency (i.e., mean)

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Shape of a Distribution

• Skewed
• extreme high or low scores can skew the
  distribution in either direction

Negative skew
Positive skew
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Shape of a Distribution

• Unimodal
• Multimodal

Minor Mode
Major Mode
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Distribution

• So, ordering our data and understanding the
  shape of the distribution organizes our data
• Now, we must simplify and describe the
  distribution
• What value best represents our distribution?
  (central tendency)

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Central Tendency

• Mode the most frequent score
• good for nominal scales (eye color)
• a must for multimodal distributions
• Median the middle score
• separates the bottom 50 and the top 50 of the
  distribution
• good for skewed distributions (net worth)

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Central Tendency

• Mean the arithmetic average
• add all of the scores and divide by total number
  of scores
• This the preferred measure of central tendency
  (takes all of the scores into account)

population
sample
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Computing a Mean
10 scores 8, 4, 5, 2, 9, 13, 3, 7, 8, 5
?? 64
??/n 6.4
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Central Tendency

• Is the mean always the best measure of central
  tendency?
• No, skew pulls the mean in the direction of the
  skew

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Central Tendency and Skew
Mode
Median
Mean
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Central Tendency and Skew
Mode
Median
Mean
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Distribution

• So, central tendency simplifies and describes
  our distribution by providing a representative
  score
• What about the difference between the individual
  scores and the mean?
• (variability)

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Variability

• Range maximum value minimum value
• only takes two scores from the distribution into
  account
• easily influenced by extreme high or low scores
• Standard Deviation/Variance
• the average deviation of scores from the mean of
  the distribution
• takes all scores into account
• less influenced by extreme values

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Standard Deviation

• most popular and important measure of
  variability
• a measure of how far all of the individual
  scores in the distribution are from a standard
  (mean)

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Standard Deviation
low variability small SD
high variability large SD
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Computing a Standard Deviation
10 scores 8, 4, 5, 2, 9, 13, 3, 7, 8, 5
??/n 6.4
8 6.4 4 6.4 5 6.4 2 6.4 9 6.4
13 6.4 3 6.4 7 6.4 8 6.4 5
6.4
1.6 - 2.4 - 1.4 - 4.4 2.6 6.6 - 3.4 0.6
1.6 - 1.4
2.56 5.76 1.96 19.36 6.76 43.56 11.56
0.36 2.56 1.96
SS 96.4
10.71
3.27
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Standard Deviation

• In a perfectly symmetrical (i.e. normal)
  distribution 2/3 of the scores will fall within
  /- 1 standard deviation

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6.4
9.67
3.13
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Variance vs. SD

• So, SD simplifies and describes the distribution
  by providing a measure of the variability of
  scores
• If we only ever report SD, then why would
  variance be considered a separate measure of
  variability?
• Variance will be an important value in many
  calculations in inferential statistics

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Review

• Descriptive statistics organize, simplify, and
  describe the important aspects of a distribution
• This is the first step toward testing hypotheses
  with inferential statistics
• Distributions can be described in terms of
  shape, central tendency, and variability
• There are small differences in computation for
  populations vs. samples
• It is often useful to graphically represent a
  distribution