Edpsy 511

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Edpsy 511

• Exploratory Data Analysis
• Homework 1 Due 9/20

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Landmarks in the data

• Quartiles
• Were often interested in the 25th, 50th and 75th
  percentiles.
• 39, 38, 38, 36, 36, 31, 29, 29, 28, 19
• Steps
• First, order the scores from least to greatest.
• Second, Add 1 to the sample size.
• Why?
• Third, Multiply sample size by percentile to find
  location.
• Q1 (10 1) .25
• Q2 (10 1) .50
• Q3 (10 1) .75
• If the value obtained is a fraction take the
  average of the two adjacent X values.

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Box-and-Whiskers Plots (a.k.a., Boxplots)
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Shapes of Distributions

• Normal distribution
• Positive Skew
• Or right skewed
• Negative Skew
• Or left skewed

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How is this variable distributed?
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How is this variable distributed?
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How is this variable distributed?
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Descriptive Statistics
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Statistics vs. Parameters

• A parameter is a characteristic of a population.
• It is a numerical or graphic way to summarize
  data obtained from the population
• A statistic is a characteristic of a sample.
• It is a numerical or graphic way to summarize
  data obtained from a sample

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Types of Numerical Data

• There are two fundamental types of numerical
  data
• Categorical data obtained by determining the
  frequency of occurrences in each of several
  categories
• Quantitative data obtained by determining
  placement on a scale that indicates amount or
  degree

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Techniques for Summarizing Quantitative Data

• Frequency Distributions
• Histograms
• Stem and Leaf Plots
• Distribution curves
• Averages
• Variability

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Summary Measures
Summary Measures
Variation
Central Tendency
Quartile
Arithmetic Mean
Median
Mode
Range
Variance
Standard Deviation
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Measures of Central Tendency
Central Tendency
Average (Mean)
Median
Mode
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Mean (Arithmetic Mean)

• Mean (arithmetic mean) of data values
• Sample mean
• Population mean

Sample Size
Population Size
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Mean

• The most common measure of central tendency
• Affected by extreme values (outliers)

0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12
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Mean 5
Mean 6
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Mean of Grouped Frequency
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Weighted Mean

• A form of mean obtained from groups of data in
  which the different sizes of the groups are
  accounted for or weighted.

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Median

• Robust measure of central tendency
• Not affected by extreme values
• In an Ordered array, median is the middle
  number
• If n or N is odd, median is the middle number
• If n or N is even, median is the average of the
  two middle numbers

0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12
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Median 5
Median 5
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Mode

• A measure of central tendency
• Value that occurs most often
• Not affected by extreme values
• Used for either numerical or categorical data
• There may may be no mode
• There may be several modes

0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14
No Mode
Mode 9
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The Normal Curve
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Different Distributions Compared
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Variability

• Refers to the extent to which the scores on a
  quantitative variable in a distribution are
  spread out.
• The range represents the difference between the
  highest and lowest scores in a distribution.
• A five number summary reports the lowest, the
  first quartile, the median, the third quartile,
  and highest score.
• Five number summaries are often portrayed
  graphically by the use of box plots.

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Variance

• The Variance, s2, represents the amount of
  variability of the data relative to their mean
• As shown below, the variance is the average of
  the squared deviations of the observations about
  their mean

• The Variance, s2, is the sample variance, and is
  used to estimate the actual population variance,
  s 2

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Standard Deviation

• Considered the most useful index of variability.
• It is a single number that represents the spread
  of a distribution.
• If a distribution is normal, then the mean plus
  or minus 3 SD will encompass about 99 of all
  scores in the distribution.

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Calculation of the Variance and Standard
Deviation of a Distribution
Raw Score Mean X X (X X)2
85 54 31 961 80 54 26 676 70 54 16 256 60 54 6 36
55 54 1 1 50 54 -4 16 45 54 -9 81 40 54 -14 196 30
54 -24 576 25 54 -29 841

404.44
Standard deviation (SD)
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Comparing Standard Deviations
Data A
Mean 15.5 S 3.338
11 12 13 14 15 16 17 18
19 20 21
Data B
Mean 15.5 S .9258
11 12 13 14 15 16 17 18
19 20 21
Data C
Mean 15.5 S 4.57
11 12 13 14 15 16 17 18
19 20 21
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Facts about the Normal Distribution

• 50 of all the observations fall on each side of
  the mean.
• 68 of scores fall within 1 SD of the mean in a
  normal distribution.
• 27 of the observations fall between 1 and 2 SD
  from the mean.
• 99.7 of all scores fall within 3 SD of the mean.
  
• This is often referred to as the 68-95-99.7 rule

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Fifty Percent of All Scores in a Normal Curve
Fall on Each Side of the Mean
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Probabilities Under the Normal Curve
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Standard Scores

• Standard scores use a common scale to indicate
  how an individual compares to other individuals
  in a group.
• The simplest form of a standard score is a Z
  score.
• A Z score expresses how far a raw score is from
  the mean in standard deviation units.
• Standard scores provide a better basis for
  comparing performance on different measures than
  do raw scores.
• A Probability is a percent stated in decimal form
  and refers to the likelihood of an event
  occurring.
• T scores are z scores expressed in a different
  form (z score x 10 50).

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Probability Areas Between the Mean and Different
Z Scores
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Examples of Standard Scores