Frequency Distributions

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Frequency Distributions Central Tendency
Unit 2

• Homework 1, 3, 5-7, 9, 11, 13, 17, 21 (pp.
  52-54) 1-7, 9, 11, 13, 15, 17, 19, 21, 23 (pp.
  81-83)

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Organizing Data

• Describing distribution of variables
• enumeration list raw data
• Frequency distributions
• organize ---gt tables or graphs
• highlight important characteristics
• range, most frequent value

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Distributions as Tables

• f Frequency
• of times a value of variable occurs
• Sf n
• calculate proportions percentages
• Tabular frequency distributions
• ordered list of all values of variable their
  frequencies
• logical order (usually descending)

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Enumeration
of presentations to be able to recall 100
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Grouped Frequency Distribution

• Group by class intervals
• report f for intervals
• Lose information exact values
• General rules
• each interval same width
• consecutive do not overlap

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Distributions as graphs

• Summarizes data
• focus on clear communication
• see box 2.1 (p. 41)
• Bar Graphs
• nominal or ordinal data
• Histograms Frequency Polygons
• Interval/ratio data
• continuous discrete variables

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Bar Graphs
Ordinal
Nominal
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Histograms

• X-axis
• Class intervals of variables
• Y-axis
• Frequencies vertical bars

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Frequency polygons

• Frequency represented as points
• Contains same info as histogram

f
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Shapes of distributions

• Unimodal distribution
• single value is most frequent
• Bimodal
• 2 most frequently occurring values
• also multimodal

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Symmetry of distributions

• Symmetric
• if right side mirror-image of left
• Skewed - asymmetric
• a few extreme values
• Positively skewed right tail longer
• Negatively skewed left tail longer

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The Normal Distribution

• Bell-shaped
• 3 characteristics
• Unimodal
• symmetric
• asymptotic
• Characteristic of many distributions in nature

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Measures of Central Tendency
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3 essential characteristics of distributions

• Conveys most info for most distributions
• 1. Where is middle of distribution?
• 2. How wide is distribution?
• 3. What is shape of the distribution?

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Central Tendency

• Describes middle of distribution
• Mode
• All levels of measurement
• Median
• ordinal interval/ratio only
• Mean
• interval/ratio only

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Mode

• Most frequently occurring value
• appropriate for any measurement level
• nominal, ordinal, interval/ratio

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Mode

• Most frequently occurring value

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Median

• Midpoint of a data set
• values ½ smaller, ½ larger

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Finding the Median

• 1. List all values from largest---gt smallest
• if f3, then list 3 times
• 2. Odd entries median middle value
• 3. Even entries half way b/n middle 2 values

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Mean

• Average
• may not be actual value in data set
• Computing the mean
• statistic vs. parameter

Sample mean
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Calculating Weighted Means

• Combining means from 2 or more samples
• If same number scores in each group
• n1 n2 n3 .. nn

• If n1 ? n2

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Supplementary Material
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Relative Frequency

• Frequencies represented as proportion or
  percentage
• Large of data points
• Larger sample size ---gt more intervals
• narrower bars ---gt smoother curve

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Relative Frequency
f
of presentations
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Relative Frequency
f
of presentations
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10
20
30
40
50
60
70
80
90
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positively skewed mode lt median lt mean
f
10
20
30
40
50
60
70
80
90
X
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negatively skewed mode gt median gt mean
mode
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Bimodal (symmetric) distribution?