Describing Variation in Data Chapter 9

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Describing Variation in DataChapter 9

• Kimberly R. Barber, PhD
• HED 547

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Variation

• Variation is present in every aspect of every
  characteristic
• In test measures, behaviors, environment.
• All characteristics have some level of variation.
• Level of variation depends on
• Source
• Population
• Accuracy of measure

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Variation Defined

• Differences in the values of a characteristic.
• Variation inherently always exists
• Because individuals differ,
• Because individuals are not all going to have the
  same exact value for every variable.
• Student height
• Student age
• Student blood pressure

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Sources of Variation

• Biological differences
• Measurement errors
• Differences in measurement technique
• Differences in measurement conditions
• Random variation

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Biological Variation

• Differences in
• Genes,
• Nutrition,
• Environmental exposures.
• Although people tend to be similar through
  genetics, they are not exactly the same.
• In addition, differing exposures create different
  outcomes
• i.e., tall parents tend to have tall children,
• But malnutrition can result in shortness even
  with tall parents.

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Disease Variation

• Differences in the presence / absence of disease.
  
• Differences in the stages of disease.
• Differences in co-morbidity.
• Exercise
• Using Diabetes as an example explain how
  individuals can differ on all of the above
  factors.

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Measurement Variation

• Differences in conditions during measurement
• Ambient factors (i.e., temp, noise)
• Environment and blood pressure
• Patient factors (i.e., fatigue, anxiety)
• White coat syndrome and blood pressure.
• Differences in methods of measurement
• Instrument, technique, and operator.
• Exercise how can weight differ according to
  instrument, technique, or operator?

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Measurement Error

• Differences in instrument recordings.
• Machine error, survey typos, etc.
• Differences in instrument observation.
• Operator interpretation errors,
• Observer errors,
• These measurement variations are all systematic
• They occur for a reason.
• We can control for them.

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Random Error

• Unexplained variation.
• Also called background noise.
• Unsystematic differences in values
• For unknown, random reasons.
• We cannot control for random errors.
• We can estimate a level of random error
• Poll results ( or 5).

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Random Error, continued.

• Random error produces a distribution of values
  even if all systematic error is eliminated.
• Statistical tests look for systematic differences
  between samples beyond the random variation
  within a sample.

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Statistics

• Statistical methods explain variation in data
• Describe the pattern of spread (variation) in
  your data set,
• Compare the pattern of two or more groups,
• Determine whether the differences between two
  patterns are real (significant) or random (non
  significant).

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Data

• A variable is the characteristic
• Skin color, blood type, age, cholesterol level.
• The data is the value of that characteristic
• White, pink, yellow, brown.
• A, B, O, AB.
• 20, 35, 50 years.
• 136, 201, 400ml.
• Exercise which two are qualitative data and
  which two are quantitative?

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Types of Variables

• Nominal
• Naming (categorical)
• No measurement scaling
• Blood type A, B, AB, O
• Dichotomous (binary)
• Categorical but only two levels
• Indicates a direction (normal abnormal, good -
  bad)
• Cancer Yes / No
• Health status Well / Sick

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Types of Variables, cont.

• Ordinal (ranked)
• Naming with an order.
• From better to worse.
• Illness scale no illness / dizzy / nauseated /
  vomiting.
• 1 2
  3 4
• Test scale excellent / good / fair / poor
• A B C
  D
• Contains more information than nominal variables
  (ill / not ill) (passed / failed)

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Continuous Variables

• Measured on a scale
• Height, weight, age, glucose level.
• Provides even more information than ordinal
  variables
• Shows position relative to each other,
• Shows extent each observation differs from the
  other.
• i.e., with age we know just how much older we are
  than the average student.

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Units of Observation

• Counts
• Counts of a characteristic in persons, things.
• Patterns presented in a frequency table (2 X 2).
• Can compare proportions within or across groups
  (female, male).
• Proportions (risk)
• Number of persons with a characteristic.
• Number of persons who died, who ill, etc.
• Can compare the ratio of counts between groups.

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Collapsing Data in Variables(Combining data)

• Continuous variable my be converted to an ordinal
  variable
• By grouping values together,
• To form categories.
• Are shrinking many values into a few values
  (collapsing).

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Collapsing data, continued.

• Disadvantage
• Information is lost because,
• Individual values are no longer apparent.
• (500g through1200g) (lt501g, gt500g)
• Advantage
• Percentages can be created
• Relationships easier to show.

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Frequency Distributions

• The number of persons with each value in a
  variable.
• Age Distribution
• 5 people are 22y
• 10 people are 23y
• 12 people are 24y
• 15 people are 25y
• 12 people are 26y
• 10 people are 27y
• 5 people are 28y

- - - 22 23 24 25 26 27 28
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Frequency Distributions, cont.

• Real distribution
• The pattern obtained from the actual data in a
  population or sample population.
• Can be one or may shaped curves.
• Gaussian distribution
• The distribution in a population expected
  (calculated) under normal or average conditions.
• Is a smooth, bell-shaped curve.

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Distributions

• Discuss range of values histogram
• Refer to Table 9-2 and Figure 9-2
• Textbook page 141
• Discuss normal distribution
• Refer to Figure 9-3 and 9-4
• Textbook page 142

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Parameters of Frequency Distribution

• Ways to summarize and define the distribution.
• Measures of central tendency
• Where among the values the commonest value lies.
• Measures of dispersion
• How widely the values are spread out.

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Measures of Central Tendency

• In a normal distribution
• The density of observed values is greatest near
  the center.
• Each tail of the curve diminishes in similar
  frequency toward zero.
• The mean, median, and mode are located in the
  center of the bell curve.

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Measures of Central Tendency, continued.

• Distribution is not normal if
• mean, medial, mode are in different locations
  on the curve.

Mean median
mode
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Skewed Distributions

• Refer to Figure 9-8 of text.
• Curve is pushed (- skew) to the right (Fig.
  9-8A).
• Curve is pushed ( skew) to the left (Fig. 9-8B).
• Curve is abnormally peaked is leptokurtic (Fig.
  9-8C).
• Curve is abnormally flat is platykurtic (Fig.
  9-8D).

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Measures of Dispersion
Range of values Amount of spread in the
distribution
20 30 40 50 60
20 30 40 50 60
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Variance

• How far each value is from the mean.
• How far in both directions.
• Example Age of a sample.
• Mean is 35 years
• Mary is 20 yrs.
• John is 50 yrs.
• Mean is 35 years
• Mary is 30 yrs.
• John is 40 yrs.

20 35 50
30 35 40
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How Variation is Measured

• S2 ? (Xi µ)2
• N - 1
• See Box 9-3.
• Degrees of freedom (N-1)
• A way to make up for small sample sizes which
  through off the estimates.
• 200 1 199 (a .05 difference)
• 2 1 1 (a 50 difference)

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Standard Deviation

• Also, describes the amount of spread in the
  frequency distribution.
• Is the square root of the variance.
• ________
• S v ?(xi - µ)2
• N 1

• Refer to Box 9-3

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Standard Deviation, continued.

• For normal distribution
• 99 of values fall between 2.5 SD.
• 95 of values fall between 1.96 SD.
• 68 of values fall between 1.0 SD.
• Refer to Figure 9-6.
• Symbols
• µ mean of the population (theoretical).
• s standard deviation of population.
• _
• X sample mean, s sample std deviation.

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Normalized Dataset

• Choice of units effect mean and std dev.
• Cant compare two groups using two different
  measuring units.
• For example
• Weight of a group of elephants (µ 5000lbs).
• Weight of a group of mice (µ 5 mg).
• To compare we have to equalize both into the same
  units.

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Normalized Data, continued

• To eliminate the effects produced by the choice
  of units
• Data is put into a unit-free form (normalized).
• Calculate a z-score
• Z distributions have a mean 0 and sd 1.
• Values in terms of how many std deviations each
  value is away from the mean.

Zi xi µ sd from the group mean.
s
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Assumptions of Normal Distributions

• In order to test whether a sample differs from
  the population its drawn from
• The data must meet some assumptions
• That the values are normally distributed,
• The sample size is sufficiently large,
• Bias and error are minimized.
• When these assumptions are met, you may conduct
  inferential analysis.

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Effect of Error on Distributions

• Random error effects the distribution differently
  than systematic error.
• How well we can compare two distributions
• How accurate our conclusions are,
• Depend on how much error is in the data.
• Random error does not effect the group average
  (regardless of the value)
• Systematic error does effect the average and can
  really through off the estimates.

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Reducing Measurement Error

• Pilot test instruments.
• Train the data collectors.
• Double check or verify data entries.
• Use multiple instrument measures.
• Consult statistician about adjusting for
  measurement error.

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Nonparametric Distributions

• Data from categorical (nominal ordinal)
  variables are not normally distributed.
• Statistical tests based on assumptions of the
  normal curve cannot be used.
• Their parameters are not the mean and standard
  deviation.
• Their tests do not require that the data follow a
  particular distribution.
• There are special tests called nonparametric
  statistics.

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Nonparametric Distributions

• The nonparametric distribution
• Is not normal,
• Is based on a distribution of counts,
• There are special tests for nonparametric data,
• These tests only require that the data be
  ordinal.

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Summary

• Fundamental to any analysis (descriptive or
  inferential) is
• Understanding your variables and the type of data
  they are,
• Continuous data is normally distributed and has
  two parameters
• Measures of central tendency (mean)
• Measures of dispersion (variance)

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Summary, continued

• The normal distribution has
• Mean, median, mode that coincide.
• 95 of observations are within 1.96 standard
  deviations from the mean.
• Some distributions are skewed
• Outliers pull the mean away from the center.
• Non-normal distributions require special
  nonparametric statistical tests.