Statistics: Data Analysis and Presentation

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StatisticsData Analysis and Presentation

• Fr Clinic II

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Overview

• Tables and Graphs
• Populations and Samples
• Mean, Median, and Standard Deviation
• Standard Error 95 Confidence Interval (CI)
• Error Bars
• Comparing Means of Two Data Sets
• Linear Regression (LR)

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Warning

• Statistics is a huge field, Ive simplified
  considerably here. For example
• Mean, Median, and Standard Deviation
• There are alternative formulas
• Standard Error and the 95 Confidence Interval
• There are other ways to calculate CIs (e.g., z
  statistic instead of t difference between two
  means, rather than single mean)
• Error Bars
• Dont go beyond the interpretations I give here!
• Comparing Means of Two Data Sets
• We just cover the t test for two means when the
  variances are unknown but equal, there are other
  tests
• Linear Regression
• We only look at simple LR and only calculate the
  intercept, slope and R2. There is much more to
  LR!

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Tables
Table 1 Average Turbidity and Color of Water
Treated by Portable Water Filters
4 5 12
Consistent Format, Title, Units, Big
Fonts Differentiate Headings, Number Columns
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Figures
Consistent Format, Title, Units Good Axis Titles,
Big Fonts
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Figure 1 Turbidity of Pond Water, Treated and
Untreated
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Populations and Samples

• Population
• All of the possible outcomes of experiment or
  observation
• US population
• Particular type of steel beam
• Sample
• A finite number of outcomes measured or
  observations made
• 1000 US citizens
• 5 beams
• We use samples to estimate population properties
• Mean, Variability (e.g. standard deviation),
  Distribution
• Height of 1000 US citizens used to estimate mean
  of US population

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Mean and Median

• Turbidity of Treated Water (NTU)

Mean Sum of values divided by number of
samples (1336810)/6 5.2 NTU
1 3 3 6 8 10
Median The middle number Rank - 1 2 3
4 5 6 Number - 1 3 3 6 8 10
For even number of sample points, average middle
two (36)/2 4.5
Excel Mean AVERAGE Median - MEDIAN
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Variance

• Measure of variability
• sum of the square of the deviation about the mean
  divided by degrees of freedom

n number of data points
Excel variance VAR
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Standard Deviation, s

• Square-root of the variance
• For phenomena following a Normal Distribution
  (bell curve), 95 of population values lie within
  1.96 standard deviations of the mean
• Area under curve is probability of getting
  value within specified range

Excel standard deviation STDEV
Standard Deviations from Mean
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Standard Error of Mean

• Standard deviation of mean
• Of sample of size n
• taken from population with standard deviation s
• Estimate of mean depends on sample selected
• As n ?, variance of mean estimate goes down,
  i.e., estimate of population mean improves
• As n ?, mean estimate distribution approaches
  normal, regardless of population distribution

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95 Confidence Interval (CI) for Mean

• Interval within which we are 95 confident the
  true mean lies
• t95,n-1 is t-statistic for 95 CI if sample size
  n
• If n ? 30, let t95,n-1 1.96 (Normal
  Distribution)
• Otherwise, use Excel formula TINV(0.05,n-1)
• n number of data points

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Error Bars

• Show data variability on plot of mean values
• Types of error bars include
• Standard Deviation, Standard Error, 95 CI
• Maximum and minimum value

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Using Error Bars to compare data

• Standard Deviation
• Demonstrates data variability, but no comparison
  possible
• Standard Error
• If bars overlap, any difference in means is not
  statistically significant
• If bars do not overlap, indicates nothing!
• 95 Confidence Interval
• If bars overlap, indicates nothing!
• If bars do not overlap, difference is
  statistically significant
• Well use 95 CI

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Example 1
Create Bar Chart of Name vs Mean. Right click on
data. Select Format Data Series.
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Example 2
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What can we do?

• Plot mean water quality data for various filters
  with error bars
• Plot mean water quality over time with error bars

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Comparing Filter Performance

• Use t test to determine if the mean of two
  populations are different.
• Based on two data sets
• E.g., turbidity produced by two different filters

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Comparing Two Data Sets using the t test

• Example - You pump 20 gallons of water through
  filter 1 and 2. After every gallon, you measure
  the turbidity.
• Filter 1 Mean 2 NTU, s 0.5 NTU, n 20
• Filter 2 Mean 3 NTU, s 0.6 NTU, n 20
• You ask the question - Do the Filters make water
  with a different mean turbidity?

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Do the Filters make different water?

• Use TTEST (Excel)
• Fractional probability of being wrong if you
  answer yes
• We want probability to be small ? 0.01 to 0.10
  (1 to 10 ). Use 0.01

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t test Questions

• Do two filters make different water?
• Take multiple measurements of a particular water
  quality parameter for 2 filters
• Do two filters treat difference amounts of water
  between cleanings?
• Measure amount of water filtered between
  cleanings for two filters
• Does the amount of water a filter treats between
  cleaning differ after a certain amount of water
  is treated?
• For a single filter, measure the amount of water
  treated between cleanings before and after a
  certain total amount of water is treated

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Linear Regression

• Fit the best straight line to a data set

Right-click on data point and use trendline
option. Use options tab to get equation and R2.
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R2 - Coefficient of multiple Determination
yi Predicted y values, from regression
equation yi Observed y values
R2 fraction of variance explained by
regression (variance standard deviation
squared) 1 if data lies along a straight line