Statistics: Data Presentation

1
Statistics Data Presentation Analysis

• Fr Clinic I

2
Overview

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

3
Warning

• Statistics is a huge field, Ive simplified
  considerably here. For example
• Mean, Median, and Standard Deviation
• There are alternative formulas
• 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!

4
Tables
Table 1 Average Turbidity and Color of Water
Treated by Portable Water Filters
Consistent Format, Title, Units, Big
Fonts Differentiate Headings, Number Columns
5
Figures
Consistent Format, Title, Units Good Axis Titles,
Big Fonts
11
Figure 1 Turbidity of Pond Water, Treated and
Untreated
6
Populations and Samples

• Population
• All possible outcomes of experiment or
  observation
• US population
• Particular type of steel beam
• Sample
• Finite number of outcomes measured or
  observations made
• 1000 US citizens
• 5 beams
• Use samples to estimate population properties
• Mean, Variance
• E.g., Height of 1000 US citizens used to estimate
  mean of US population

7
Central Tendency

• Mean and Median

Mean xbar Sum of values divided by sample
size (1336810)/6 5.2 NTU
1 3 3 6 8 10
Median m 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
8
Variability

• Variance, s2
• sum of the square of the deviation about the mean
  divided by degrees of freedom
• s2 n(xi xbar)2/(n-1)
• Where xi a data point and n number of data
  points
• Example (cont.)
• s2 (1-5.2)2 (3-5.2)2 (3-5.2)2 6-5.2)2
  (8-5.2)2 (10-5.2)2 /(6-1) 11.8 NTU2

Excel Variance VAR
9
Error Bars

• Show data variability on plot of mean values
• Types of error bars include
• Max/min, Standard Deviation, Standard Error,
  95 CI

10
Standard Deviation, s

• Square-root of variance
• If phenomena follows Normal Distribution (bell
  curve), 95 of population lies within 1.96
  standard deviations of the mean
• Error bar is s above below mean

Excel standard deviation STDEV
Standard Deviations from Mean
11
Standard Error of Mean

• Also called St-Err or sxbar
• For sample of size n taken from population with
  standard deviation estimated as s
• As n ?, sxbar estimate?, i.e., estimate of
  population mean improves
• Error bar is St-Err above below mean

12
95 Confidence Interval (CI) for Mean

• A 95 Confidence Interval is expected to contain
  the population mean 95 of the time (i.e., of
  95-CIs from 100 samples, 95 will contain pop
  mean)
• t95,n-1 is a statistic for 95 CI from sample of
  size n
• t95,n-1 TINV(0.05,n-1)
• If n ? 30, t95,n-1 1.96 (Normal Distribution)
• Error bar is above below
  mean

13
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 in this class
• Any time you have 3 or more data points,
  determine mean, standard deviation, standard
  error, and t95,n-1, then plot mean with error
  bars showing the 95 confidence interval

14
Adding Error Bars to an Excel Graph

• Create Graph
• Column, scatter,
• Select Data Series
• In Layout Tab-Analysis Group, select Error Bars
• Select More Error Bar Options
• Select Custom and Specify Values and select cells
  containing the values

15
Example 1 95 CI
16
What can we do?

• Lift weight multiple times using different solar
  panel combinations (or hyrdoturbines, or gear
  boxes) and plot mean and 95 Confidence interval
  error bars.
• If error bars overlap between to different test
  conditions, indicates nothing!
• If error bars do not overlap, difference is
  statistically significant

17
T Test

• A more sophisticated way to compare means
• Use t test to determine if means of two
  populations are different
• E.g., lift times with different solar panel
  combinations or turbines or

18
Comparing Two Data Sets using the t test

• Example - You lift weight with two panels in
  series and two in parallel.
• Series Mean 2 min, s 0.5 min, n 20
• Parallel Mean 3 min, s 0.6 min, n 20
• You ask the question - Do the different panel
  combinations result in different lift times?
• Different in a statistically significant way

19
Are the Lift Times Different?

• Use TTEST (Excel)
• Fractional probability of being wrong if you
  claim the two populations are different
• Well say they are significantly different if
  probability is 0.05

20
Marbles
21
Linear Regression

• Fit the best straight line to a data set

Right-click on data point and select trendline.
Select options to show equation and R2.
22
R2 - Coefficient of multiple Determination

• R2 n(yi - ybar)2 / n(yi - ybar)2
• yi Predicted y values, from regression equation
• yi Observed y values
• Ybar mean of y
• R2 fraction of variance explained by
  regression
• R2 1 if data lies along a straight line