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Statistics II
Freshman Engineering Clinic II
Course Reminders & Deadlines
• Pathfinder
▫ Before exercises (on Intellectual Property) due by 10:30
am Wed. March 1st
• 3D Game Lab
▫ 2nd deadline of 700 XP midnight Fri. March 10th
• Heart Lung Project
▫ Re-write of Literature Review due by Mon. February 27th
Review of Last Class – Key Concepts
• Statistics I
▫ Area under the bell curve always equals 1
▫ More similar your data the larger the peak of the bell
curve
▫ Z-statistic is used to determine probabilities with
normally distributed populations
𝑋−𝜇
𝑍=
𝜎
Review of Last Class – Example Problem
• Porphyrin is a pigment in blood protoplasm and other
body fluids that is significant in body energy and
storage. In healthy Alaskan brown bears, the amount
of porphyrin in the bloodstream (in mg/dl) has
approximate normal distribution with a mean of 38
mg/dl and a standard deviation of 12. What
proportion of these bears have between 27.5 and 67.5
mg/dl porphyrin in their bloodstream?
▫ Z1 = (67.5-38)/12 = 2.46; Probability from Z-table =
0.9931
▫ Z2 = (27.5-38)/12 = -0.875; Probability from Z-table =
0.1908
▫ Final answer = 0.9931-0.1908 = 0.8023
Class Overview
▫ Mean, Median, Variance, Standard Deviation, Standard
Error
▫ 95% Confidence Interval (C.I.)
▫ Error Bars
▫ Comparing Means of Two Data Sets
Basic Stats Review
Using the data to the left, calculate the following:
(NTU)
1
3
3
6
8
10
•
•
•
•
•
Mean
Median
Variance
Standard Deviation
Standard Error
95% Confidence Interval (C.I.) for Mean
• A 95% C.I. is expected to contain the population
mean 95 % of the time (from 100 samples, 95 will
contain population mean if expressed as x ± 95% C.I. )
X ± (t 95%,n -1)(sX )
• t95%,n-1 is a statistic for 95% C.I. from sample of size n
▫ In EXCEL: t95%,n-1 = TINV(0.05,n-1)
 Where 0.05 = (100-95)/100 & n = sample size
 For a sample size of 6, t95%,5 = TINV(0.05, 5) = 2.57
▫ If n ≥ 30, then t95%,n-1 ≈ 1.96 (Normal Distribution)
Mean
(NTU)
1
3
3
6
8
10
± Confidence Interval
X ± (t 95%,n -1)(sX )
X ± 2.57*1.4 NTU
= 5.2 NTU ± 3.6 NTU
Note: 95% confidence intervals is typically larger
than +/- standard error interval
Filter Example
Determine the 95% C.I. for each filter
Turbidity Data
Test 1
Test 2
Test 3
NTU
NTU
NTU
Filter 1
2.1
2.1
2.2
Filter 2
3.2
4.4
5
Filter 3
4.3
4.2
4.5
Error Bars
• Show data variability on plot of mean values
• Types of error bars include:
 Max/min, ± Standard Deviation, ± Standard Error,
± 95% C.I.
• “Significant Difference”
Using Error Bars to compare data
• Standard Deviation
▫ Demonstrates data variability
• 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
• We’ll use 95 % CI by default
▫ 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
• But if you want to conclude samples are the same or
different, you have to use the right error bar!
Standard Error Bars
No overlap: cannot be sure
that the difference is
statistically significant.
Overlap: can be sure
that the difference is not
statistically significant.
Confidence Interval Error Bars
Mean ± 95% C.I., n = 5
No overlap: Can be
sure that the difference
is statistically
significant.
Mean ± 95% C.I., n = 5
Overlap: Can not be
sure that the difference
is not statistically
significant.
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 t95%, n 1 s X values
Example 1: 95% CI
Turbidity Data
1
2
3 mean St Dev
NTU NTU NTU NTU NTU
2.1
2.1 2.2 2.1
0.06
3.2
4.4
5
4.2
0.92
4.3
4.2 4.5 4.3
0.15
Filter 1
Filter 2
Filter 3
7.0
6.0
Turbidity (NTU)
5.0
4.2
4.3
Filter 2
Filter 3
4.0
3.0
2.1
2.0
1.0
0.0
Filter 1
Portable Water Filter
n
3
3
3
St-Err
NTU
0.03
0.53
0.09
t95%,2
+/- 95% CI
t95%,2St-Err
4.30
4.30
4.30
0.14
2.28
0.38
Key Takeaways:
•
•
•
•
•
How to calculate confidence intervals
How to read the t-test chart
How to calculate variance
How to calculate standard error
Difference between error bars with standard
error and confidence interval
• What confidence interval means
Review: Measures of Central Tendency
Mean = x =
(NTU)
1
3
3
6
8
10
åx
i
n
= (1 + 3 + 3 + 6 + 8 + 10) / 6
= 5.2 NTU
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
= (3+6) NTU/2 = 4.5 NTU
Excel: Mean – AVERAGE; Median - MEDIAN
Variability
• Variance, s2
▫ sum of the square of the deviation about the mean
divided by degrees of freedom
s
2
x-x)
å
(
=
2
i
n -1
• 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
Standard Deviation, s
s s
2
• 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
Normal Distribution
above & below mean
95%
Excel: standard deviation –
STDEV
-4
-1.96
-2
0
1.96
2
Standard Deviations
from Mean
Standard Deviation
4
Standard Deviation
s s
2
s = s = 11.8 NTU
2
2
= 3.43 NTU
Standard Error of Mean
sX
• Also called St-Err
• For sample of size n taken from population with
standard deviation estimated as s
s
sX 
n
estimate↓, i.e., estimate of population
• As n ↑, sxbar
mean improves
• Error bar is St-Err above & below mean
Standard Error
s
sX 
n
s
sX =
= 3.43 NTU / 6 =1.4 NTU
n