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![Page 1: Statistics Dealing With Uncertainty. Objectives Describe the difference between a sample and a population Learn to use descriptive statistics (data sorting,](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649ebd5503460f94bc6de6/html5/thumbnails/1.jpg)
Statistics
Dealing With Uncertainty
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Objectives Describe the difference between a sample
and a population Learn to use descriptive statistics (data
sorting, central tendency, etc.) Learn how to prepare and interpret
histograms State what is meant by normal distribution
and standard normal distribution. Use Z-tables to compute probability.
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Statistics
“There are lies, d#$& lies, and then there’s statistics.”
Mark Twain
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Statistics is... a standard method for...
- collecting, organizing, summarizing, presenting, and analyzing data - drawing conclusions - making decisions based upon the
analyses of these data. used extensively by engineers (e.g.,
quality control)
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Populations and Samples Population - complete set of all of
the possible instances of a particular object e.g., the entire class
Sample - subset of the population e.g., a team
We use samples to draw conclusions about the parent population.
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Why use samples? The population may be large
all people on earth, all stars in the sky. The population may be dangerous to
observe automobile wrecks, explosions, etc.
The population may be difficult to measure subatomic particles.
Measurement may destroy sample bolt strength
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Team Exercise: Sample Bias
To three significant figures, estimate the average age of the class based upon your team.
When would a team not be a representative sample of the class?
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Measures of Central Tendency
If you wish to describe a population (or a sample) with a single number, what do you use?
Mean - the arithmetic average Mode - most likely (most common)
value. Median - “middle” of the data set.
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What is the Mean? The mean is the sum of all data
values divided by the number of values.
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Sample Mean
Where: is the sample mean xi are the data points n is the sample size
n
iixn
x1
1
x
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Population Mean
Where: μ is the population mean
xi are the data points
N is the total number of observations in the population
N
iixN 1
1
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What is the Mode? mode - the value that occurs the
most often in discrete data (or data that have been grouped into discrete intervals)
Example, students in this class are most likely to get a grade of B.
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Mode continued
Example of a grade distribution with mean C, mode B
0
5
10
15
20
25
F D C B A
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What is the Median?
Median - for sorted data, the median is the middle value (for an odd number of points) or the average of the two middle values (for an even number of points). useful to characterize data sets
with a few extreme values that would distort the mean (e.g., house price,family incomes).
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What Is the Range? Range - the difference between
the lowest and highest values in the set. Example, driving time to Houston is 2
hours +/- 15 minutes. Therefore... Minimum = 105 min Maximum = 135 minutes Range = 30 minutes
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Standard Deviation
Gives a unique and unbiased estimate of the scatter in the data.
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Standard Deviation Population
Sample
2
1
)(1
N
iix
N
2
1
)()1(
1xx
ns
n
ii
Deviation
Variance = 2
Variance = s2
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The Subtle Difference Between and σ
N versus n-1n-1 is needed to get a better
estimate of the population from the sample s.
Note: for large n, the difference is trivial.
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A Valuable Tool Gauss invented standard deviation circa 1700 to explain the error observed in measured star positions.
Today it is used in everything from quality control to measuring financial risk.
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Team Exercise In your team’s bag of M&M
candies, count the number of candies for each color the total number of candies in the bag
When you are done counting, have a representative from your team enter your data on the board
Using Excel, enter the data gathered by the entire classMore
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Team Exercise (con’t)
For each color, and the total number of candies, determine the following:
maximum modeminimum medianrange standard deviationmean variance
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Individual Exercise: Histograms
Flip a coin EXACTLY ten times. Count the number of heads YOU get.
Report your result to the instructor who will post all the results on the board
Open Excel Using the data from the entire class,
create bar graphs showing the number of classmates who get one head, two heads, three heads, etc.
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Data Distributions The “shape” of the data is described
by its frequency histogram. Data that behaves “normally”
exhibit a “bell-shaped” curve, or the “normal” distribution.
Gauss found that star position errors tended to follow a “normal” distribution.
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The Normal Distribution The normal distribution is
sometimes called the “Gauss” curve. 22 /
2
1
2
1RF
xe
mean
x
RF
RelativeFrequency
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Standard Normal Distribution
Define:
Then
/ xz
2RF
2
2
1z
e
0.0
0.1
0.2
0.3
0.4
0.5
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
Area = 1.00
z
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Some handy things to know.
50% of the area lies on each side of the mid-point for any normal curve.
A standard normal distribution (SND) has a total area of 1.00.
“z-Tables” show the area under the standard normal distribution, and can be used to find the area between any two points on the z-axis.
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Using Z Tables (Appendix C, p. 624)
Question: Find the area between z= -1.0 and z= 2.0 From table, for z = 1.0, area = 0.3413 By symmetry, for z = -1.0, area = 0.3413 From table, for z= 2.0, area = 0.4772 Total area = 0.3413 + 0.4772 = 0.8185 “Tails” area = 1.0 - 0.8185 = 0.1815
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“Quick and Dirty” Estimates of and
(lowest + 4*mode + highest)/6 For a standard normal curve, 99.7%
of the area is contained within ± 3 from the mean.
Define “highest” = Define “lowest” = Therefore, (highest - lowest)/6
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Example: Drive time to Houston
Lowest = 1 h Most likely = 2 h Highest = 4 h (including a flat tire,
etc.) = (1+4*2+4)/6 = 2.16 (2 h 12 min) = (4 - 1)/6= 0.5 h
This technique (Delphi) was used to plan the moon flights.
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Team Exercise You want to put a scale on your
rubber-band car to relate a given scale setting and an expected distance traveled.
Design an experiment to establish a scale for your car.
More
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Team Exercise continued. Some Issues to consider:
Sample size Range of distances Desired accuracy
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Review Central tendency
mean mode median
Scatter range variance standard deviation
Normal Distribution