Transcript What is Error? - University of California, Irvine
Introduction to
Statistics
Statistic [stuh-tis-tik] noun .
A numerical fact or datum, especially one computed from a sample
How long does the ball take to fall?
Measured values: See Board • How do we decide which of these measured values is correct?
• How do we discuss the variation in our measurements?
Mean Also known as “Average” Add all results, and divide by the number of measurements. Equation form: m =
x
=
x
1 +
x
2 + ...
+
x n
=
n
1
n n
å
i
= 1
x i
Propagation of Uncertainty Accuracy Sources of Inaccuracy: Broken measurement device Parallax Random error ?
Low bias, high variability Precision Sources of Imprecision: Multiple measurement methods Systematic error?
High bias, low variability
Variance and Standard Deviation Squared deviation: How much variation is there from the mean?
s
2
x
1
x
2
x
2
x
2
n
....
x n
x
2 1
n i n
1
x i
x
2 Variance: measures the absolute distance observations are from the mean s @
s
=
s
2
Error Error is the difference between the measured and
expected value
Error is how we make sense of differences between two measurements that should be the same Error is NOT mistakes! If you made a mistake, do it again.
Types of Error Descriptions For a true mean, µ, and standard deviation, σ, the sample mean has an uncertainty of the mean over the square root of the number of samples. Gives a measure of reliability of the mean. D
x
= s
N
Sample standard error tells you how close your sample mean should be to the true mean.
D
x
@
s N
Using the Standard Error This is the simplest way of using data to confirm or refute a hypothesis.
inside
x
x
outside
x
x
This is also what is used to create the error bars. confirmed not confirmed
x x
x
Example with data Set of values: 2, 4, 4, 4, 5, 5, 7, 9 Mean: 2 4 4 4 5 5 7 9 8 5 Standard Deviation:
mean
measuremen t
2 #
samples
(2 5) 2 + (4 5) 2 + (4 5) 2 + (4 5) 2 + (5 5) 2 + (5 5) 2 + (7 5) 2 + (9 5) 2 8 2 0 2 0 2 2 2 4 2 8 2
Error Types False positive: Say things are different when they are the same False negative: Say things are the same when they are different. Reject null hypothesis Accept null hypothesis No effect, null hypothesis true Type I error (false positive) Correct Effect exists, Null hypothesis false Correct Type II error (false negative)
Group Discussion: What happens to Standard Deviation as sample size increases?
What does that imply about sample error? Define standard deviation and sample error in your own words?
Summary Mean Standard Deviation m =
x
=
x
1 +
x
2 + ...
+
x n n
= 1
n n
å
i
= 1
x i s
2
x
1
x
2
x
2
x
2 ....
x n
x
2
n
1
n i n
1
x i
x
2 Variance s @
s
=
s
2
x
Sample Standard Error D
x
= s
N x
x
Summary: Often used alternative Mean m =
x
=
x
1 +
x
2 + ...
+
x n n
= 1
n n
å
i
= 1
x i
Variance Standard Deviation Sample Standard Error D
x
= s
N
s @
s
=
s
2
x
x
x
Types of Graphs: Continuous vs. Catagorical Examples?
Options: Times of ball rolling down ramp with increasing steepness Sales of coffee, tea and soft drinks at a restaurant Time it takes students to commute to UCI SAT scores of varying ethnic groups
Density Curve Low values indicate a small spread (all values close to the mean) high values indicate a large spread (all values far from the mean)
Normal Distribution • Particularly important class of density curve • Symmetric, unimodal, • bell-shaped • Mean, μ, is at the center of the curve • Probabilities are the area under the curve • Total area = 1
The Empirical Rule In a normal distribution with mean μ and standard deviation of σ: • 68% of observations fall within 1 σ of the mean • 95% of observation fall within 2 σ of the mean • 99.7% observations fall within 3 σ of the mean F D C B A
Example with data Set of values: 2, 4, 4, 4, 5, 5, 7, 9 Mean: 2 4 4 4 5 5 7 9 8 5 Standard Deviation:
mean
measuremen t
2 #
samples
(2 5) 2 + (4 5) 2 + (4 5) 2 + (4 5) 2 + (5 5) 2 + (5 5) 2 + (7 5) 2 + (9 5) 2 8 2 0 2 0 2 2 2 4 2 8 2
Data Distribution 5-6 5-4 5-2 5 5+2 5+4 5+6
Confidence Interval 5-6 5-4 5-2 5 5+2 5+4 5+6
Central Limit Theorem If X follows a normal distribution with mean μ and standard deviation σ, then x̄ is also normally distributed with mean What if X is not normally distributed? When sampling from any population with mean μ and standard deviation σ, when n is large, the sampling distribution of x̄ is approximately normal: As the number of measurements increase, they will approach a normal distribution (Gaussian).
P
e
2
x
2
N
2 2 2
N
e
x
2
x
2 2 2
x
2 Visit This webpage to play with the numbers
e
x
2 http://www.intuitor.com/statistic s/CLAppClasses/CentLimApplet.
htm
Applications Simulated examples: Dice rolling, coin flipping ect… Exit polling
Non-normal Distributions
Central Limit Theorem Summary For large N of sample, the distribution of those mean values will be: µ
e
-
x
2 which is a normal distribution.
Normal distribution of CLT is independent of the type of distribution of data.
Where else would this become problematic? Where can it still be used, but issues should be considered?
Questions?
Effective Statistics You might have strong association, but how do you prove causation? (that x causes y?) Good evidence for causation: a well designed experiment where all other variables that cause changes in the response variable are controlled
The Scientific/Statistic Process 5.
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Formulating a scientific question Decide on the population you are interested in Select a sample Observational study or experiment? Collect data Analyze data State your conclusion
Ways to collect information from sample Anecdotal evidence Available data Observational study Experiment
Sampling and Inference
population σ μ
sampling inference
sample s x̄
Some Cautions Statistics can not account for poor experimental design There is no sharp border between “significant” and “non-significant” correlation, only increasing and decreasing evidence Lack of significance may be due to poorly designed experiment
Fit Tests
t
-test,
z-
test, and χ 2 test
z-Test
z-test • • • • • All normal distributions are the same if we standardize our data: Units of size σ Mean μ as center If x is an observation from a normal distribution, the standardized value of x is called the z-score Z-scores tell how many standard deviations away from the mean an observation is
z- test procedure • To use: find the mean, standard deviation, and standard error • Use these statistics along with the observed value to find Z value • Consult the z-score table to find P(Z) the determined z
Equation for hypothesis testing: z
= s
x
/ m
n
Example Jacob scores 16 on the ACT. Emily scores 670 on the SAT. Assuming that both tests measure scholastic aptitude, who has the higher score? The SAT scores for 1.4 million students in a recent graduating class were roughly normal with a mean of 1026 and standard deviation of 209. The ACT scores for more than 1 million students in the same class were roughly normal with mean of 20.8 and standard deviation of 4.8.
Example Continued
Jacob – ACT Emily - SAT Score: 16 Mean: 20.8 Standard Dev.: 4.8
Score: 670 Mean: 1026 Standard Dev.: 209
Interpreting Results
“Backwards” z-test What if we are given a probability (P(Z)) and we are interested in finding the observed value corresponding to the probability.?
Find the Z-score Set up the probability (could be 2 sided) P(-z 0
x
=
z
´ æ ç è s
n
ö ø ÷+ m
t Tests
Necessary assumptions for t-Test 1. Population is normally distributed.
2. Sample is randomly selected from the unknown population.
3. Standard deviation of the unknown population is the same as the known population.
So, we can take the sample standard deviation as an estimate of the known population.
t
x s
/
n
Probability that fish populations are the same average length in each lake
1 0,8 0,6 0,4 0,2 T Test Accumulating Data (N) Progressively 0 1 11 21 31 41 51 61 71 81 91
# of samples included in analysis from each lake
101 111 121 This is typical of the kind of data many of you may generate. Let ’ s take a quick Look at how this T Test calculated from the data, using Excel.
z versus t procedures Use z procedures if you know the population standard deviation Use t procedure if you don’t know the population standard deviation Usually we don’t know the population standard deviation, unless told otherwise Central Limit Theorem
χ
2
-test (kai)
• • • • • χ 2 -test (Goodness-of-fit) Users Guide χ 2 -test tells us whether distributions of categorical variables differ from one another Can use to determine if your data conforms to a functional fit.
Compares multiple means to multiple expected values.
Can only use when you have multiple data sets that cannot be combined into one mean.
Use when comparing means to expected values.
χ 2 -test • • X i µ i is each individual mean is each expected value ΔX i = uncertainty in X i d = # of mean values χ 2 /d table gives probability that data matches expected values.
In χ 2 /d , d is count of independent measurements. 2
d
i
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X i
X i
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i
2
χ 2 - (Goodness-of-fit) Test Procedure Find averages and uncertainty for each average.
Calculate χ 2 using averages, uncertainties, and expected values.
Count number of independent variables.
Use table to find probability of fit accuracy based on χ 2 /d and number of independent variables (d).
Example • • • • Launch a bottle rocket with several different volumes of water.
Measure height of flight multiple times for each volume.
You decide you have a fit of: Plot of fit with data on left.
y 0.204
V (m/ml) 10 -4 V 2 (m/ml 2 )
Example • This does not mean that other fits might not match the data better, so try other fits and see which one is closest.
7 degrees of freedom Probability of fit ≈50% 50% of the time, chance alone could produce a larger χ 2 value.
No reason to reject fit.
Interpreting Results Probability is how similar data is to expected value.
Large P means data is similar to expected value.
Small P means data is different than expected value.
Summary Propagation of uncertainty Mean Accuracy vs. Precision Error Standard deviation Central Limit Theorem Fit Tests z-test t-test χ 2 -test