Statistics

1
Statistics

• Introduction
• 1.) All measurements contain random error
• results always have some uncertainty
• 2.) Uncertainty are used to determine if two or
  more experimental results are equivalent or
  different
• Statistics is used to accomplish this task

Is the mutant (transgenic) mouse significantly
fatter than the normal (wild-type) mouse?
Statistical Methods Provide Unbiased Means to
Answer Such Questions.
Masuzaki, H., et. al Science (2001), 294(5549),
2166
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Statistics

• Gaussian Curve
• 1.) For a series of experimental results with
  only random error
• (i) A large number of experiments done under
  identical conditions will yield a distribution of
  results.
• (ii) Distribution of results is described by a
  Gaussian or Normal Error Curve

High population about correct value
low population far from correct value
Number of Occurrences
Value
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Statistics

• Gaussian Curve
• 2.) Any set of data (and corresponding Gaussian
  curve) can be characterized by two parameters
• (i) Mean or Average Value ( )
• where
• n number of data points
• xi value of data point number i
• value1 value2 value3 valuen
• (ii) Standard Deviation (s)

Smaller the standard deviation is, more precise
the measurement is.
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Statistics

• Gaussian Curve
• 3.) Other Terms Used to Describe a Data Set
• (i) Variance Related to the standard deviation
• Used to describe how wide or precise a
  distribution of results is
• variance (s)2
• where s standard deviation
• (ii) Range difference in the highest and lowest
  values in a set of data
• Example measurments of 4 light bulb lifetimes
• 821, 783, 834, 855
• High Value 855 hours
• Low Value 783 hours
• Range High Value Low Value

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Statistics

• Gaussian Curve
• 3.) Other Terms Used to Describe a Data Set
• (iii) Median The value in a set of data which
  has an equal number of data values above it and
  below it
• For odd number of data points, the median is
  actually the middle value
• For even number of data points, the median is the
  value halfway between the two middle values
• Example
• Data Set 1.19, 1.23, 1.25, 1.45 ,1.51
  mean( ) 1.33
• Data Set 1.19, 1.23, 1.25, 1.45 mean( )
  1.28
• median 1.24

Median value
Median value
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Statistics

• Gaussian Curve
• (iii) Example

For the following bowling scores 116.0, 97.9,
114.2, 106.8 and 108.3, find the mean, median,
range and standard deviation.
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Statistics

• Gaussian Curve
• 4.) Relating Terms Back to the Gaussian Curve
• (i) Formula for a Gaussian curve
• where e base of natural logarithm (2.71828)
• m (mean)
• s s (standard deviation)

mean
Entire area under curve is normalized to one
standard deviation
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Statistics

• Standard Deviation and Probability
• 1.) By knowing the standard deviation (s) and the
  mean ( ) of a set of result (and the
  corresponding Gaussian curve)
• (i) The probability of the next result falling
  in any given range can be calculated by
• (ii) The probability of a result falling in that
  portion of the Gaussian curve is equal to the
  normalized area of the curve in that portion.
• (iii) Example
• 68.3 of the area of a Gaussian curve occurs
  between the values -1s and 1s (
  1s)
• Thus, any new result has a 68.3 chance of
  falling within this range.

Probability of Measuring a value in a certain
range is equal to the area of that range
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Statistics
Standard Deviation and Probability

• - Area under curve from mean value and result.
• Total ½ area is 0.5.
• Remaining area is 0.5 Area.
• Example
• z 1.3?area from mean to 1.3 is 0.403
• ? area from infinity to 1.3 is 0.5
  0.403 0.097

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Statistics

• Standard Deviation and Probability
• (iii) Example
• A bowler has a mean score of 108.6 and a
  standard deviation of 7.1.
• What fraction of the bowlers scores will be
  less than 80.2?

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Statistics

• Standard Deviation and Probability
• 2.) Knowing the standard deviation (s) of a data
  set indicates the precision of a measurement
• (i) Common intervals used for expressing
  analytical results are shown below
• (ii) The precision of many analytical
  measurements is expressed as
• There is only a 5 chance (1 out of 20) that any
  given measurement on the sample will be outside
  of this range

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Statistics

• Standard Deviation and Probability
• 4.) The precision of a mean (average) result is
  expressed using a confidence interval
• (i) Relationship between the true mean value (m)
  and the measured mean ( ) is given by
• where s standard deviation
• n number of measurements
• t students t value
• degrees of freedom (n-1)

Confidence interval
Note As n increases, the confidence interval
becomes smaller (m
becomes more precisely known)
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Statistics

• Standard Deviation and Probability
• 4.) The precision of a mean (average) result is
  expressed using a confidence interval
• (ii) Students t
• Statistical tool frequently used to express
  confidence intervals

From number of measurements (n-1)
A probability distribution that addresses the
problem of estimating the mean of a normally
distributed population when the sample size is
small.
Population standard deviation (s) is unknown and
has to be estimated from the data using s.
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Statistics

• Standard Deviation and Probability
• 4.) The precision of a mean (average) result is
  expressed using a confidence interval
• (iii) The meaning of Confidence Interval
• To determine the true mean need to collect an
  infinite number of data points.
• - obviously not possible
• Confidence interval tells us the probability that
  the range of numbers contains the true mean.
• 50 confidence interval ? range of numbers only
  contains true mean 50 of the time
• 90 confidence interval ? range of numbers
  contains true mean 90 of the time.

true mean
50 of data sets do not contain true mean
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Statistics

• Standard Deviation and Probability
• (iii) Example
• For the following bowling scores 116.0, 97.9,
  114.2, 106.8 and 108.3, a bowler has a mean score
  of 108.6 and a standard deviation of 7.1.
• What is the 90 confidence interval for the
  mean?

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Statistics

• Standard Deviation and Probability
• 5.) Comparison of Two Data Sets
• (i) To determine if two results obtained by the
  same method are statistically the same, use the
  following formula to determine a calculated t
• where
• mean results of samples 1 2
• n1, n2 number of measurements of samples 1
  2
• spooled pooled standard deviation

Requires standard deviation from the two data
sets be similar.
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Statistics

• Standard Deviation and Probability
• 5.) Comparison of Two Data Sets
• (ii) Compare calculated t to the corresponding
  value in the Students t probability table.
• Use the desired confidence level at the
  appropriate Degrees of freedom
• Degrees of Freedom (n1 n2 -2)
• (iii) If calculated t is greater than the value
  in the Students t probability table, then the
  two results are significantly different at the
  given confidence level.
• Easier to achieve for lower confidence level

Calculated t needs to be less than table value
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Statistics

• Standard Deviation and Probability
• 5.) Comparison of Two Data Sets
• (iv) Example

The amount of 14CO2 in a plant sample is measured
to be 28, 32, 27, 39 40 counts/min (mean
33.2). The amount of radioactivity in a blank is
found to be 28, 21, 28, 20 counts/min (mean
24.2). Are the mean values significantly
different at a 95 confidence level?
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Statistics

• Standard Deviation and Probability
• 5.) Comparison of Two Data Sets
• (iv) Example

Degrees of Freedom (5 4 2) 7
From Students t probability table
Degrees of Freedom (7)
95 Confidence level
Calculated t (2.48) gt 2.365 The results are
significantly different at a 95 confidence
level, but not at 98 or higher confidence levels
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Statistics

• Standard Deviation and Probability
• 6.) Comparison of Two Methods
• (i) To determine if the results of two methods
  for the same sample are the same, use the
  following formula to determine a calculated t
• where
• difference in the mean values of the
  two methods
• n number of samples analyzed by each
  method
• sd
• (ii) Degree of Freedom (n - 1)
• (iii) If calculated t is greater than the value
  in the Students t probability table, then the
  two methods are significantly different at the
  given confidence level.

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Statistics

• Standard Deviation and Probability
• 6.) Comparison of Two Methods
• (iv) Example

Two methods for measuring cholesterol in blood
provide the following results Are
these methods significantly different at the 95
confidence level?
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Statistics

• Standard Deviation and Probability
• 6.) Comparison of Two Methods
• (iv) Example

Degrees of Freedom (6-1 5)
95 Confidence level
Calculated t (1.20) 2.571 The results are not
significantly different at a 95 confidence level.
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Statistics

• Dealing with Bad Data
• 1.) Q Test
• (i) Method used to decide whether or not to
  reject a bad data point.
• (ii) Procedure
• Arrange Data in order of increasing value.
• Determine the lowest and highest values and the
  total range of values.
• Example
• 12.47 12.48 12.53 12.56 12.67
• Determine the difference between the bad data
  point and the nearest value.
• - Calculate the Q value

Questionable point
Range 0.20
gap 0.11
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Statistics

• Dealing with Bad Data
• 1.) Q Test
• (ii) Procedure
• 4. Compare the calculated Q value to those in
  Tables at the same value of n and the desired
  confidence level.
• - n total number of values or observations
• - For example, at n 5 and 90 confidence, the
  value of Q is 0.64
• - Since
• Q (calculated) Q (table)
• 0.55 0.64
• - data point 12.67 can not be rejected at the
  90 confidence level
• (iii) Although the Q-test is valuable in
  eliminating bad data, common sense and repeating
  experiments with questionable results are usually
  more helpful.

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Statistics

• Dealing with Bad Data
• 1.) Q Test
• (ii) Example
• For the following bowling scores 116.0, 97.9,
  114.2, 106.8 and 108.3, a bowler has a mean score
  of 108.6 and a standard deviation of 7.1.
• Using the Q test, decide whether the number 97.9
  should be discarded.